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+{"text":"\\section{Introduction}\n\nLet $f : X \\to S$ be a smooth projective family defined over $\\mathcal{O}_{K, N} = \\mathcal{O}_{K}[N^{-1}]$, with $K \\subset \\mathbb{C}$ a number field, $\\mathcal{O}_{K}$ its ring of integers, and $N$ any non-zero element of $\\mathbb{Z}$. In a recent paper \\cite{LV}, Lawrence and Venkatesh proposed a strategy for bounding the dimension of the Zariski closure $\\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}}$ of $S(\\mathcal{O}_{K,N})$. Since for any point $s \\in S(\\mathcal{O}_{K,N})$ the fibre $X_{s}$ has good reduction away from $N$, the problem of bounding the dimension of $\\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}}$ can be interpreted as Shafarevich-type problem for the family $f$. In particular, the Shaferevich conjectures for curves, abelian varieties, and K3 surfaces are equivalent to showing that $\\dim \\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} = 0$ for particular choices of $f$.\n\nThe Lawrence-Venkatesh strategy has been implemented to reprove various classical diophantine finiteness results, including the Mordell conjecture \\cite{LV}, the finiteness of $S$-units \\cite{LV}, and Seigel's theorem \\cite{noordman2021siegels}. However it is of particular interest to apply the strategy in situations where $\\dim S > 1$. In this situation the problem becomes much harder, because one needs some understanding of the monodromy of the family $f$ over essentially arbitrary geometrically irreducible subvarieties $Z \\subset S_{\\mathbb{C}}$. To explain what we mean, let $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$ and define for each such $Z \\subset S_{\\mathbb{C}}$:\n\\begin{defn}\nThe algebraic monodromy group $\\mathbf{H}_{Z}$ of $Z$ is the identity component of Zariski closure of the monodromy representation associated to $\\restr{\\mathbb{V}}{Z^{\\textrm{nor}}}$, where $Z^{\\textrm{nor}} \\to Z$ is the normalization.\n\\end{defn}\n\\noindent The variation $\\mathbb{V}$ determines a flag variety $\\ch{L}$, on which the group $\\mathbf{H}_{Z}$ can be said to act after identifying $\\ch{L}$ with the variety of Hodge flags on some fibre $\\mathbb{V}_{s}$ for $s \\in Z(\\mathbb{C})$. Let $\\varphi : S \\to \\Gamma \\backslash D$ be a period map determined by $\\mathbb{V}$ with $D \\subset \\ch{L}$ the complex submanifold of polarized Hodge flags. Then the key quantity relevant for the Lawrence-Venkatesh method is\n\\begin{equation}\n\\Delta = \\min_{Z, s} \\left[ \\dim (\\mathbf{H}_{Z} \\cdot F^{\\bullet}_{s}) - \\dim \\varphi(Z) \\right] , \n\\end{equation}\nwhere the minimum is taken over all positive-dimensional geometrically irreducible subvarieties $Z \\subset S_{\\mathbb{C}}$, points $s \\in Z(\\mathbb{C})$, and where $F^{\\bullet}_{s}$ is the Hodge flag on $\\mathbb{V}_{s}$.\\footnote{We not here that $\\varphi(Z)$ is analytically constructible, for instance by applying the main result of \\cite{OMINGAGA}, so its dimension makes sense.} In particular, the Lawrence-Venkatesh method produces an integer $k$, and shows that if $\\Delta \\geq k$, then the Shafarevich conjecture holds for $f$.\n\nFor successful applications of the Lawrence-Venkatesh strategy for the Shafarevich problem in situations when $\\dim S > 1$ we know of only the paper \\cite{lawrence2020shafarevich} of Lawrence and Sawin, who are able to apply this strategy beyond the first induction step to prove a Shafarevich conjecture for hypersurfaces lying inside a fixed abelian variety $A$. Their methods require the auxilliary use of a Tannakian category associated to $A$, and it seems unclear what to do without this abelian variety structure present.\n\nOur main result is as follows:\n\n\\begin{thm}\n\\label{mainthm}\nConsider a smooth projective family $f : X \\to S$ defined over $\\mathcal{O}_{K,N}$ and with $S$ smooth and quasi-projective, and given an integer $d$, define\n\\begin{equation}\n\\Delta_{d} = \\min_{Z, s} \\left[ \\dim (\\mathbf{H}_{Z} \\cdot F^{\\bullet}_{s}) - \\dim \\varphi(Z) \\right] , \n\\end{equation}\nwhere the minimum ranges over all geometrically-irreducible subvarieties $Z \\subset S_{\\mathbb{C}}$ with $\\dim Z > d$ and points $s \\in Z(\\mathbb{C})$. Then there exists an effective procedure which outputs an infinite sequence of integers\n\\[ \\kappa(1) \\leq \\kappa(2) \\leq \\cdots \\leq \\kappa(r) \\leq \\cdots < \\Delta_{d} \\]\nsuch that for some $r = r_{0}$ we have $\\kappa(r_{0}) = \\Delta_{d} - 1$.\n\nIf moreover the period map $\\varphi$ is quasi-finite, one can determine $r_{0}$.\n\\end{thm}\n\n\\begin{rem}\nLet us make absolutely clear what is meant by the term ``effective procedure'' in \\autoref{mainthm}. We mean that there exists an infinite-time algorithm (for instance, a non-halting Turing machine), which outputs a sequence of integers $\\{ \\kappa(r) \\}_{r = 1}^{\\infty}$, with the integer $\\kappa(r)$ being outputted at some finite time $t_{r}$ depending on $r$. Moreover, one also has at time $t_{r}$ a proof that $\\kappa(r) < \\Delta_{d}$. Therefore, after time $t_{r}$, one can stop the algorithm and use the bound $\\kappa(r) < \\Delta_{d}$ as input to the Lawrence-Venkatesh method. One is also guaranteed that there is some $r_{0}$ so that at time $t_{r_{0}}$ the bound $\\kappa(r_{0}) < \\Delta_{d}$ is best possible, however one does not necessarily have a method to determine $r_{0}$ unless $\\varphi$ is quasi-finite. Finally, the algorithm can be described entirely in terms of algebro-geometric computations involving algebraically constructible sets, and implicit in the proof is a description of how to implement it.\n\\end{rem}\n\nThere is no fundamental obstruction which requires us to restrict to quasi-finite $\\varphi$ for the second part of \\autoref{mainthm}. Rather, the second part of \\autoref{mainthm} references some delicate arguments in \\cite{urbanik2021sets} which are only enough to handle the quasi-finite case directly, and recalling enough of the machinery of \\cite{urbanik2021sets} to carry out the argument for the general case would lead us too far astray from the main ideas. We note that one does not actually need to determine the integer $r_{0}$ referenced in \\autoref{mainthm} to apply the machinery of Lawrence and Venkatesh: one wants to be able to compute the best possible lower bound for $\\Delta_{d}$, but one is not required to actually prove that the bound one has is optimal in order to deduce diophantine finiteness results.\n\n\\subsection{The Approach of Lawrence and Venkatesh}\n\nWe begin with a preliminary observation. To show that $\\dim \\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} \\leq d$, it suffices to show, for any irreducible subscheme $T \\subset S$ of dimension $> d$ and defined over $\\mathcal{O}_{K,N}$, that the Zariski closure $\\overline{T(\\mathcal{O}_{K,N})}^{\\textrm{Zar}}$ is a proper algebraic subscheme of $T$. We therefore fix such a subscheme $T \\subset S$ with $\\dim T > d$, and seek to show that $\\dim \\overline{T(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} < \\dim T$.\n\nFix a prime $p \\in \\mathbb{Z}$ not dividing $N$, and let $t \\in T(\\mathcal{O}_{K,N})$ be a point. It is conjectured that for any $i$ the representation of $\\textrm{Gal}(\\overline{K}\/K)$ on $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$ is semisimple. If this result were to be known for all such $t$, an argument of Faltings shows that for $t \\in T(\\mathcal{O}_{K,N})$ there are at most finitely many possibilities for the isomorphism class of the representation of $\\textrm{Gal}(\\overline{K}\/K)$ on $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$. To establish the non Zariski-density of $T(\\mathcal{O}_{K,N})$ it would then suffice to show that the fibres of the map\n\\[ t \\in T(\\mathcal{O}_{K,N})  \\hspace{1em} \\xrightarrow{\\tau} \\hspace{1em} \\big\\{ \\textrm{Gal}(\\overline{K}\/K)\\textrm{-rep. on }H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p}) \\big\\}\\hspace{0.5em} \\big\/ \\hspace{0.5em} \\textrm{iso}.  \\]\nare not Zariski dense. As explained by Lawrence and Venkatesh in \\cite{LV}, this is essentially the original argument for the Mordell conjecture due to Faltings.\n\nThe problem with applying this strategy in general is twofold. First, the semisimplicity of $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$ is not known, and for most choices of $f$ appears out of reach. Secondly, without a good geometric interpretation of the \\'etale cohomology $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$ it is difficult to understand $\\tau$. The key insight in the paper of Lawrence and Venkatesh is that one may potentially overcome both problems by passing to a $p$-adic setting where they are more managable.\n\nInstead of considering the global Galois representation $\\rho_{t} : \\textrm{Gal}(\\overline{K}\/K) \\to H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$, we consider its semisimplification $\\rho^{\\textrm{ss}}_{t}$, and restrict $\\rho_{t}$ along the map $\\textrm{Gal}(\\overline{K_{v}}\/K_{v}) \\to \\textrm{Gal}(\\overline{K}\/K)$ induced by a fixed embedding $\\overline{K} \\hookrightarrow \\overline{K_{v}}$ to obtain $\\rho_{t,v}$, where $v$ is a fixed place above $p$. The functors of $p$-adic Hodge theory tell us that the representation $\\rho_{t,v}$ determines a triple $(H^{i}_{\\textrm{dR}}(X_{t,K_{v}}), \\phi_{t}, F^{\\bullet}_{t})$, where $F^{\\bullet}_{t}$ is the Hodge filtration on de Rham cohomology and $\\phi_{t}$ is the crystalline Frobenius at $t$. If we somehow manage to consider the data $(H^{i}_{\\textrm{dR}}(X_{t,K_{v}}), \\phi_{t}, F^{\\bullet}_{t})$ up to ``semisimplification'' (in the sense that we identify such triples when the associated \\emph{global} Galois representations have isomorphic semisimplifications), our problem is then to study the fibres of the map\n\\[ t \\in T(\\mathcal{O}_{K,N}) \\hspace{0.5em} \\xrightarrow{\\tau_{p}} \\hspace{0.5em} \\big\\{\\textrm{``semisimplifications'' of } (H^{i}_{\\textrm{dR}}(X_{t,K_{v}}), \\phi_{t}, F^{\\bullet}_{t}) \\big\\} \\hspace{0.5em} \\big\/ \\hspace{0.5em} \\textrm{iso}. \\]\nand show they lie in a Zariski closed subscheme of smaller dimension.\n\nNext, we make the elementary observation that to bound the dimension of the Zariski closure of $T(\\mathcal{O}_{K,N})$, it suffices to cover $T(\\mathcal{O}_{K,v})$ by finitely many $v$-adic disks $D_{1}, \\hdots, D_{k}$ and bound the dimension of the Zariski closure of $D_{i} \\cap T(\\mathcal{O}_{K,N})$ for each $i$; here $\\mathcal{O}_{K,v}$ is the ring of $v$-adic integers. It can then be shown that there exists such a cover for which the Hodge bundle $\\mathcal{H} = R^{i} f_{*} \\Omega^{\\bullet}_{X\/S}$ can be trivialized rigid-analytically over each $D_{i}$, moreover with respect to each such trivialilzation the Frobenius operator $\\phi_{t}$ is independent of $t \\in D_{i}$. The problem then reduces to studying a varying filtration $F^{\\bullet}_{t}$ on a fixed vector space $V_{p} = H^{i}_{\\textrm{dR}}(X_{t_{0}})$ for some $t_{0} \\in D_{i}$. In particular, we obtain a rigid-analytic map\n\\[ D_{i} \\hspace{0.5em} \\xrightarrow{\\psi_{p}} \\hspace{0.5em} \\underbrace{\\{ \\hspace{0.3em} \\textrm{Hodge filtrations on }V_{p} \\hspace{0.3em} \\}}_{\\ch{L}_{p}} , \\]\nand those points of $\\ch{L}_{p}$ arising from points $t \\in T(\\mathcal{O}_{K,N}) \\cap D_{i}$ lie inside finitely many subvarieties $O_{i1}, \\hdots, O_{i\\ell}$ of $\\ch{L}_{p}$ corresponding to the finitely many possible isomorphism classes of $\\rho^{\\textrm{ss}}_{t}$.\n\nWe are now faced with the problem of understanding the intersections $\\psi_{p}(D_{i}) \\cap O_{ij}$, and showing that their inverse images under $\\psi_{p}^{-1}$ lie in an algebraic subscheme of smaller dimension. One part of this problem is to understand the dimensions of the varieties $O_{ij}$, and to show that they are sufficiently small: this step is carried out successfully for the families of hypersurfaces studied both by Lawrence-Venkatesh and Lawrence-Sawin, and appears to be managable in general. The more difficult object to control is $\\psi_{p}(D_{i})$, for which we need to understand the variation of the filtration $F^{\\bullet}$ over $D_{i}$. This, in turn, is governed by the Gauss-Manin connection $\\nabla : \\mathcal{H} \\otimes \\Omega^{1}_{T} \\to \\mathcal{H}$, which exists universally over $S$ after possibly increasing $N$; we may adjust $p$ so that it does not divide $N$ if necessary. The fact that $\\nabla$ exists universally over $\\mathcal{O}_{K,N}$ means that the same system of differential equations satisfied by $\\psi_{p}$ at $t \\in T(\\mathcal{O}_{K,N}) \\cap D_{i}$ is also satisfied by a Hodge-theoretic period map $\\psi : B \\to \\ch{L}$ on a sufficiently small analytic neighbourhood $B \\subset T(\\mathbb{C})$ containing $t$, where $\\ch{L}$ is a variety of Hodge flags. In particular, one can prove that the Zariski closures of $\\psi_{p}(D_{i})$ and $\\psi(B)$ have the same dimension.\n\nThe final step, which is to show that the Zariski closure of $T(\\mathcal{O}_{K,N}) \\cap D_{i}$ in $T$ has smaller dimension, is completed as follows. The Ax-Schanuel Theorem \\cite{AXSCHAN} for variations of Hodge structures due to Bakker-Tsimerman shows that if $V$ is an algebraic subvariety of $\\ch{L}$ of dimension at most $\\dim \\overline{\\psi(B)}^{\\textrm{Zar}} - \\dim \\psi(B)$,\\footnote{The dimension $\\dim \\psi(B)$ can once again, at least for open neighbourhoods $B$ with a sufficiently mild geometry, be made sense of the dimension of a locally constructible analytic set. Alternatively one can replace $\\dim \\psi(B)$ with $\\dim \\varphi(Z)$, where $\\varphi$ is as before and $Z \\subset T_{\\mathbb{C}}$ is a component containing $B$.} then the inverse image under $\\psi$ of the intersection $\\psi(B) \\cap V$ lies in an algebraic subvariety of $T_{\\mathbb{C}}$ of smaller dimension. Choosing an isomorphism $\\mathbb{C} \\cong \\overline{K_{v}}$ one can transfer this fact to the same statement for the map $\\psi_{p}$ and in particular for $V = O_{ij}$. Our problem is finally reduced to giving a lower bound for the difference $\\dim \\overline{\\psi(B)}^{\\textrm{Zar}} - \\dim \\psi(B)$. Our main result now reads:\n\n\\begin{thm}\n\\label{mainthm2}\nDefine the quantity\n\\[ \\Delta_{d} := \\min_{Z, \\psi} \\left[ \\dim \\overline{\\psi(B)}^{\\textrm{Zar}} - \\dim \\psi(B) \\right] , \\]\nwhere $Z$ ranges over all irreducible complex algebraic subvarieties $Z \\subset S_{\\mathbb{C}}$ of dimension greater than $d$, and where $\\psi$ is any complex analytic period map determined by the variation of Hodge structures $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$ and defined on a neighbourhood $B \\subset Z(\\mathbb{C})$. Then there exists an effective procedure which outputs an infinite sequence of lower bounds\n\\[ \\kappa(1) \\leq \\kappa(2) \\leq \\cdots \\leq \\kappa(r) \\leq \\cdots < \\Delta_{d} \\]\nsuch that for some $r = r_{0}$ we have $\\kappa(r_{0}) = \\Delta_{d} - 1$.\n\nIf moreover the period map $\\varphi$ is quasi-finite, one can determine $r_{0}$.\n\\end{thm}\n\n We note that by \\cite[Lem. 4.10(ii)]{urbanik2021sets} it is also a consequence of the Bakker-Tsimerman Theorem that when $Z$ is geometrically irreducible, we have $\\overline{\\psi(B)}^{\\textrm{Zar}} = \\mathbf{H}_{Z} \\cdot \\psi(t)$ for any point $t \\in Z(\\mathbb{C})$, which recovers the statement of \\autoref{mainthm}. \n\n\\subsection{Basic Idea of the Method}\n\\label{methodsketch}\n\nWe may observe that the computation of the bound described in \\autoref{mainthm2} is a purely Hodge-theoretic problem, i.e., it concerns only properties of the integral variation $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$ of Hodge structures on $S_{\\mathbb{C}}$. Let $\\mathcal{Q} : \\mathbb{V} \\otimes \\mathbb{V} \\to \\mathbb{Z}$ be a polarization of $\\mathbb{V}$, and let $(V, Q)$ be a fixed polarized lattice isomorphic to one (hence any) fibre of $(\\mathbb{V}, \\mathcal{Q})$; as it causes no harm, we will assume that $V = \\mathbb{Z}^{m}$ for some $m$, and therefore sometimes write $\\textrm{GL}_{m}$ for $\\textrm{GL}(V)$. Let $D$ be the complex manifold parametrizing polarized Hodge structures on $(V, Q)$ with the same Hodge numbers as $(\\mathbb{V}, \\mathcal{Q})$. A point $h \\in D$ we may view as a morphism $h : \\mathbb{S} \\to \\textrm{GL}(V)_{\\mathbb{R}}$, where $\\mathbb{S}$ is the Deligne torus, and the Mumford-Tate group $\\textrm{MT}(h)$ is the $\\mathbb{Q}$-Zariski closure of $h(\\mathbb{S})$. \n\nTo present our method, we introduce some terminology:\n\n\\begin{notn}\nWe denote by $\\ch{L}$ the $\\mathbb{Q}$-algebraic variety of all Hodge flags on the lattice $V$, not necessarily polarized. We note that $D$ is an open submanifold of a closed $\\mathbb{Q}$-algebraic subvariety $\\ch{D} \\subset \\ch{L}$.\n\\end{notn}\n\n\\begin{defn}\nGiven two subvarieties $W_{1}, W_{2} \\subset \\ch{L}$, we say that $W_{1} \\sim_{\\textrm{GL}} W_{2}$ if there exists $g \\in \\textrm{GL}_{m}(\\mathbb{C})$ such that $g \\cdot W_{1} = W_{2}$. Given a variety $W \\subset \\ch{L}$, we call the equivalence class $\\mathcal{C}(W)$ under $\\sim_{\\textrm{GL}}$ a \\emph{type}. The dimension of a type $\\mathcal{C}(W)$ is the dimension of $W$.\n\\end{defn}\n\n\\begin{defn}\nWe say that a type $\\mathcal{C}$ is \\emph{Hodge-theoretic} if $\\mathcal{C} = \\mathcal{C}(W)$, where $W = N(\\mathbb{C}) \\cdot h$ for $h \\in D$ and $N$ a $\\mathbb{Q}$-algebraic normal subgroup of $\\textrm{MT}(h)$.\n\\end{defn}\n\n\\vspace{0.5em}\n\nThe first step in our algorithm is:\n\n\\begin{quote}\n\\textbf{Step One:} Compute a finite list of types $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ such that every Hodge-theoretic type appears somewhere in the list.\n\\end{quote}\n\n\\noindent When we say to compute a type $\\mathcal{C}$, we mean to compute a representative $W \\subset \\ch{L}$ such that $\\mathcal{C} = \\mathcal{C}(W)$. That there are only finitely many Hodge-theoretic types is shown in \\autoref{finmantypes} below. \n\n\nThe problem given in Step One is solved in \\cite[Prop. 5.4]{urbanik2021sets}; we will say little about it here. It is related to the problem of classifying Mumford-Tate groups up to conjugacy by $\\textrm{GL}_{m}(\\mathbb{C})$, for which one can use a constructive version of the proof in \\cite[Thm. 4.14]{hodgelocivoisin}. It is also similar to the problem of classifying Mumford-Tate domains as studied in \\cite[Chap. VII]{GGK}. We note that the methods of \\cite[Chap. VII]{GGK}, when they can be carried out effectively, result in an approach for which $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ will be exactly the set of Hodge-theoretic types.\n\n\\vspace{0.5em}\n\nThe second step is more involved, and is the crux of our method. To describe it we need to introduce some terminology. \n\n\\begin{defn}\n\\label{locperdef}\nA \\emph{local period map} is a map $\\psi : B \\to \\ch{L}$ obtained as a composition $\\psi = q \\circ A$, where:\n\\begin{itemize}\n\\item[(i)] The set $B \\subset S(\\mathbb{C})$ is a connected analytic neighbourhood on which $\\mathbb{V}$ is constant and $F^{k} \\mathcal{H}$ is trivial for each $k$, where $\\mathcal{H} = \\mathbb{V} \\otimes \\mathcal{O}_{\\an{S}}$.\n\\item[(ii)] The map $A : B \\to \\textrm{GL}_{m}(\\mathbb{C})$ is a varying filtration-compatible period matrix over $B$. More precisely, there exists a basis $v^{1}, \\hdots, v^{m}$ for $\\mathcal{H}(B)$, compatible with the filtration in the sense that $F^{k} \\mathcal{H}(B)$ is spanned by $v^{1}, \\hdots, v^{i_{k}}$ for some $i_{k}$, and a flat frame $b^{1}, \\hdots, b^{m}$ for $\\mathbb{V}_{\\mathbb{C}}(B)$, such that $A(s)$ is the change-of-basis matrix from $v^{1}_{s}, \\hdots, v^{m}_{s}$ to $b^{1}_{s}, \\hdots, b^{m}_{s}$.\n\\item[(iii)] The map $q : \\textrm{GL}_{m} \\to \\ch{L}$ sends a matrix $M$ to the Hodge flag $F_{M}^{\\bullet}$ defined by the property that $F^{k}_{M}$ is spanned by the first $i_{k}$ columns.\n\\end{itemize}\n\\end{defn}\n\n\\noindent To summarize the preceding definition: a local period map is exactly a period map on $B$ except one does not necessarily compute periods with respect to the integral lattice $\\mathbb{V}(B) \\subset \\mathbb{V}_{\\mathbb{C}}(B)$ but is instead allowed to consider periods with respect to a more general complex flat frame. There is a natural $\\textrm{GL}_{m}(\\mathbb{C})$-action on the set of germs of local period maps at a point $s \\in S(\\mathbb{C})$, where $M \\in \\textrm{GL}_{m}(\\mathbb{C})$ acts on the map $\\psi = q \\circ A$ to give $M \\cdot \\psi = q \\circ (M \\cdot A)$. This action corresponds exactly to a change of the flat frame $b^{1}, \\hdots, b^{m}$, and all germs of local period maps at $s$ lie in a single $\\textrm{GL}_{m}(\\mathbb{C})$-orbit.\n\n\nThe construction of a local period map $\\psi : B \\to \\ch{L}$ involves picking a basis $b^{1}, \\hdots, b^{m}$ of $\\mathbb{V}_{\\mathbb{C}}(B)$, and hence choosing an isomorphism $\\mathbb{V}_{\\mathbb{C}}(B) \\simeq \\mathbb{C}^{m}$. When working with a local period map, we will always assume that such a basis has been choosen, and hence identify subgroups of $\\textrm{GL}(\\mathbb{V}_{\\mathbb{C}}(B))$ with subgroups of $\\textrm{GL}_{m}(\\mathbb{C})$. In particular, if $Z \\subset S_{\\mathbb{C}}$ is a geometrically irreducible subvariety which intersects $B$, we have an induced action of $\\mathbf{H}_{Z}$ on $\\ch{L}$.\n\nLastly, we need:\n\n\\begin{defn}\nGiven two types $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$, we say that $\\mathcal{C}_{1} \\leq \\mathcal{C}_{2}$ if there exists $W_{i} \\subset \\ch{L}$ for $1 = 1, 2$ such that $\\mathcal{C}_{i} = \\mathcal{C}(W_{i})$ and $W_{1} \\subset W_{2}$.\n\\end{defn}\n\n\\begin{defn}\n\\label{vartypedef}\nGiven a local period map $\\psi : B \\to \\ch{L}$ and a geometrically irreducible subvariety $Z \\subset S_{\\mathbb{C}}$ intersecting $B$ at $s$, we call $\\mathcal{C}(\\overline{\\psi(B \\cap Z)}^{\\textrm{Zar}}) = \\mathcal{C}(\\mathbf{H}_{Z} \\cdot \\psi(s))$ the \\emph{type} of $Z$, and denote it by $\\mathcal{C}(Z)$.\n\\end{defn}\n\n\\noindent For well-definedness, see \\autoref{welldeflem} below. From Step One, we have computed a finite list $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ of types containing all types that can arise from the variation $\\mathbb{V}$. Our next task is then:\n\n\\begin{quote}\n\\textbf{Step Two:} For each type $\\mathcal{C}_{i}$ appearing in the list, compute a differential system $\\mathcal{T}(\\mathcal{C}_{i})$ on $S$ characterized by the property that an algebraic subvariety $Z \\subset S_{\\mathbb{C}}$ is an integral subvariety for $\\mathcal{T}(\\mathcal{C}_{i})$ if and only if $\\mathcal{C}(Z) \\leq \\mathcal{C}_{i}$, and determine the dimension of a maximal integral subvariety for this system.\n\\end{quote}\n\n\\noindent We explain precisely what we mean by ``differential system'' in \\autoref{secthree}; actually our method does something more subtle than Step Two due to the fact that we can only approximate $\\mathcal{T}(\\mathcal{C}_{i})$ up to some finite order, but for expository purposes this is the essential point. After this, we will see the problem is reduced to analyzing which of the differential systems $\\mathcal{T}(\\mathcal{C}_{i})$ admit algebraic solutions of ``exceptional'' dimension, which can be carried out using tools from functional transcendence.\n\n\\subsection{Acknowledgements}\n\nThe author thanks Brian Lawrence, Akshay Venkatesh, and Will Sawin for comments on a draft of this manuscript.\n\n\n\\section{Algebraic Monodromy Orbits up to Conjugacy}\n\nIn this section we describe an effective method for solving ``Step One'' as posed in \\autoref{methodsketch}. We will also prove some preliminary facts about types used in the introduction, and we continue with the notation established there. We will work in the context of a general polarizable integral variation of Hodge structure $\\mathbb{V}$ on the complex algebraic variety $S$, not necessarily coming from a projective family as in the introduction.\n\n\\subsection{Basic Properties of Types}\n\n\\begin{lem}\n\\label{finmantypes}\nFor any geometrically irreducible subvariety $Z \\subset S$ and any local period map $\\psi : B \\to \\ch{L}$ with $Z \\cap B$ non-empty, we have\n\\[ \\overline{\\psi(Z \\cap B)}^{\\textrm{Zar}} = \\mathbf{H}_{Z} \\cdot \\psi(s) , \\]\nfor any point $s \\in Z(\\mathbb{C})$.\n\\end{lem}\n\n\\begin{proof}\nIt suffices to show that \n\\[ \\overline{\\psi(C)}^{\\textrm{Zar}} = \\mathbf{H}_{Z} \\cdot \\psi(s) , \\]\nfor each analytic component $C \\subset Z \\cap B$ separately, with $s$ a point of $C$. By acting on $\\psi$ by an element of $\\textrm{GL}_{m}(\\mathbb{C})$, the claim can be reduced to the situation where the periods which determine $\\psi$ are computed with respect to a basis for the integral lattice $\\mathbb{V}(B)$, and then the claim follows from \\cite[Lem. 4.10(ii)]{urbanik2021sets}.\n\\end{proof}\n\n\\begin{lem}\n\\label{welldeflem}\nThe equivalence class under $\\sim_{\\textrm{GL}}$ of $\\overline{\\psi(B \\cap Z)}^{\\textrm{Zar}}$ is independent of $\\psi$; i.e., the type of $Z \\subset S$ is well-defined.\n\\end{lem}\n\n\\begin{proof}\nLet $p : Z^{\\textrm{sm}} \\to Z$ be a smooth resolution, and consider the variation $p^{*} \\mathbb{V}$. From the fact that germs of local period maps on $Z^{\\textrm{sm}}$ with respect to the variation $p^{*} \\mathbb{V}$ factor through germs of local period maps on $S$, we may reduce to the same problem for $Z^{\\textrm{sm}}$ and the variation $p^{*} \\mathbb{V}$, i.e., we may assume $Z = S$. By analytically continuing a fixed local period map $\\psi$ to the universal covering $\\widetilde{S} \\to S$, we learn from the irreducibility of $\\widetilde{S}$ that at each point $s \\in S$, there exists a local period map $\\psi_{s} : B_{s} \\to \\ch{L}$ such that $\\overline{\\psi_{s}(B_{s})}^{\\textrm{Zar}} = \\overline{\\psi(B)}^{\\textrm{Zar}}$. Since the Zariski closure of $\\psi_{s}(B_{s})$ is determined by the germ of $\\psi_{s}$ at $s$, and because all germs of local period maps at $s$ lie in a single $\\textrm{GL}_{m}(\\mathbb{C})$-orbit, the result follows.\n\\end{proof}\n\n\\begin{lem}\n\\label{fintypesarise}\nThere are only finitely many Hodge-theoretic types.\n\\end{lem}\n\n\\begin{proof}\nWe observe that the problem reduces to the following: show they are finitely many $\\textrm{GL}_{m}(\\mathbb{C})$-equivalence classes of pairs $(h, N)$, where\n\\begin{itemize}\n\\item[(i)] $h \\in D$ is a polarized Hodge structure; and\n\\item[(ii)] $N$ is a $\\mathbb{Q}$-algebraic connected normal subgroup of $\\textrm{MT}(h)$;\n\\end{itemize}\nwhere we regard $\\textrm{GL}_{m}(\\mathbb{C})$ as acting on $h$ through its action on $\\ch{L}$, and on $N$ by conjugation. Note that two such equivalent pairs will generate orbits in $\\ch{L}$ equivalent under $\\sim_{\\textrm{GL}}$. Since the groups $\\textrm{MT}(h)$ are reductive and have finitely many connected normal algebraic factors, this reduces to the same problem for pairs of the form $(h, \\textrm{MT}(h))$. We recall that $D$ is an open submanifold of $\\ch{D}$, the flag variety of flags satisfying the first Hodge-Riemann bilinear relation (the isotropy condition), and that $\\ch{D}$ is an algebraic subvariety of $\\ch{L}$. We then use the fact that there are finitely many Mumford-Tate groups up to $\\textrm{GL}_{m}(\\mathbb{C})$-conjugacy (see \\cite[Thm. 4.14]{hodgelocivoisin}), and that for a fixed Mumford-Tate group $M$ the Hodge structures in $D$ with Mumford-Tate contained in $M$ lie inside finitely many $M(\\mathbb{C})$-orbits in $\\ch{D}$, see \\cite[VI.B.9]{GGK}.\n\\end{proof}\n\n\\subsection{Computing Types up to Conjugacy}\n\nIn this section we give some references for carrying out Step One as described in the introduction.\n\n\\begin{prop}\n\\label{MTgroupequivalgo}\nThere exists an algorithm to compute subvarieties $W_{1}, \\hdots, W_{\\ell} \\subset \\ch{L}$ such that the set of Hodge-theoretic types is a (possibly proper) subset of $\\{ \\mathcal{C}(W_{1}), \\hdots, \\mathcal{C}(W_{\\ell}) \\}$.\n\\end{prop}\n\n\\begin{proof}\nThis is solved in \\cite[Prop. 5.4]{urbanik2021sets}.\n\\end{proof}\n\nLet us comment briefly on a different approach to Step One given in \\cite[Chap. VII]{GGK}. In \\cite[Chap. VII]{GGK}, the authors describe a method for classifying both Mumford-Tate groups and Mumford-Tate domains (orbits of points $h \\in D$ under $\\textrm{MT}(h)(\\mathbb{R})$ and $\\textrm{MT}(h)(\\mathbb{C})$). Given an appropriate such classification, one can easily solve Step One by computing the decompositions of the groups $\\textrm{MT}(h)$ that arise into $\\mathbb{Q}$-simple factors. The method of \\cite[Chap. VII]{GGK} is to first classify CM Hodge structures $h_{\\textrm{CM}} \\in D$, and then give a criterion for deciding when a Lie subalgebra of $\\mathfrak{gl}(V)$ corresponds to a Mumford-Tate group generating a Mumford-Tate domain containing $h_{\\textrm{CM}}$. They carry out this classification procedure successfully when $\\dim V = 4$, and so for variations with Hodge numbers $(2, 2)$ and $(1, 1, 1, 1)$.\n\nThe method given for classifying CM Hodge structures given in \\cite[Chap. VII]{GGK} is to observe that CM Hodge structures up to isogeny are determined by certain data associated to embeddings of CM fields, and hence the first step of the procedure in \\cite[Chap. VII]{GGK} is to ``classify all CM fields of rank up to [$\\dim V$] by [their] Galois group''. We are not aware of an effective method for carrying out this step.\\footnote{The paper \\cite{dodson1984structure} gives a potential approach by giving a method to classify certain abstract structures associated with Galois groups of CM fields. However one still needs to determine which such structures are actually associated to a concrete CM field.} It is also not clear to us precisely the sense in which the term ``classify'' is being used; i.e., we do not know what form the data of a ``classification of CM Hodge structures'' takes, and consequently what form the resulting classification of Mumford-Tate domains will have. For this reason, we were unable to apply the methods of \\cite[Chap. VII]{GGK} to prove \\autoref{MTgroupequivalgo}. \n\n\n\n\n\\section{Differential Tools and a Jet Criterion}\n\\label{secthree}\n\nIn this section we introduce a collection of effectively computable algebro-geometric correspondences which can be used for studying systems of differential equations on $S$ induced by the variation $\\mathbb{V}$, and then use it to solve the main problem. We have already carried out most of the work in two preceding papers \\cite{periodimages} and \\cite{urbanik2021sets}, so we will first need to collect some results. In this section we assume that $S$ is a $K$-variety for $K \\subset \\mathbb{C}$ a number field, and that $\\mathbb{V}$ is a polarizable integral variation of Hodge structure on $S_{\\mathbb{C}}$ such that the vector bundle $\\mathcal{H} = \\mathbb{V} \\otimes_{\\mathbb{Z}} \\mathcal{O}_{\\an{S_{\\mathbb{C}}}}$, the filtration $F^{\\bullet}$, and the connection $\\nabla : \\mathcal{H} \\to \\mathcal{H} \\otimes \\Omega^{1}_{S}$ all admit $K$-algebraic models. Moreover, we assume that we may effectively compute a description of these objects in terms of finitely-presented $K$-modules over an affine cover of $S$; for a justification of this assumption in the situation where $\\mathbb{V}$ comes from a smooth projective $K$-algebraic family $f : X \\to S$ see \\cite[\\S2]{urbanik2021sets}.\n\n\\subsection{The Constructive Period-Jet Correspondence}\n\nOur algebro-geometric correspondences will be formulated using the language of \\emph{jets}. Let $A^{d}_{r} = K[t_{1}, \\hdots, t_{d}]\/\\langle t_{1}, \\hdots, t_{d} \\rangle^{r+1}$, and define $\\mathbb{D}^{d}_{r} = \\textrm{Spec} \\hspace{0.15em}  A^{d}_{r}$ to be the $d$-dimensional disk of order $r$; we suppress the field $K$ in the notation. A \\emph{jet space} associated to a space $X$ is a space which parametrizes maps $\\mathbb{D}^{d}_{r} \\to X$. More formally, for $X$ a finite-type $K$-scheme, we have:\n\n\n\n\\begin{defn}\n\\label{jetspacedef}\nWe define $J^{d}_{r} X$ to be the scheme representing the contravariant functor $\\textrm{Sch}_{K} \\to \\textrm{Set}$ given by \\vspace{-0.2em}\n\\[ T \\mapsto \\textrm{Hom}_{K}(T \\times_{K} \\mathbb{D}^{d}_{r}, X), \\hspace{1.5em} [T \\to T'] \\mapsto [\\textrm{Hom}_{K}(T' \\times_{K} \\mathbb{D}^{d}_{r}, X) \\to \\textrm{Hom}_{K}(T \\times_{K} \\mathbb{D}^{d}_{r}, X)] , \\]\nwhere the natural map $\\textrm{Hom}_{K}(T' \\times_{K} \\mathbb{D}^{d}_{r}, X) \\to \\textrm{Hom}_{K}(T \\times_{K} \\mathbb{D}^{d}_{r}, X)$ obtained by pulling back along $T \\times_{K} \\mathbb{D}^{d}_{r} \\to T' \\times_{K} \\mathbb{D}^{d}_{r}$. \n\\end{defn}\n\n\\vspace{0.5em}\n\n\\noindent That the functor defining $J^{d}_{r} X$ in \\autoref{jetspacedef} is representable is handled by \\cite[\\S2]{periodimages}. Moreover, $J^{d}_{r}$ is itself a functor, sending a map $g : X \\to Y$ to the map $J^{d}_{r} g : J^{d}_{r} X \\to J^{d}_{r} Y$ that acts on points by post-composition. For $X$ an analytic space, there is an analogous construction that appears in \\cite[\\S2.3]{periodimages}. If $K \\subset \\mathbb{C}$ is a subfield, this construction is compatible with analytification.\n\nThe purpose of introducing jets is the following result, proven in \\cite{urbanik2021sets}, building on \\cite{periodimages}.\n\n\\begin{thm}\n\\label{jetcorresp}\nFor each $d, r \\geq 0$, a variation of Hodge structure $\\mathbb{V}$ on $S$ gives rise to a canonical map \n\\[ \\eta^{d}_{r} : J^{d}_{r} S \\to \\textrm{GL}_{m} \\backslash J^{d}_{r} \\ch{L} , \\] \nof algebraic stacks characterized by the property that for any local period map $\\psi : B \\to \\ch{L}$ and any jet $j \\in J^{d}_{r} B$ we have $\\psi \\circ j = \\eta^{d}_{r}(j)$ modulo $\\textrm{GL}_{m}(\\mathbb{C})$.\n\nMoreover, if the data $(\\mathcal{H}, F^{\\bullet}, \\nabla)$ associated to the variation $\\mathbb{V}$ admits a $K$-algebraic model, the map $\\eta^{d}_{r}$ is defined over $K$, and there exists an algorithm to compute the $\\textrm{GL}_{m}$-torsor $p^{d}_{r} : \\mathcal{P}^{d}_{r} \\to J^{d}_{r} S$ and the $\\textrm{GL}_{m}$-invariant map $\\alpha^{d}_{r} : \\mathcal{P}^{d}_{r} \\to J^{d}_{r} \\ch{L}$ which defines $\\eta^{d}_{r}$ from a presentation of the data $(\\mathcal{H}, F^{\\bullet}, \\nabla)$ in terms of finitely-presented $K$-modules.\n\\end{thm}\n\nWe note that the computability of the torsor $\\mathcal{P}^{d}_{r}$ in \\autoref{jetcorresp} has in particular the following consequence: if $\\mathcal{S} \\subset (\\textrm{GL}_{m} \\backslash J^{d}_{r} \\ch{L})(\\mathbb{C})$ is a subset which is the image under the quotient of a constructible $L$-algebraic set $\\mathcal{F} \\subset J^{d}_{r} \\ch{L}$, where $K \\subset L$ is a computable extension, then we can compute $(\\eta^{d}_{r})^{-1}(\\mathcal{S})$ by computing $p^{d}_{r}((\\alpha^{d}_{r})^{-1}(\\mathcal{F}))$. Thus if we define \\vspace{0.5em}\n\\begin{defn}\n\\label{Tconstdef}\nFor a constructible $L$-algebraic set $\\mathcal{F} \\subset J^{d}_{r} \\ch{L}$, with $K \\subset L$ an extension, we write\n\\[ \\mathcal{T}^{d}_{r}(\\mathcal{F}) := (\\eta^{d}_{r})^{-1}(\\textrm{GL}_{m} \\cdot \\mathcal{F}) . \\]\nMoreover, for a type $\\mathcal{C} = \\mathcal{C}(W)$, we will write either $\\mathcal{T}^{d}_{r}(\\mathcal{C})$ or $\\mathcal{T}^{d}_{r}(W)$ for the set $\\mathcal{T}^{d}_{r}(J^{d}_{r} W)$.\n\\end{defn} \\vspace{0.3em}\n\\noindent then the main consequence of the preceding discussion for our situation is the following, which is immediate from what we have said:\n\\begin{prop}\n\\label{diffconstprop}\nFor each $d, r \\geq 0$ there exists an algorithm which, given a constructible $L$-algebraic set $\\mathcal{F} \\subset J^{d}_{r} \\ch{L}$ with $K \\subset L$ a computable extension, computes $\\mathcal{T}^{d}_{r}(\\mathcal{F}) \\subset J^{d}_{r} S$. \\qed\n\\end{prop}\n\n\\subsection{Jet Conditions and Types}\n\nLet us now try to understand how computing the ``differential constraints'' induced by types $\\mathcal{C}(W)$ as in \\autoref{diffconstprop} can help us carry out Step Two of \\autoref{methodsketch}. Let $\\Gamma = \\textrm{Aut}(V, Q)(\\mathbb{Z})$, and let $\\varphi : S_{\\mathbb{C}} \\to \\Gamma \\backslash D$ be the canonical period map which sends a point $s \\in S(\\mathbb{C})$ to the isomorphism class of the polarized Hodge structure on $\\mathbb{V}_{s}$. By \\cite{OMINGAGA}, the map $\\varphi$ factors as $\\iota \\circ p$, where $p : S_{\\mathbb{C}} \\to T$ is a dominant map of algebraic varieties and $\\iota$ is a closed embedding of analytic spaces; it follows that for each subvariety $Z \\subset S_{\\mathbb{C}}$ the dimension of the image $\\varphi(Z)$ makes sense as the dimension of a constructible algebraic set.\n\nFix a sequence of compatible embeddings \n\\[ \\textrm{Spec} \\hspace{0.15em}  K = \\mathbb{D}^{0}_{r} \\xhookrightarrow{\\iota_{0}} \\mathbb{D}^{1}_{r} \\xhookrightarrow{\\iota_{1}} \\mathbb{D}^{2}_{r} \\xhookrightarrow{\\iota_{2}} \\mathbb{D}^{3}_{r} \\xhookrightarrow{\\iota_{3}} \\mathbb{D}^{4}_{r} \\xhookrightarrow{\\iota_{4}} \\cdots \\]\nof formal disks. By acting on points via pullback, we obtain natural transformations of functors $\\textrm{res}^{d}_{e} : J^{d}_{r} \\to J^{e}_{r}$ which produce maps $J^{d}_{r} X \\to J^{e}_{r} X$ that take jets $j : \\mathbb{D}^{d}_{r} \\to X$ to their restrictions $j \\circ \\iota_{d-1} \\circ \\cdots \\circ \\iota_{e}$. We are now ready to present the key proposition for our method:\n\n\\begin{defn}\nFor a scheme $X$ (resp. analytic space $X$) denote by $J^{d}_{r,nd} X \\subset J^{d}_{r} X$ the subscheme (resp. the analytic subspace) parametrizing those maps $j : \\mathbb{D}^{d}_{r} \\to X$ which are injective on the level of tangent spaces. We call such $j$ \\emph{non-degenerate} jets.\n\\end{defn}\n\n\\begin{prop}\n\\label{mainjetprop}\nLet $\\mathcal{S}$ be a set of types containing all the Hodge-theoretic types, and let $e$ and $k$ be non-negative integers. Then the following are equivalent:\n\\begin{itemize}\n\\item[(i)] there exists a geometrically irreducible subvariety $Z \\subset S_{\\mathbb{C}}$ with $\\dim Z > d$, $\\dim \\varphi(Z) \\geq e$, and such that $\\dim \\mathcal{C}(Z) - \\dim \\varphi(Z) \\leq k$;\n\\item[(ii)] there exists $\\mathcal{C} \\in \\mathcal{S}$ with $\\dim \\mathcal{C} - e \\leq k$, and such that the intersection \n\\[ \\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k) := \\mathcal{T}^{d+1}_{r}(\\mathcal{C}) \\cap \\mathcal{T}^{d+1}_{r}((\\textrm{res}^{d+1}_{e})^{-1}(J^{e}_{r,nd} \\ch{L})) \\cap J^{d+1}_{r,nd} S \\] \nis non-empty for each $r \\geq 0$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{rem}\nIn the situation that the variation $\\mathbb{V}$ admits a local Torelli theorem, one can drop the distinction between $\\dim Z$ and $\\dim \\varphi(Z)$ and consider instead the intersections $\\mathcal{T}^{d+1}_{r}(\\mathcal{C}) \\cap J^{d+1}_{r,nd} S$ in part (ii), ignoring the middle term.\n\\end{rem} \\vspace{0.5em}\n\nThe rest of this section we devote to proving \\autoref{mainjetprop}, identifying $S$ with $S_{\\mathbb{C}}$ for ease of notation. To begin with, let us check that (i) implies (ii) by applying the definitions. If $g : S' \\to S$ is an \\'etale cover and we consider the variation $\\mathbb{V}' = g^{*} \\mathbb{V}$, then the maps $\\eta^{d}_{r}$ and $\\eta'^{d}_{r}$ obtained from \\autoref{jetcorresp} are related by $\\eta'^{d}_{r} = \\eta^{d}_{r} \\circ (J^{d}_{r} g)$. Choosing a finite index subgroup $\\Gamma' \\subset \\Gamma$ and passing to such a cover, we can reduce to the case where we have a period map $\\varphi : S \\to \\Gamma \\backslash D$ with $D \\to \\Gamma \\backslash D$ a local isomorphism. Applying \\cite{OMINGAGA} the map $\\varphi : S \\to \\Gamma \\backslash D$ factors as $\\varphi = \\iota \\circ p$, where $p : S \\to T$ is a dominant map of algebraic varieties and $\\iota$ is an analytic closed embedding. Then via $p$, the variety $Z$ is dominant over a closed subvariety $Y \\subset T$ of dimension $\\dim \\varphi(Z) \\geq e$. Shrinking $S$ (and hence $Z$) we may assume that $Z$ is smooth, and that $Z$ is surjective onto a dense open subset $Y^{\\circ} \\subset Y$. Shrinking $S$ even further we may assume that $Z \\to Y^{\\circ}$ is smooth. The smoothness of $Z \\to Y^{\\circ}$ implies in particular that the induced jet space maps $J^{d}_{r} Z \\to J^{d}_{r} Y^{\\circ}$ for all choices of $d$ and $r$ are surjective.\n\nWe may choose neighbourhoods $B \\subset S(\\mathbb{C})$ and $U \\subset D$ such that $\\restr{\\pi}{U} : U \\to \\pi(U)$ is an isomorphism, both $B \\cap Z$ and $\\pi(U) \\cap Y^{\\circ}$ are non-empty, and we have a local lift $\\psi : B \\to U$ of $\\varphi$. Choose a jet $\\sigma \\in J^{e}_{r,nd} (Y^{\\circ} \\cap \\pi(U))$ and lift it along $p$ to a jet $\\widetilde{\\sigma} \\in J^{e}_{r,nd} (Z \\cap B)$ landing at the point $s \\in S(\\mathbb{C})$. Using the fact that the germ $(Z, s)$ is smooth of dimension $\\dim Z > d$ the jet $\\widetilde{\\sigma}$ can be extended to a jet $j \\in J^{d+1}_{r, nd} (Z \\cap B)$ such that $\\textrm{res}^{d+1}_{e} (j) = \\widetilde{\\sigma}$, and hence $\\textrm{res}^{d+1}_{e} (\\varphi \\circ j) = \\sigma$. From the fact that $\\restr{\\varphi}{B} = \\pi \\circ \\psi$ and the defining property of the map $\\eta^{d}_{r}$ it follows that $j$ lies inside $\\mathcal{T}^{d+1}_{r}((\\textrm{res}^{d+1}_{e})^{-1}(J^{e}_{r,nd} \\ch{L})) \\cap J^{d+1}_{r,nd} S$. We can then take $\\mathcal{C} = \\mathcal{C}(Z)$, and the fact that $j$ factors through $Z$ implies that $j \\in \\mathcal{T}^{d+1}_{r}(\\mathcal{C})$ as well.\n\nTo prove the reverse implication, we review some preliminary facts relating to jets.\n\n\\begin{defn}\nWe say a sequence $\\{ j_{r} \\}_{r \\geq 0}$ with $j_{r} \\in J^{d}_{r} X$ is \\emph{compatible} if the projections $J^{d}_{r} X \\to J^{d}_{r-1} X$ map $j_{r}$ to $j_{r-1}$.\n\\end{defn}\n\n\\begin{lem}\n\\label{compseqlem}\nSuppose that $\\mathcal{T}_{r} \\subset J^{d}_{r} X$ is a collection of non-empty constructible algebraic sets such that the projections $J^{d}_{r} X \\to J^{d}_{r-1} X$ map $\\mathcal{T}_{r}$ into $\\mathcal{T}_{r-1}$. Then there exists a compatible sequence $\\{ j_{r} \\}_{r \\geq 0}$ with $j_{r} \\in \\mathcal{T}_{r}$ for all $r \\geq 0$.\n\\end{lem}\n\n\\begin{proof}\nSee \\cite[Lem. 5.3]{periodimages}.\n\\end{proof}\n\n\\begin{defn}\nGiven a variety $Z$ (algebraic or analytic) and $z \\in Z$ a point, we denote by $(J^{d}_{r} Z)_{z}$ the fibre above $z$ of the natural projection map $J^{d}_{r} Z \\to Z$.\n\\end{defn}\n\n\\begin{lem}\n\\label{factorthrough}\nIf $g : (Z, z) \\to (Y, y)$ is a map of analytic germs with $\\dim (Z, z) = d$ and $(Z, z)$ smooth, we have an infinite compatible family $j_{r} \\in (J^{d}_{r,nd} Z)_{z}$, and $g \\circ j_{r} \\in (J^{d}_{r} X)_{y}$ for some germ $(X, y) \\subset (Y, y)$ and all $r \\geq 0$, then $g$ factors through the inclusion $(X, x) \\subset (Y, y)$. \n\\end{lem}\n\n\\begin{proof}\nSee \\cite[Lem. 4.5]{urbanik2021sets}.\n\\end{proof}\n\n\\begin{lem}\n\\label{jetdimlem}\nSuppose that $X$ is an algebraic variety (resp. analytic space) and $x \\in X$ is a point for which the fibre $(J^{d}_{r,nd} X)_{x}$ above $x$ is non-empty for all $r \\geq 0$. Then the germ $(X, x)$ has dimension at least $d$. \n\\end{lem}\n\n\\begin{proof}\nSee \\cite[Prop. 2.7]{periodimages}.\n\\end{proof}\n\n\\begin{proof}[Proof of \\ref{mainjetprop}:]\nBy what we have said, we are reduced to showing that (ii) implies (i). The statement is unchanged by replacing $S$ with a finite \\'etale covering $g : S' \\to S$ and the variation $\\mathbb{V}$ with $g^{*} \\mathbb{V}$; as before this does not affect the hypothesis (ii) since the maps $\\eta^{d}_{r}$ and $\\eta'^{d}_{r}$ associated to $S$ and $S'$ are related by $\\eta'^{d}_{r} = \\eta^{d}_{r} \\circ (J^{d}_{r} g)$. Choosing a finite index subgroup $\\Gamma' \\subset \\Gamma$ and choosing $g$ so the monodromy of $g^{*} \\mathbb{V}$ lies in $\\Gamma'$ we may reduce to the case where $D \\to \\Gamma \\backslash D$ is a local isomorphism. Moreover, taking a futher finite \\'etale cover we may apply \\cite[Cor. 13.7.6]{CMS} to reduce to the case where $\\varphi$ is proper; this requires possibly extending $S'$ to a variety $S''$ by adding a closed subvariety at infinity, but as long as we are careful to work only with jets that factor through $S'$ our proof will produce a variety $Z$ intersecting $S'$; in particular, we now assume that $\\varphi : S \\to \\Gamma \\backslash D$ is proper but redefine the sets $\\mathcal{K}^{d}_{r}$ to equal\n\\[ \\mathcal{T}^{d+1}_{r}(\\mathcal{C}) \\cap \\mathcal{T}^{d+1}_{r}((\\textrm{res})^{d+1}_{e})^{-1}(J^{d}_{r,nd} \\ch{L})) \\cap J^{d+1}_{r,nd} S^{\\circ} , \\]\nfor some open subvariety $S^{\\circ} \\subset S$.\n\nApplying the main result of \\cite{OMINGAGA}, the map $\\varphi$ once again factors as $\\varphi = \\iota \\circ p$ with $p : S \\to T$ a dominant (now proper) map of algebraic varieties. We can then consider the Stein factorization $S \\xrightarrow{q} U \\xrightarrow{r} T$ of $p$; note that $q$ is proper with connected fibres, $U$ is normal, and $r$ is finite. One can define the type of a subvariety $Y \\subset U$ exactly as in \\autoref{vartypedef} with respect to the period map $U \\to \\Gamma \\backslash D$. From \\autoref{compseqlem} above, the assumption (ii) entitles us to a compatible sequence $\\{ j_{r} \\}_{r \\geq 0}$ of jets such that $j_{r} \\in \\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ for all $r \\geq 0$. Let us write $\\mathcal{C} = \\mathcal{C}(W)$ for some subvariety $W \\subset \\ch{L}$.\n\nBy construction, the jets $\\sigma_{r} = \\textrm{res}^{d+1}_{e} j_{r}$ are non-degenerate, and remain so after composing with any local period map $\\psi : B \\to D$ for which $\\sigma_{r}$ factors through $B$. This in particular implies (since $D \\to \\Gamma \\backslash D$ is a local isomorphism) that the jets $\\varphi \\circ \\sigma_{r}$ are non-degenerate, and hence so are the jets $q \\circ \\sigma_{r}$. Let $Y \\subset U$ be the smallest algebraic subvariety such that $q \\circ j_{r} \\in J^{d+1}_{r} Y$ for all $r$. We observe that there exists a component $Z$ of $q^{-1}(Y)$ of dimension at least $d+1$ that contains the image of the jets $\\{ j_{r} \\}_{r \\geq 0}$: one can see this by picking a neighbourhood of $j_{0}$ of the form $\\mathbb{C}^{\\ell} \\times \\mathbb{C}^{d+1}$ such that $j_{r}$ is constant on the first factor, and applying \\autoref{factorthrough} above to see that the restriction of $q$ to $\\{ 0 \\} \\times \\mathbb{C}^{d+1}$ factors through $Y$. Moreover, we must have $q(Z) = Y$ by minimality, and by applying \\autoref{jetdimlem} to the non-degenerate sequence $\\{ q \\circ \\sigma_{r} \\}_{r \\geq 0}$ that $\\dim Y \\geq e$. Since $r$ is finite, this means $\\dim \\varphi(Z) \\geq e$. From the fact that local period maps on $S$ factor through local period maps on $U$ we learn that $\\mathcal{C}(Z) = \\mathcal{C}(Y)$, so the result will follow if we can show that $\\dim \\mathcal{C}(Y) - \\dim Y \\leq k$. For ease of notation let us now write $\\tau_{r} = q \\circ j_{r}$.\n\nFix a local lift $\\psi : B \\to D$ of the period map $U \\to \\Gamma \\backslash D$ with $B \\subset U(\\mathbb{C})$ an analytic ball such that the jets $\\tau_{r}$ factor through $B$. Consider the set $\\mathcal{G}_{r} \\subset \\textrm{GL}_{m}(\\mathbb{C})$ consisting of those $g \\in \\textrm{GL}_{m}(\\mathbb{C})$ for which $\\psi \\circ \\tau_{r} \\in g \\cdot (J^{d+1}_{r} W)$. Then for each $r$ the set $\\mathcal{G}_{r}$ is algebraically-constructible, and using the fact that $j_{r} \\in \\mathcal{T}^{d+1}_{r}(W)$ the set $\\mathcal{G}_{r}$ is necessarily non-empty. Let $g_{\\infty}$ be an element of this intersection. Extend $\\psi$ to a lift $\\widetilde{\\varphi}_{Y} : \\widetilde{Y} \\to D$ of $Y \\to \\Gamma \\backslash D$ to the universal covering. Then $\\widetilde{\\varphi}_{Y}(\\widetilde{Y}) \\subset D$ is a closed analytic set containing the jets $\\psi \\circ \\tau_{r}$, and hence the non-degenerate jets $\\textrm{res}^{d+1}_{e}(\\psi \\circ \\tau_{r})$. Letting $A \\subset \\widetilde{\\varphi}_{Y}(\\widetilde{Y}) \\cap (g_{\\infty} \\cdot W)$ be the minimal analytic germ through which $\\psi \\circ \\tau_{r}$ (and hence $\\textrm{res}^{d+1}_{e}(\\psi \\circ \\tau_{r})$) factors, it follows from \\autoref{jetdimlem} that $A$ has dimension at least $e$.\n\nConsider the Zariski closure $V \\subset Y$ of $\\psi^{-1}(A)$. We claim that $V = Y$. Because $Y$ was chosen minimal containing the compatible family of jets $\\{ \\tau_{r} \\}_{r \\geq 0}$, it suffices to show that each $\\tau_{r}$ factors through $V$. Consider the component of $q^{-1}(B)$ containing $j_{0}$; by choosing coordinates we may assume $q^{-1}(B) \\subset \\mathbb{C}^{\\ell} \\times \\mathbb{C}^{d+1}$ is an open neighbourhood and identify $j_{0}$ with the origin. After a further change of coordinates we may assume $j_{r}$ is constant on the first factor, and let $F = q^{-1}(B) \\cap (\\{ 0 \\} \\times \\mathbb{C}^{d+1})$. Applying \\autoref{factorthrough} we find that $\\psi(q(F)) \\subset A$, and hence $q(F) \\subset V$. Using proper base change the map $q^{-1}(B) \\to B$ is proper, so $q(F)$ is an analytic subvariety of $B$, and by construction the jets $\\{ \\tau_{r} \\}_{r \\geq 0}$ factor through it, hence through $V$.\n\nWe are now ready to apply the Bakker-Tsimerman transcendence theorem; the jets are no longer needed. It follows from the structure theorem for period mappings \\cite[III.A]{GGK}, the closed analytic set $\\widetilde{\\varphi}_{Y}(\\widetilde{Y})$ lies inside an orbit $\\ch{D}' = \\mathbf{H}_{Y} \\cdot \\psi(\\tau_{0})$ of the algebraic monodromy of $Y$. Consider the graph $E$ of $\\widetilde{\\varphi}_{Y}$ in $Y \\times \\ch{D}'$. Then as $A$ has dimension $e$ and $\\psi^{-1}(A)$ is Zariski dense, there exists a component $C$ of $E \\cap (Y \\times (\\ch{D}' \\cap g_{\\infty} \\cdot W))$ of dimension at least $e$ and projecting to a Zariski dense subset of $Y$. Applying the main theorem of \\cite{AXSCHAN} we learn that\n\\begin{align*}\n\\textrm{codim}_{Y \\times \\ch{D}'} (Y \\times (\\ch{D}' \\cap g_{\\infty} \\cdot W)) + \\textrm{codim}_{Y \\times \\ch{D}'} E &\\leq \\textrm{codim}_{Y\n \\times \\ch{D}'} C \\\\\n(\\dim \\ch{D}' - \\dim W) + \\dim \\ch{D}' &\\leq \\dim Y + \\dim \\ch{D}' - \\dim C \\\\\n\\dim \\ch{D}' - \\dim Y &\\leq \\dim W - \\dim C \\\\\n\\dim \\mathcal{C}(Y) - \\dim Y &\\leq \\dim \\mathcal{C} - e \\\\\n&\\leq k\n\\end{align*}\n\\end{proof}\n\n\\section{Main Results}\n\n\\subsection{Computing Bounds on $\\Delta_{d}$}\n\n\\subsubsection{Computing Lower Bounds}\n\nLet us explain the significance of \\autoref{mainjetprop} in proving \\autoref{mainthm}, i.e., giving an effective method to compute bounds for \\[ \\Delta_{d} = \\min_{\\dim Z > d} [ \\dim \\mathcal{C}(Z) - \\dim \\varphi(Z) ] , \\]\nwhere we have used \\autoref{mainthm2} and \\autoref{vartypedef} to give this equivalent expression for $\\Delta_{d}$. Since $\\Delta_{d}$ is a integer bounded by $\\dim D$, giving an effective method to compute it amounts to developing a procedure to decide, for any integer $k$, whether we have $\\Delta_{d} \\leq k$. This in turn amounts to deciding, for any integer $0 \\leq e \\leq \\dim \\varphi(S)$, whether (ii) holds in \\autoref{mainjetprop}.\n\nLet us take $\\mathcal{S}$ to be the set up types computed by Step One, and let us suppose that in fact $\\Delta_{d} > k$. Then by the equivalence in \\autoref{mainjetprop}, we should find that for any $\\mathcal{C} \\in \\mathcal{S}$ with $\\dim \\mathcal{C} - e \\leq k$, there must be some $r = r(\\mathcal{C}, e, k)$ such that $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ is empty. Moreover, verifying that such an $r$ exists for each such $\\mathcal{C}$ and $e$ proves, again by the same equivalence, that $\\Delta_{d} > k$. Consequently, we obtain the following result, which is the first half of \\autoref{mainthm}:\n\n\\begin{prop}\n\\label{lowerboundcomp}\nBy computing the sets $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ described in \\autoref{mainjetprop} in parallel, we may compute a non-decreasing sequence of lower bounds\n\\[ \\kappa(1) \\leq \\kappa(2) \\leq \\cdots \\leq \\kappa(r) \\leq \\cdots < \\Delta_{d} \\]\nsuch that for some $r = r_{0}$ we have $\\kappa(r_{0}) = \\Delta_{d} - 1$.\n\\end{prop}\n\n\\begin{proof}\nAt the $r$'th stage we compute all the sets $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ for all applicable choices of $\\mathcal{C}$, $e$ and $k$, and set $\\kappa(r)$ to be the smallest $k$ for which all the sets $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ are empty. From the discussion preceding the Proposition, the result follows.\n\\end{proof}\n\n\\subsubsection{Computing Upper Bounds}\n\n\\autoref{lowerboundcomp} does not actually give an algorithm for computing $\\Delta_{d}$, since no way is given to decide when $r = r_{0}$. For applications to the Lawrence-Venktesh method this doesn't matter: one wants to be able to compute an optimal lower bound for $\\Delta_{d}$, but one does not actually have to prove that this lower bound actually equals $\\Delta_{d}$ in order to apply the diophantine finiteness machinery. Nevertheless, let us explain how one can do this in the case where $\\varphi$ is quasi-finite; under this assumption, we may drop the distinction between $\\dim Z$ and $\\dim \\varphi(Z)$, and we are instead interested in computing\n\\[ \\min_{\\dim Z > d} \\, \\left[ \\dim \\mathcal{C}(Z) - \\dim Z \\right] . \\]\n\n What is needed is the following:\n\n\\begin{prop}\n\\label{upperboundcomp}\nSuppose that $S$ is quasi-projective and $\\varphi$ is quasi-finite. Then there exists a procedure that outputs an infinite sequence of upper bounds\n\\[ \\tau(1) \\geq \\tau(2) \\geq \\cdots \\geq \\tau(i) \\geq \\cdots \\geq \\Delta_{d} \\] \nsuch that for some $i = i_{0}$ we have $\\tau(i) = \\Delta_{d}$.\n\\end{prop}\n\n\\noindent Given both \\autoref{lowerboundcomp} and \\autoref{upperboundcomp} we obtain an algorithm for computing $\\Delta_{d}$ by running both procedures in parallel and terminating when $\\kappa(r) + 1 = \\tau(i)$. \n\n\\subsection{Finding Varieties that Exhibit $\\Delta_{d}$}\n\nIn this section we prove \\autoref{upperboundcomp}, assuming throughout that $S$ is quasi-projective and $\\varphi$ is quasi-finite. Let us fix a projective compactification $S \\subset \\overline{S}$ of $S$ and consider the Hilbert scheme $\\textrm{Hilb}(\\overline{S})$. There exist algorithms, for instance by working with the Pl\\\"uker coordinates of the appropriate Grassmannian, for computing any finite subset of components of $\\textrm{Hilb}(\\overline{S})$. By \\cite[Lem. 5.10]{urbanik2021sets} we obtain the same fact for the open locus $\\textrm{Var}(S) \\subset \\textrm{Hilb}(\\overline{S})$ consisting of just those points $[\\overline{Z}]$ for which $Z = S \\cap \\overline{Z}$ is a non-empty geometrically irreducible algebraic subvariety of $S$. What we will show is that there exists a procedure which outputs an infinite sequence $\\{ \\mathcal{W}_{i} \\}_{i = 1}^{\\infty}$ of constructible algebraic loci $\\mathcal{W}_{i} \\subset \\textrm{Var}(S)$, with the following two properties:\n\\begin{itemize}\n\\item[(i)] for each $i$, the type $\\mathcal{C}(Z)$ and dimension $\\dim Z$ are constant over all $[Z] \\in \\mathcal{W}_{i}$;\n\\item[(ii)] there exists some $i = i_{0}$ such that \n\\[ \\Delta_{d} = \\dim \\mathcal{C}(Z) - Z , \\]\nfor some (hence any) point $[Z] \\in \\mathcal{W}_{i}$.\n\\end{itemize}\nGiven such an algorithm the problem of computing the bound $\\tau(i)$ that appears in \\autoref{upperboundcomp} reduces to choosing a point $[Z] \\in \\mathcal{W}_{i}$, computing $\\dim \\mathcal{C}(Z) - \\dim Z$, and setting \n\\[ \\tau(i) := \\textrm{min} \\{ \\tau(i-1), \\dim \\mathcal{C}(Z) - \\dim Z \\} . \\] \n(We note that the problem of computing $\\dim \\mathcal{C}(Z)$ from $Z$ and the restriction $\\restr{(\\mathcal{H}, F^{\\bullet}, \\nabla)}{Z}$ of the algebraic data on $S$ is solved for us by \\cite[Lem. 5.8]{urbanik2021sets} by taking the family $g$ in the statement of \\cite[Lem. 5.8]{urbanik2021sets} to be a trivial family; we will say little about this problem here.)\n\nIn fact, an algorithm for computing the sets $\\mathcal{W}_{i}$ has already been given in a previous paper by the author. We begin by recalling the necessary background. We regard $S$ as a complex algebraic variety in what follows. Given two (geometrically) irreducible subvarieties $Z_{1}, Z_{2} \\subset S$ with $Z_{1} \\subset Z_{2}$, the algebraic monodromy group $\\mathbf{H}_{Z_{1}}$ may be naturally regarded as a subgroup of $\\mathbf{H}_{Z_{2}}$ (after choosing a base point $s \\in Z_{1}(\\mathbb{C})$). Using this we define:\n\\begin{defn}\nAn irreducible complex subvariety $Z \\subset S$ is said to be \\emph{weakly special} if it is maximal among such subvarieties for its algebraic monodromy group.\n\\end{defn}\nThe key fact is then the following:\n\\begin{lem}\nFor each integer $d > 0$, there exists a weakly special subvariety $Z \\subset S$ such that \n\\[ \\Delta_{d} = \\dim \\mathcal{C}(Z) - \\dim Z . \\]\n\\end{lem}\n\n\\begin{proof}\nBy \\cite[Prop. 4.18]{urbanik2021sets}, the condition that $Z$ be weakly special is equivalent to $Z$ being a maximal irreducible complex subvariety of $S$ of type $\\mathcal{C}(Z)$. Thus if we have any $Y$ which is not weakly special, there exists a weakly special $Z$ properly containing $Y$ with $\\mathcal{C}(Y) = \\mathcal{C}(Z)$, hence\n\\[ \\dim \\mathcal{C}(Z) - \\dim Z < \\dim \\mathcal{C}(Y) - \\dim Y . \\]\nIt follows that the value of $\\Delta_{d}$ can only be achieved by a weakly special variety.\n\\end{proof}\n\n\\begin{proof}[Proof of \\ref{upperboundcomp}]\nBy the \\emph{degree} of a subvariety $Z \\subset S$ we will mean the degree of its closure $\\overline{Z}$ inside $\\overline{S}$. For any integer $b$, denote by $\\textrm{Var}(S)_{b} \\subset \\textrm{Var}(S)$ the finite-type subscheme parametrizing varieties of degree at most $b$. Denote by $\\mathcal{W} \\subset \\textrm{Var}(S)$ the locus of weakly special subvarieties. Then given an integer $b$, the algorithm that appears in \\cite[Thm. 5.15]{urbanik2021sets} computes the intersection $\\mathcal{W} \\cap \\textrm{Var}(S)_{b}$ as a constructible algebraic locus.\n\nLet us describe the algorithm appearing in \\cite[Thm. 5.15]{urbanik2021sets} more precisely. Consider the types $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ computed by Step One, and define for each such type $\\mathcal{C}_{j}$ the locus\n\\[ \\mathcal{W}(\\mathcal{C}_{j}) := \\{ [Z] \\in \\textrm{Var}(S) : \\mathcal{C}(Z) \\leq \\mathcal{C}_{j} \\} . \\] \nIt is shown in \\cite[Prop. 4.31]{urbanik2021sets} that for each $j$ the locus $\\mathcal{W}(\\mathcal{C}_{j})$ is closed algebraic. We can then consider the sublocus $\\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}} \\subset \\mathcal{W}(\\mathcal{C}_{j})$ consisting of just those components $C \\subset \\mathcal{W}(\\mathcal{C}_{j})$ for which a generic point $[Z] \\in C$ satisfies $\\mathcal{C}(Z) = \\mathcal{C}_{j}$.\n\nIn \\cite[Prop. 5.14]{urbanik2021sets}, an algorithm is given for computing $\\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}} \\cap \\textrm{Var}(S)_{b}$ for each $j$. Using this, one can compute all the finitely many closed algebraic loci $C_{1}, \\hdots, C_{i_{b}}$ which arise as a component of $\\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}} \\cap \\textrm{Var}(S)_{b}$ for some $j$. The problem of computing $\\mathcal{W} \\cap \\textrm{Var}(S)_{b}$ is reduced to computing constructible algebraic conditions on each component $C_{i} \\subset \\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}}$ which define the locus $\\mathcal{W}_{i} \\subset C_{i}$ of points $[Z] \\in C_{i}$ that are weakly special of type $\\mathcal{C}_{j}$. This is taken care of by the proof of \\cite[Thm. 5.15]{urbanik2021sets}. By construction, the points in $\\mathcal{W}_{i}$ all have the same type and the same dimension, so we complete the proof by computing these loci for increasing values of $b$.\n\\end{proof}\n\n\n\\section{Application to Lawrence-Venkatesh}\n\nWe now show how the bound of \\autoref{mainthm} can be used to establish diophantine finiteness results. Similar arguments appear in \\cite{LV} and \\cite{lawrence2020shafarevich}, but as they are not precisely adapted to our setup, we give our own version. We recall the situation: we have a smooth projective family $f : X \\to S$ over the smooth base $S$, with everything defined over $\\mathcal{O}_{K,N}$.\\footnote{Note in particular we are assuming now that $S$ is smooth over $\\mathcal{O}_{K,N}$, which we can achieve by increasing $N$ if necessary.} The relative algebraic de Rham cohomology $\\mathcal{H} = R^{i} f_{*} \\Omega^{\\bullet}_{X\/S}$ gives a model for the Hodge bundle $\\mathbb{V} \\otimes \\mathcal{O}_{\\an{S}}$, where $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$. By a result of Katz and Oda \\cite{katz1968}, the flat connection associated to the local system $\\mathbb{V}_{\\mathbb{C}}$ by the Riemann-Hilbert correspondence admits a model $\\nabla : \\mathcal{H} \\to \\Omega^{1}_{S} \\otimes \\mathcal{H}$ after possibly increasing $N$. Likewise, we may also assume the Hodge filtration $F^{\\bullet}$ gives a filtration of $\\mathcal{H}$ by vector subbundles. \n\nFix a prime $p$ not dividing $N$, and a place $v$ of $K$ above $p$. Then for each integral point $s \\in S(\\mathcal{O}_{K,N})$, we have a Galois representation $\\rho_{s} : \\textrm{Gal}(\\overline{K}\/K) \\to \\textrm{Aut}(H^{i}_{\\textrm{\\'et}}(X_{\\overline{K}, s}, \\mathbb{Q}_{p}))$, and an argument of Faltings \\cite[Lem 2.3]{LV} shows that the semisimplifications of the representations $\\rho_{s}$ belong to a finite set of isomorphism classes. From crystalline cohomology, each $s \\in S(\\mathcal{O}_{K,N})$, viewed as a point of $S(\\mathcal{O}_{K,v})$ where $\\mathcal{O}_{K,v}$ is the $v$-adic ring of integers, gives rise to a triple $(H^{i}_{\\textrm{dR}}(X_{s}), \\phi_{s}, F^{\\bullet}_{s})$ where $\\phi_{s}$ is the crystalline Frobenius. Moreover, using the functor $D_{\\textrm{cris}}$ of $p$-adic Hodge theory \\cite[Expos\\'e III]{fontaine1994corps}, the triple $(H^{i}_{\\textrm{dR}}(X_{s}), \\phi_{s}, F^{\\bullet}_{s})$ is determined up to isomorphism by the restriction $\\rho_{s,v}$ along the map $\\textrm{Gal}(\\overline{K_{v}}\/K_{v}) \\to \\textrm{Gal}(\\overline{K}\/K)$ determined by a fixed embedding $\\overline{K} \\hookrightarrow \\overline{K_{v}}$. We denote by $\\mathcal{I}(s)$ all those triples $(V, \\phi, F^{\\bullet})$ which are of the form $D_{\\textrm{cris}}(\\restr{\\rho}{\\mathbb{Q}_{p}})$, where $\\rho$ is a global Galois representation whose semisimplification is isomorphic to the semisimplification of $\\rho_{s}$. \n\n\nRecall that we have fixed the integral lattice $V = \\mathbb{Z}^{m}$, where $m$ is the dimension of the cohomology of the fibres of $f$, and a $\\mathbb{Q}$-algebraic flag variety $\\ch{L}$ of Hodge flags on $V$. In what follows we write $V_{p}$ for $V \\otimes \\mathbb{Q}_{p}$, and $\\ch{L}_{p}$ for $\\ch{L}_{\\mathbb{Q}_{p}}$. Then the key idea of the Lawrence-Venkatesh method is the following:\n\n\\begin{prop}\n\\label{LVprop}\nSuppose that for each $s \\in S(\\mathcal{O}_{K,N})$, whenever we have an endomorphism $\\phi_{s} : V_{p} \\to V_{p}$ and a flag $F^{\\bullet}_{s}$ on $V_{p}$ such that $(V_{p}, \\phi_{s}, F^{\\bullet}_{s})$ represents $\\mathcal{I}(s)$, the Hodge flags $F^{\\bullet}$ on $V_{p}$ for which $(V_{p}, \\phi_{s}, F^{\\bullet}) \\in \\mathcal{I}(s)$ lie in an algebraic subvariety $O_{s} \\subset \\ch{L}_{p}$ satisfying $\\Delta_{d} \\geq \\dim O_{s}$. Then $\\dim \\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} \\leq d$.  \n\\end{prop}\n\nTo prove \\autoref{LVprop} we will need a rigid-analytic version of the Bakker-Tsimerman transcendence theorem, which we will see can be deduced formally from the complex analytic one. To set things up, let us revisit the term \\emph{local period map}, this time in the rigid analytic setting (c.f. \\autoref{locperdef}). We will denote by $\\mathbb{C}_{p}$ the completion of the algebraic closure $\\overline{K_{v}}$. In what follows we sometimes identify algebraic varieties with their rigid-analytifications when the context is clear.\n\n\\begin{defn}\n\\label{padiclocperdef}\nLet $K_{p}$ be a local field containing $K_{v}$, let $\\an{S_{K_{p}}}$ be the rigid-analytification of the base-change $S_{K_{p}}$ of $S$, and suppose that $B_{p} \\subset \\an{S_{K_{p}}}$ is an affinoid subdomain. Then a (rigid-analytic) local period map $\\psi : B_{p} \\to \\an{\\ch{L}_{K_{p}}}$ is a rigid-analytic map obtained as a composition $\\psi = \\an{q_{K_{p}}} \\circ A_{p}$, where:\n\\begin{itemize}\n\\item[(i)] The rigid analytifications $F^{k} \\an{\\mathcal{H}_{K_{p}}}$ are all trivial on $B_{p}$.\n\\item[(ii)] The map $A_{p} : B_{p} \\to \\an{\\textrm{GL}_{m, K_{p}}}$ is a varying filtration-compatible $p$-adic period matrix over $B_{p}$. More precisely, there exists a basis $v^{1}, \\hdots, v^{m}$ for $\\an{\\mathcal{H}_{K_{p}}}(B_{p})$, compatible with the filtration in the sense that $F^{k} \\an{\\mathcal{H}_{K_{p}}}(B_{p})$ is spanned by $v^{1}, \\hdots, v^{i_{k}}$ for some $i_{k}$, and a flat (for $\\an{\\nabla_{K_{p}}}$) frame $b^{1}, \\hdots, b^{m}$ such that $A_{p}$ gives a varying change-of-basis matrix from $v^{1}, \\hdots, v^{m}$ to $b^{1}, \\hdots, b^{m}$.\n\\item[(iii)] The map $q : \\textrm{GL}_{m} \\to \\ch{L}$ is the map that sends a matrix $M$ to the Hodge flag $F^{\\bullet}_{M}$ defined by the property that $F^{k}_{M}$ is spanned by the first $i_{k}$ columns.\n\\end{itemize}\n\\end{defn}\n\nTo prove \\autoref{LVprop} we will need a version of the Bakker-Tsimerman transcendence result for rigid-analytic local period maps, which we prove by formally transferring the same result for complex analytic local period maps. To avoid certain minor pathologies that can occur in the complex analytic case we will restrict to local period maps $\\psi : B \\to \\ch{L}$ which are definable in the structure $\\mathbb{R}_{\\textrm{an}, \\textrm{exp}}$; for background on definability and definable analytic spaces we refer to \\cite{van1996geometric} and \\cite{OMINGAGA}. We note that this is not a serious restriction: given any local period map $\\psi$ and any point $s \\in B$ there exists a definable restriction of $\\psi$ to a neighbourhood of $s$, a fact which is for instance easily deduced from \\cite[Prop. 4.27]{urbanik2021sets}.\n\n\\begin{lem}\n\\label{padicaxschanlem}\n~\\begin{itemize}\n\\item[(i)] Suppose that $\\psi : B \\to \\an{\\ch{L}_{\\mathbb{C}}}$ is a definable analytic local period map on $\\an{S_{\\mathbb{C}}}$. Let $V \\subset \\ch{L}_{\\mathbb{C}}$ be an algebraic subvariety satisfying $\\Delta_{d} \\geq \\dim V$. Then $\\psi^{-1}(V)$ lies in an algebraic subvariety of $S_{\\mathbb{C}}$ of dimension at most $d$.\n\\item[(ii)] Suppose that $\\psi_{p} : B_{p} \\to \\an{\\ch{L}_{\\mathbb{C}_{p}}}$ is a rigid-analytic local period map on $\\an{S_{\\mathbb{C}_{p}}}$. Let $V_{p} \\subset \\ch{L}_{p, \\mathbb{C}_{p}}$ be an algebraic subvariety satisfying $\\Delta_{d} \\geq \\dim V_{p}$. Then $\\psi_{p}^{-1}(V_{p})$ lies in an algebraic subvariety of $S_{\\mathbb{C}_{p}}$ of dimension at most $d$.\n\\end{itemize}\n\\end{lem}\n\n\\paragraph{Proof of \\autoref{padicaxschanlem}(i):} ~ \\\\\n\n\\vspace{-0.5em}\n\nThis is an application of the Bakker-Tsimerman transcendence theorem. Let $Z \\subset S_{\\mathbb{C}}$ be the Zariski closure of $\\psi^{-1}(V)$. We assume for contradiction that $\\dim Z > d$, and let $Z_{0} \\subset Z$ be a component of maximal dimension. Let $\\varphi : \\an{S_{\\mathbb{C}}} \\to \\Gamma \\backslash D$ be the canonical period map with $\\Gamma = \\textrm{Aut}(V,Q)(\\mathbb{Z})$. The statement is invariant under replacing $\\psi$ with a $\\textrm{GL}_{m}(\\mathbb{C})$-translate $g \\cdot \\psi$ and $V$ with $g \\cdot V$, so we may assume that $\\psi$ is a local lift of $\\varphi$. Arguing as in \\cite[Cor. 13.7.6]{CMS} we may assume that $\\varphi$ is proper, hence the image $T = \\varphi(S)$ is algebraic by \\cite{OMINGAGA}, and we may consider the Stein factorization $S \\xrightarrow{q} U \\xrightarrow{r} T$ of the map $S \\to T$. \n\nLet $Y = q(Z_{0})$, and note that $\\dim Y = \\dim \\varphi(Z_{0})$. By assumption we have $\\Delta_{d} \\leq \\dim \\mathcal{C}(Z_{0}) - \\dim \\varphi(Z_{0}) = \\dim \\mathcal{C}(Y) - \\dim Y$, where the type of $Y$ is taken with respect to the period map $U \\to \\Gamma \\backslash D$. Moreover, this continues to hold if we replace $Y$ with a smooth resolution $Y'$. The variation of Hodge structure on $S$ descends to $U$, and hence shrinking $B$ if necessary we may factor $\\psi$ through a definable local lift on $U$. By pulling back along the resolution we obtain a definable local lift $\\psi' : B' \\to D$ of the period map $\\varphi' : Y' \\to \\Gamma \\backslash D$ such that $\\psi'^{-1}(V)$ is Zariski dense in $Y'$. We are reduced to the following situation: we have smooth variety $Y'$ with a period map $\\varphi' : Y' \\to \\Gamma \\backslash D$, a local lift $\\psi' : B' \\to D$ such that $\\psi'^{-1}(V)$ is Zariski dense, and such that $\\dim \\mathcal{C}(Y') - \\dim Y' \\geq \\dim V$. \n\nWe now contradict the Bakker-Tsimerman theorem. In particular, we may extend the local lift $\\psi'$ to a lift $\\widetilde{\\varphi'} : \\widetilde{Y'} \\to D$ of $\\varphi'$ to the universal cover, and consider the graph $W \\subset Y' \\times \\ch{D}'$ of the map $\\widetilde{\\varphi'}$, where $\\ch{D}'$ is the orbit $\\mathbf{H}_{Y'} \\cdot \\psi'(y)$ for some $y \\in Y'(\\mathbb{C})$. We then have that\n\\begin{align*}\n\\textrm{codim}_{Y' \\times \\ch{D}'} (Y' \\times (V \\cap \\ch{D}')) + \\textrm{codim}_{Y' \\times \\ch{D}'} W &\\geq \\dim \\ch{D}' - \\dim V + \\dim \\ch{D}' \\\\\n&= \\dim \\mathcal{C}(Y') - \\dim V + \\dim \\mathcal{C}(Y') \\\\\n&\\geq \\dim Y' + \\dim \\mathcal{C}(Y') .\n\\end{align*}\nSince $\\psi'$ is definable, $\\psi'^{-1}(V)$ has finitely many components, and hence there exists an analytic component $C \\subset B'$ of $\\psi'^{-1}(V)$ such that $C$ is Zariski dense in $Y'$. Let $\\widetilde{C} \\subset Y' \\times \\ch{D}'$ be its graph under $\\psi'$. If $\\dim Y' = 0$ there is nothing to show, so we may assume that $\\dim \\mathcal{C}(Y') > 0$. Hence we find that $\\dim Y' + \\dim \\mathcal{C}(Y') > \\textrm{codim}_{Y' \\times \\ch{D}'} \\widetilde{C}$, and by the Bakker-Tsimerman theorem \\cite{AXSCHAN} the component $C$ lies in a proper subvariety of $Y'$, giving a contradiction. \\qed\n\n\\vspace{1em}\n\nTo prove \\autoref{padicaxschanlem}(ii) we first translate \\autoref{padicaxschanlem}(i) into a claim about rings of formal power series. In particular let $\\psi : B \\to \\ch{L}$ be a local period map with $V \\subset \\ch{L}$ an algebraic subvariety, and choose a point $s \\in B$ such that $\\psi(s) = t \\in V$. Then $\\psi$ induces a map on formal power series rings $\\widehat{\\psi}^{\\sharp} : \\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t} \\to \\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s}$. The claim of \\autoref{padicaxschanlem}(i) then says that if $I_{V} \\subset \\mathcal{O}_{\\ch{L}_{\\mathbb{C}}, t}$ is the ideal defining $V$ with extension $\\widehat{I}_{V}$ inside $\\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t}$, then the ideal generated by $\\widehat{\\psi}^{\\sharp}(\\widehat{I}_{V})$ contains an ideal $\\widehat{I}_{Z}$ which is the extension of an ideal $I_{Z} \\subset \\mathcal{O}_{S_{\\mathbb{C}}, s}$ defining the germ of a subvariety of dimension at most $d$.\n\n\n\\paragraph{Proof of \\autoref{padicaxschanlem}(ii):} ~ \\\\\n\n\\vspace{-0.5em}\n\nThe claim is Zariski-local on $S$, so we can in particular assume that the bundles $F^{k} \\mathcal{H}$ for varying $k$ are algebraically trivial over $S$, that $S$ is affine, and by smoothness that $\\Omega^{1}_{S}$ is free. By definition, the map $\\psi : B_{p} \\to \\an{\\ch{L}_{\\mathbb{C}_{p}}}$ is associated to the following data: a filtration-compatible frame $v^{1}, \\hdots, v^{m}$, where $v^{1}, \\hdots, v^{i_{k}}$ spans $F^{k} \\an{\\mathcal{H}_{\\mathbb{C}_{p}}} (B_{p})$, and a flat frame $b^{1}, \\hdots, b^{m}$ spanning $\\an{\\mathcal{H}_{\\mathbb{C}_{p}}}(B_{p})$, where flatness means $\\an{\\nabla_{\\mathbb{C}_{p}}} b_{i} = 0$ for all $1 \\leq i \\leq m$. This data satisfies the property that $\\psi = \\an{q_{\\mathbb{C}_{p}}} \\circ A_{p}$, where $A_{p}$ is the change-of-basis matrix from the frame  $v^{1}, \\hdots, v^{m}$ to the frame $b^{1}, \\hdots, b^{m}$, and $q$ is the map $\\an{\\textrm{GL}_{m,\\mathbb{C}_{p}}} \\to \\an{\\ch{L}_{\\mathbb{C}_{p}}}$ sending a matrix to the Hodge flag it represents. We note that changing the frame $v^{1}, \\hdots, v^{m}$ to another filtration-compatible frame $v'^{1}, \\hdots, v'^{m}$ does not change the local period map: such a change has the effect of replacing the map $A_{p}$ with $A_{p} \\cdot C$, where $C$ is a varying matrix over $B_{p}$ whose right-action on $A_{p}$ preserves the span of the first $i_{k}$ columns for each $k$, and hence $q \\circ (A_{p} \\cdot C) = q \\circ A_{p}$. We threfore lose no generality by assuming the filtration-compatible frame is the restriction to $B_{p}$ of an algebraic filtration-compatible frame over $S$. \n\nThe affinoid neighbourhood $B_{p}$ is of the form $\\textrm{Sp} \\, T$, where $T$ is an affinoid $\\mathbb{C}_{p}$-algebra. The inverse image $\\psi^{-1}(V)$ is then a closed affinoid subdomain of $B_{p}$, i.e., it corresponds to an ideal $I \\subset T$ such that $\\psi^{-1}(V)$ may be identified with $\\textrm{Sp} \\, T\/I$. If $R$ is the coordinate ring of $S_{\\mathbb{C}_{p}}$, then the map $B_{p} \\hookrightarrow \\an{S_{\\mathbb{C}_{p}}} \\to S_{\\mathbb{C}_{p}}$ induces a map $\\iota : R \\to T$, and the claim to be shown is that there exists an ideal $J \\subset R$ defining a subvariety of dimension at most $d$ such that $\\iota(J) \\subset I$. The ring $T$ is Noetherian, so the ideal $I$ admits a primary decomposition. Taking radicals, we obtain finitely many prime ideals $I_{1}, \\hdots, I_{\\ell}$ containing $I$ such that the problem reduces, for each $1 \\leq j \\leq \\ell$, to finding $J_{j} \\subset R$ defining subvarieties of dimension at most $d$ such that $\\iota(J_{j}) \\subset I_{j}$ for each $j$. The analytification map $\\an{S_{\\mathbb{C}_{p}}} \\to S_{\\mathbb{C}_{p}}$ is bijective onto the set of closed points of $S_{\\mathbb{C}_{p}}$ and induces isomorphisms on completed local rings [see whatever]. It follows that if we choose a maximal ideal $\\mathfrak{m} \\subset T$ containing $I_{j}$ we obtain a commuting diagram\n\\begin{center}\n\\begin{tikzcd}\nR \\arrow[r,\"\\iota\"] \\arrow[d, hook] & \\arrow[d, hook] T \\\\\n\\widehat{R}_{\\iota^{-1}(\\mathfrak{m})} \\arrow[r,\"\\sim\"] & \\widehat{\\mathcal{O}}_{B_{p}, \\mathfrak{m}} ,\n\\end{tikzcd}\n\\end{center}\nwhere the bottom arrow is an isomorphism of completed local rings, and the vertical arrows are injections. In particular, if we denote by $\\widehat{I}_{j}$ the extension of $I_{j}$ in $\\widehat{\\mathcal{O}}_{B_{p}, \\mathfrak{m}}$, it suffices to show that $\\widehat{\\iota}(J_{j}) \\subset \\widehat{I}_{j}$, where $\\widehat{\\iota}$ is the composition of the left and bottom arrow; here we have used the fact that $I_{j} = \\widehat{I}_{j} \\cap T$. \n\nFix an isomorphism $\\tau : \\mathbb{C}_{p} \\xrightarrow{\\sim} \\mathbb{C}$, which we choose to preserve the embeddings $K \\subset \\mathbb{C}$ and $K \\subset \\mathbb{C}_{p}$. Using the model for $S$ over $K$, the isomorphism $\\tau$ allows us to identify $R$ with the coordinate ring of $S_{\\mathbb{C}}$, the ideal $\\iota^{-1}(\\mathfrak{m})$ with a complex point $s \\in S(\\mathbb{C})$, the ring $\\widehat{R}_{\\iota^{-1}(\\mathfrak{m})}$ with the completed local ring $\\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s}$. Let $t_{p}$ be the image of the point corresponding to $\\mathfrak{m}$ under $\\psi$, and let $t$ be the composition $t_{p} \\circ \\tau^{-1}$. Applying the isomorphism $\\tau$ at the level of formal power series, the rigid-analytic local period map $\\psi$ induces a map \n\\[ \\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t} \\xrightarrow{\\tau} \\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}_{p}}, t_{p}} \\xrightarrow{\\widehat{\\psi}^{\\sharp}} \\widehat{\\mathcal{O}}_{B_{p}, \\mathfrak{m}} \\xrightarrow{\\sim} \\widehat{R}_{\\iota^{-1}(\\mathfrak{m})} \\xrightarrow{\\tau} \\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s} , \\]\nwhose composition we denote by $\\widehat{\\eta}$. In what follows we identify the ideals $\\widehat{I}_{j}$ with their images in $\\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s}$; by construction they are the extensions along $\\widehat{\\eta}$ of an ideal in $\\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t}$ associated to the base-change of $V$ using $\\tau$. By part (i) of this theorem and our reformulation of it in terms of completed local rings, it suffices to show that $\\widehat{\\eta}$ is induced by a complex analytic local period map defined on a neighbourhood of $s$.\n\nRecall that we have a decomposition $\\psi = \\an{q_{\\mathbb{C}_{p}}} \\circ A_{p}$, where $q$ is the rigid-analytification of a $\\mathbb{Q}$-algebraic map, and $A_{p}$ gives a varying change-of-basis matrix between a filtration-compatible frame $v^{1}, \\hdots, v^{m}$ and a rigid-analytic flat frame. Recall also that we have chosen $v^{1}, \\hdots, v^{m}$ so that it is the rigid-analytification of a $K$-algebraic filtration-compatible frame $w^{1}, \\hdots, w^{m}$ over $S$. Using the decomposition $\\psi = \\an{q_{\\mathbb{C}_{p}}} \\circ A_{p}$ and the isomorphism $\\tau$ we may factor $\\widehat{\\eta}$ as $\\widehat{q} \\circ \\widehat{\\kappa}$, where $\\widehat{\\kappa} : \\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s} \\to \\widehat{\\mathcal{O}}_{\\textrm{GL}_{m, \\mathbb{C}}, P}$ is the base-change under $\\tau$ of the map induced by $A_{p}$. From our definition of local period map in \\autoref{locperdef}, it suffices to show that $\\widehat{\\kappa}$ is induced by a varying change-of-basis matrix $A : B \\to \\an{\\textrm{GL}_{m,\\mathbb{C}}}$ from $w^{1}, \\hdots, w^{m}$ to a complex-analytic flat frame.\n\nThe result will follow from the fact that $A$ and $A_{p}$ satisfy a common set of $K$-algebraic differential equations whose solutions are uniquely determined by the period matrix they assign to a point in $B$. To see this, let us write $\\nabla w^{i} = \\sum_{j = 1}^{m} c_{ij} \\otimes w^{j}$ for $K$-algebraic sections $c_{ij} \\in \\Omega^{1}_{S_{K}}$. Suppose then that $b^{k} = \\sum_{i = 1}^{m} f_{ik} w^{i}$ is a flat frame on some complex analytic or rigid-analytic neighbourhood. We then have that\n\\begin{align*}\n\\nabla b^{k} &= \\nabla \\left( \\sum_{i = 1}^{m} f_{ik} w^{i} \\right) \\\\\n&= \\sum_{j = 1}^{m} df_{jk} \\otimes w^{j} + \\sum_{i = 1}^{m} f_{ik} \\left( \\sum_{j = 1}^{m} c_{ij} \\otimes v^{j} \\right) \\\\\n&= \\sum_{j = 1}^{m} \\left( df_{jk} + \\sum_{i = 1}^{m} f_{ik} c_{ij} \\right) \\otimes v^{j} ,\n\\end{align*}\nfrom which we see that $b^{k}$ giving a flat frame is equivalent to $f_{jk}$ satisfying the system of differential equations $df_{jk} = -\\sum_{i = 1}^{m} f_{ik} c_{ij}$ for all $1 \\leq j, k \\leq m$. If we choose a trivialization $dz_{1}, \\hdots, dz_{n}$ of $\\Omega^{1}_{S_{K}}$, we may write the $c_{ij}$ in terms of their coefficients $c_{ij,\\ell}$ with respect to this trivialization, and the same system of differential equations becomes \n\\begin{equation}\n\\label{diffeq}\n\\partial_{\\ell} f_{jk} = -\\sum_{i} f_{ik} c_{ij,\\ell} ;\n\\end{equation} \nhere the operator $\\partial_{\\ell}$ is defined using the dual basis to $dz_{1}, \\hdots, dz_{n}$. By differentiating \\autoref{diffeq} and substituting the lower-order differential equations into the higher-order ones, we obtain, for each sequence $\\{ \\ell_{i} \\}_{i = 1}^{e}$ with $1 \\leq \\ell_{i} \\leq n$ and $e \\geq 1$, a set of $K$-algebraic polynomials $\\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}$ in the functions $f_{uv}$ for $1 \\leq u, v \\leq m$ with coefficients in the coordinate ring of $S_{K}$ such that \n\\begin{equation}\n\\label{diffeqpolys}\n\\partial_{\\ell_{1}} \\cdots \\partial_{\\ell_{e}} f_{jk} = \\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}([f_{uv}]) .\n\\end{equation}\n\nBecause $S_{K}$ is smooth, given a point $s$ of $S_{K}$ the functions $z_{1} - s_{1}, \\hdots, z_{n} - s_{n}$, where $s_{i}$ is the value of $z_{i}$ on $s$, induce a coordinate system in the local and formal power series rings associated to $S_{K}$ at $s$. In these coordinates, the map $A_{p}$ is given by $f^{-1}_{p}$, where $f_{p} = [f_{uv}]$ is a rigid-analytic matrix-valued solution to the differential equations \\autoref{diffeq}. The formal map $\\widehat{\\kappa}$ obtained using the isomorphism $\\tau$ then satisfies the same set of differential equations, and in particular its derivatives of all orders at $s$ are determined using \\autoref{diffeqpolys} by the initial condition $f^{-1}(s) = P$. If we then construct an analytic solution to the differential  system in \\autoref{diffeq} in a neighbourhood of $s$ satisfying $f^{-1}(s) = P$, the resulting analytic map induces the map $\\widehat{\\kappa}$ on formal power series rings. It follows that $\\widehat{\\eta}$ is induced by a local period map, which completes the proof. \\qed\n\n\\paragraph{Proof of \\autoref{LVprop}:} ~ \\\\\n\n\\vspace{-0.5em}\n\nWe denote by $\\mathcal{O}_{K,(v)}$ the ring of integers localized at the prime ideal $\\mathfrak{p}$ of $\\mathcal{O}_{K}$ corresponding to $v$. We begin by showing that (base changes to $K_{v}$ of) the points of $S(\\mathcal{O}_{K,(v)})$ lie inside finitely many distinguished open affinoids $B_{p} \\subset \\an{S_{K_{v}}}$ admitting local period maps $\\psi_{p} : B_{p} \\to \\an{\\ch{L}_{K_{v}}}$. This reduces to showing that there are finitely many distinguished open affinoids $B_{p} \\subset \\an{S_{K_{v}}}$ containing the points in $S(\\mathcal{O}_{K,(v)})$ over which $\\an{\\mathcal{H}_{K_{v}}}$ admits a rigid-analytic flat frame. We may cover $S$ by finitely many open subschemes $U \\subset S$ such that $\\Omega^{1}_{U}$ and the bundles $F^{k} \\mathcal{H}$ for varying $k$ are all trivial. Then any point $s \\in S(\\mathcal{O}_{K,(v)})$ factors through some element of this cover, so we may reduce to the case where $\\Omega^{1}_{S}$ and the bundles $F^{k} \\mathcal{H}$ are all trivial.\n\nProceeding as in the proof of \\autoref{padicaxschanlem}, we can choose algebraic functions $z_{1}, \\hdots, z_{n}$ on $S$ such that $d z_{1}, \\hdots, d z_{n}$ trivializes $\\Omega^{1}_{S}$. We obtain differential equations as in (\\ref{diffeqpolys}), where the polynomials $\\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}$ are functions in the coordinate ring $R$ of $S \\times \\textrm{GL}_{m}$, and so in particular we may view them after base-changing as elements of $R_{\\mathcal{O}_{K,v}}$, where $\\mathcal{O}_{K,v}$ is the ring of $v$-adic integers. Choose a point $\\overline{s_{0}} \\in S(\\mathcal{O}_{K,N}\/\\mathfrak{p} \\mathcal{O}_{K,N})$. Then as $S$ has (by assumption) good reduction modulo $\\mathfrak{p}$, we obtain by \\cite[IV. 18.5.17]{EGA} a lift $s_{0} \\in S(\\mathcal{O}_{K,v})$. Choosing an initial condition $P \\in \\textrm{GL}_{m}(\\mathcal{O}_{K,v})$, we may use (\\ref{diffeqpolys}) to construct a map $\\psi_{s_{0}} = \\an{q_{K_{v}}} \\circ f^{-1}$, where the partial derivatives of $f$ at $s_{0}$ are given by evaluating the polynomials $\\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}$ at $(s_{0}, P)$. As the coefficients of the power series defining $\\psi_{s_{0}}$ lie in $\\mathcal{O}_{K,v}$, the map $\\psi_{s_{0}}$ is defined on a residue disk $B_{p, \\overline{s_{0}}}$ of radius $|p|^{1\/[K_{v} : \\mathbb{Q}_{p}]}$, where $|\\cdot|$ is the absolute value on $\\mathbb{Q}_{p}$. Varying $s_{0}$ over the finitely many elements of $S(\\mathcal{O}_{K,N}\/\\mathfrak{p} \\mathcal{O}_{K,N})$, we obtain the desired cover.\n\nNow we wish to show that $\\dim S(\\mathcal{O}_{K,N})^{\\textrm{Zar}} \\leq d$. Recall that for each $s \\in S(\\mathcal{O}_{K,N})$, we have a set $\\mathcal{I}(s) \\subset S(\\mathcal{O}_{K,N})$ of points whose associated Galois representations have isomorphic semisimplifications. As there are finitely many possibilities for the semisimplification, it suffices to consider the sets $\\mathcal{S}(s) \\subset S(\\mathcal{O}_{K,N})$ defined by\n\\[ \\mathcal{S}(s) = \\{ s' \\in S(\\mathcal{O}_{K,N}) : \\mathcal{I}(s) = \\mathcal{I}(s') \\textrm{ and } s \\equiv s' \\textrm{ mod } \\mathfrak{p} \\} , \\]\nand show that $\\dim \\overline{\\mathcal{S}(s)}^{\\textrm{Zar}} \\leq d$. In particular, we can consider the Zariski closure of just those elements of $\\mathcal{S}(s)$ whose associated points in $S(\\mathcal{O}_{K,v})$ lie inside one of the neighbourhoods $B_{p,\\overline{s_{0}}}$ constructed above on which we have a local period map $\\psi_{s_{0}} : B_{p, \\overline{s_{0}}} \\to \\an{\\ch{L}_{K_{v}}}$. The hypothesis of the proposition tells us that the image under $\\psi_{s_{0}}$ of the points in $\\mathcal{S}(s)$ lie in a subvariety $O_{s} \\subset \\ch{L}_{\\mathbb{C}_{p}}$ satisfying $\\dim \\Delta_{d} \\geq \\dim O_{s}$; here we use the fact that the flat frame on $B_{p, \\overline{s_{0}}}$ is compatible with the Frobenius endomorphism (c.f. the discussion in \\cite[\\S3]{LV}). Base-changing to $\\mathbb{C}_{p}$ and applying \\autoref{padicaxschanlem} above, we find that $\\dim \\overline{\\psi_{s_{0}}^{-1}(O_{s})}^{\\textrm{Zar}} \\leq d$, hence the result.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Background}\\label{sec:introduction}\n\nAs a fundamental tool in discrete mathematics,  Lov\\'asz extension  has been deeply connected to  submodular analysis \\cite{Choquet54,Lovasz}, and has been applied in many  areas like  combinatorial  optimization, game  theory, matroid theory, stochastic processes, electrical networks, computer vision and machine learning \\cite{F05-book}. There are many generalizations, such as the disjoint-pair Lov\\'asz extension and the Lov\\'asz extension on distributive lattices \\cite{F05-book,Murota03book}.  Recent developments include quasi-Lov\\'asz extension on some algebraic structures and fuzzy mathematics, applications of Lov\\'asz extensions to  graph cut problems and computer science, as well as Lov\\'asz-softmax loss in deep learning.\n\n\n\n\n\\vspace{0.16cm}\n\nWe shall start by looking  at the original Lov\\'asz extension. For simplicity, we shall work throughout this paper with a finite and nonempty set $V=\\{1,\\cdots,n\\}$ and its power set $\\mathcal{P}(V)$. Also, we shall sometimes work on $\\mathcal{P}(V)^k:=\\{(A_1,\\cdots,A_k):A_i\\subset V,\\,i=1,\\cdots,k\\}$ and $\\mathcal{P}_k(V):=\\{(A_1,\\cdots,A_k)\\in\\ensuremath{\\mathcal{P}}(V)^k:A_i\\cap A_j=\\varnothing,\\,\\forall i\\ne j\\}$, as well as some restricted family $\\ensuremath{\\mathcal{A}}\\subset \\mathcal{P}(V)^k$. We denote the cardinality of a set $A$ by $\\#A$.\nGiven a function $f:\\mathcal{P}(V)\\to \\ensuremath{\\mathbb{R}}$, one identifies every $A\\in \\mathcal{P}(V){\\setminus\\{\\varnothing\\}}$ with its indicator vector $\\vec1_A\\in \\ensuremath{\\mathbb{R}}^V=\\ensuremath{\\mathbb{R}}^n$. The Lov\\'asz extension  extends the domain of $f$ to the whole Euclidean space\\footnote{Some other versions in the literature only extend the domain to the cube $[0,1]^V$ or the nonnegative orthant $\\ensuremath{\\mathbb{R}}_{\\ge0}^V$. In fact, many works on Boolean lattices  identify  $\\ensuremath{\\mathcal{P}}(V)$ with the discrete cube $\\{0,1\\}^n$.} $\\ensuremath{\\mathbb{R}}^V$. There are several equivalent\nexpressions:\n\n\\begin{itemize}\n\\item For $\\vec x =(x_1,\\dots ,x_n)\\in \\mathbb{R}^n$, let $\\sigma:V\\cup\\{0\\}\\to V\\cup\\{0\\}$ be a bijection such that $ x_{\\sigma(1)}\\le x_{\\sigma(2)} \\le \\cdots\\le x_{\\sigma(n)}$ and $\\sigma(0)=0$, where $x_0:=0$. The Lov\\'asz extension of $f$ is defined by\n\\begin{equation}\\label{eq:Lovasum}\nf^{L}(\\vec x)=\\sum_{i=0}^{n-1}(x_{\\sigma(i+1)}-x_{\\sigma(i)})f(V^{\\sigma(i)}(\\vec x)),\n\\end{equation}\nwhere   $V^0(\\vec x)=V$ and $V^{\\sigma(i)}(\\vec x):=\\{j\\in V: x_{j}> x_{\\sigma(i)}\\},\\;\\;\\;\\; i=1,\\cdots,n-1$. We can  write \\eqref{eq:Lovasum} in an integral form as\n\\begin{align}\\label{eq:Lovaintegral}\nf^{L}(\\vec x)&=\\int_{\\min\\limits_{1\\le i\\le n}x_i}^{\\max\\limits_{1\\le i\\le n}x_i} f(V^t(\\vec x))d t+f(V)\\min_{1\\le i\\le n}x_i\n\\end{align}\n\nwhere $V^t(\\vec x)=\\{i\\in V:  x_i>t\\}$. If we apply the   M\\\"obius transformation, this becomes\n \\begin{equation}\\label{eq:LovaMobuis} f^{L}(\\vec x)=\\sum\\limits_{A\\subset V}\\sum\\limits_{B\\subset A}(-1)^{\\#A-\\#B}f(B)\\bigwedge\\limits_{i\\in A} x_i,\\end{equation}\nwhere $\\bigwedge\\limits_{i\\in A} x_i$ is the minimum over $\\{x_i:i\\in A\\}$.\n\\end{itemize}\nIt is easy to see  that $f^L$ is positively one-homogeneous, PL (piecewise linear) and Lipschitz continuous \\cite{Lovasz,Bach13}. Also, $f^L(\\vec x+t\\vec 1_V)=f^L(\\vec x)+tf(V)$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$, $\\forall \\vec x\\in\\ensuremath{\\mathbb{R}}^V$, {and $f^L(\\vec1_A)=f(A)$ for any $A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$. The definition of $f^L$ does not involve the datum $f(\\emptyset)$, and thus by convention, it is natural to reset $f(\\emptyset)=0$ to match the equality  $f^L(\\vec0)=0$, unless stated otherwise. For convenience, we say that $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is a constant (resp., positive) function if $f$ is constant (resp., positive) on $\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$}. \nMoreover, a continuous function $F:\\ensuremath{\\mathbb{R}}^V\\to \\ensuremath{\\mathbb{R}}$ is the Lov\\'asz extension of some $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F(\\vec x+\\vec y)=F(\\vec x)+F(\\vec y)$ whenever $(x_i-x_j)(y_i-y_j)\\ge0$, $\\forall i,j\\in V$.\n\n\n\\vspace{0.13cm}\n\nIn this paper, we shall use the Lov\\'asz extension and its variants to study\nthe interplay between discrete and continuous aspects in  topics such as  convexity, optimization and spectral  theory.\n\n\n\n\n\\vspace{0.13cm}\n\n\\textbf{Submodular and convex functions}\n\n\nSubmodular function have emerged as a powerful concept in discrete optimization, see  Fujishige's monograph \\cite{F05-book} and {Bach's works \\cite{Bach13,Bach19}}.   We also refer the readers to some recent related works regarding \nsubmodular functions on  hypergraphs  \\cite{LM18,LM18-,LHM20}.  We recall\n  that a discrete function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$  defined on  an algebra $\\ensuremath{\\mathcal{A}}\\subset\\mathcal{P}(V)$ (i.e., $\\ensuremath{\\mathcal{A}}$ is  closed under  union and  intersection)  is submodular if $f(A)+f(B)\\ge f(A\\cup B)+f(A\\cap B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$. The Lov\\'asz extension turns a  submodular into a convex function, and we can hence minimize the former by minimizing the latter:\n\\begin{theorem}[Lov\\'asz \\cite{Lovasz}]\n\t$f:\\mathcal{P}(V)\\to\\mathbb{R}$ is submodular $\\Leftrightarrow$ $f^L$ is convex.\n\\end{theorem}\n\n\\begin{center}\n\t\\begin{tikzpicture}[node distance=6cm]\n\t\n\t\\node (convex) [startstop] {  Submodularity };\n\t\n\t\\node (submodular) [startstop, right of=convex, xshift=1.6cm]  { Convexity  };\n\t\n\t\\draw [arrow](convex) --node[anchor=south] { \\small Lov\\'asz extension } (submodular);\n\t\\draw [arrow](submodular) --node[anchor=north] {  } (convex);\n\t\\end{tikzpicture}\n\\end{center}\n\n\\begin{theorem}[Lov\\'asz \\cite{Lovasz}]If $f:\\mathcal{P}(V)\\to\\mathbb{R}$ is submodular with $f(\\varnothing)=0$, then\n\t$$\\min\\limits_{A\\subset V}f(A)=\\min\\limits_{\\vec x\\in [0,1]^V}f^L(\\vec x).$$\n\\end{theorem}\n\n\\begin{center}\n\t\\begin{tikzpicture}[node distance=6cm]\n\t\n\t\\node (convex) [process] {\n\t\tSubmodular optimization  };\n\t\n\t\\node (submodular) [process, right of=convex, xshift=1.6cm]  {\n\t\tConvex programming   };\n\t\n\t\\draw [arrow](convex) --node[anchor=south] { \\small Lov\\'asz extension } (submodular);\n\t\\draw [arrow](submodular) --node[anchor=north] { \\small} (convex);\n\t\\end{tikzpicture}\n\\end{center}\n\n\nThus, submodularity can be seen  as some kind of\n `discrete convexity', and that naturally lead to  many generalizations, such as bisubmodular, $k$-submodular, L-convex and M-convex, see \\cite{F05-book,Murota03book}.\nMoreover, the following classical result characterizes the class of all functions which can be expressed as Lov\\'asz extensions of  submodular functions. \n \\begin{theorem}\n  Theorem 7.40 in \\cite{Murota03book}]\n \tA one-homogeneous function $F:\\ensuremath{\\mathbb{R}}^V\\to \\ensuremath{\\mathbb{R}}$ is a Lov\\'asz extension of some submodular function if and only if $F(\\vec x+t\\vec 1_V)=F(\\vec x)+tF(\\vec 1_V)$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$, $\\forall \\vec x\\in\\ensuremath{\\mathbb{R}}^V$, and $F(\\vec x)+F(\\vec y)\\ge F(\\vec x\\vee \\vec y)+F(\\vec x\\wedge \\vec y)$, where the $i$-th components of $\\vec x\\vee \\vec y$ and $\\vec x\\wedge \\vec y$ are\n \t$(\\vec x\\vee \\vec y)_i=\\max\\{x_i,y_i\\}$ and $(\\vec x\\wedge \\vec y)_i=\\min\\{x_i,y_i\\}$.\n \\end{theorem}\n One may want to extend such a result to the  bisubmodular  or more general cases.\n In that direction, we shall obtain some   results such as Proposition \\ref{pro:bisubmodular-continuous} and Theorem \\ref{thm:submodular-L-equivalent} in Section \\ref{sec:SubmodularityConvexity}. It is also worth noting that Bach investigated an interesting generalization of submodular functions by a generalized Lov\\'asz extension \\cite{Bach19}. \n\n\nSo far,  research has mainly  focused  on `discrete convex' functions, leading to\n`Discrete Convex Analysis' \\cite{Murota98,Murota03book}, whereas the  discrete non-convex setting which is quite  popular in modern sciences has not yet received that much attention.\n\n\\vspace{0.16cm}\n\n\\textbf{Non-submodular cases}\n\nObviously, the non-convex case is so diverse and general that it cannot be directly studied by standard submodular tools.  Although some publications show several results on non-submodular (i.e., non-convex) minimization based on Lov\\'asz extension \\cite{HS11}, so far, these only work for special minimizations over the whole power set.  Here, we shall find applications for discrete optimization and nonlinear spectral graph theory by employing the multi-way Lov\\'asz extension on enlarged and restricted  domains. \n\n\n\\vspace{0.16cm}\n\nIn summary, we are going to initiate the study of diverse continuous extensions in non-submodular settings. This paper develops a systematic framework for many aspects around the topic. We  establish a universal discrete-to-continuous framework via multi-way extensions, by systematically utilizing integral representations. \nIn \\cite{JZ-prepare21}, we establish the links between discrete Morse theory and continuous Morse theory via the original Lov\\'asz extension. \nWe shall now discuss some connections with other various fields.\n\n\\vspace{0.19cm}\n\n\\textbf{Connections with  combinatorial optimization}\n\nBecause of the wide range of applications of discrete mathematics in computer science,  combinatorial optimization has been much studied from the mathematical perspective.\nIt is known that any combinatorial optimization can be equivalently expressed as a continuous optimization via convex (or concave) extension, but often,\nthere is the difficulty that one cannot  write down an equivalent continuous\nobject function in closed form.\nFor practical purposes, it would be very helpful if one could  transfer a\ncombinatorial optimization problem to an explicit and simple equivalent\ncontinuous optimization problem in closed form. Formally, in many concrete situations, it would be useful if one could get an identity of the  form\n\\begin{equation}\\label{eq:D-to-C-formal}\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{A}}\\cap \\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A_1,\\cdots,A_k)}{g(A_1,\\cdots,A_k)}=\\inf\\limits_{\\psi\\in {\\mathcal D}(\\ensuremath{\\mathcal{A}})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.\\end{equation}\nwhere $f,g:\\ensuremath{\\mathcal{A}}\\to [0,\\infty)$,  ${\\mathcal D}(\\ensuremath{\\mathcal{A}})$ is a feasible domain determined by $\\ensuremath{\\mathcal{A}}$ only, $\\mathrm{supp}(g)$ is the support of $g$, and $\\widetilde{f}$ and $\\widetilde{g}$ are suitable continuous extensions of $f$ and $g$.\n\nSo far, only situations where  $f:\\ensuremath{\\mathcal{P}}(V)\\to \\ensuremath{\\mathbb{R}}$ or $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ have been investigated systematically \\cite{HS11,CSZ18}, and what is lacking are situations with restrictions, that is,  incomplete data.\n\nAlso, to the best of our knowledge, the known results in the literature do not work for  combinatorial optimization directly on set-tuples.  But most of combinatorial optimization problems should be formalized in the form of set-tuples, and only a few can be represented in set form or disjoint-pair form. Whenever one can find an equivalent Lipschitz function for a combinatorial problem in the field of discrete optimization, this makes  useful tools available and leads to new connections.\nThat is, one wishes to establish   a {\\sl discrete-to-continuous transformation} like the operator $\\sim$ in \\eqref{eq:D-to-C-formal}.\nWe will show in Section \\ref{sec:CC-transfer} that the Lov\\'asz extension and its variants \nare suitable choices for such a transformation\n(see Theorems \\ref{thm:tilde-fg-equal}, \\ref{thm:tilde-H-f} and Proposition \\ref{pro:fraction-f\/g} for details).\n\n\\vspace{0.15cm}\n\n\nTo reach these goals, we need to systematically study various generalizations  of the  Lov\\'asz extension. More precisely, we shall work with the following  two different multi-way forms:\n\n\\begin{enumerate}[(1)]\n\n\\item Disjoint-pair version:\n    for a function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$, its disjoint-pair\n  Lov\\'asz extension is defined as\n\\begin{equation}\\label{eq:disjoint-pair-Lovasz-def-integral}\nf^{L}(\\vec x)=\\int_0^{\\|\\vec x\\|_\\infty} f(V_+^t(\\vec x),V_-^t(\\vec x))dt,\n\\end{equation}\nwhere $V_\\pm^t(\\vec x)=\\{i\\in V:\\pm x_i>t\\}$, $\\forall t\\ge0$. For $\\ensuremath{\\mathcal{A}}\\subset\\ensuremath{\\mathcal{P}}_2(V)$ and $f:\\ensuremath{\\mathcal{A}}\\to\\ensuremath{\\mathbb{R}}$, the feasible domain ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ of the disjoint-pair Lov\\'asz extension is $\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^V:(V_+^t(\\vec x),V_-^t(\\vec x))\\in\\ensuremath{\\mathcal{A}},\\forall t\\ge0\\}$. \n\nIt should be noted that the disjoint-pair Lov\\'asz extension introduced by\nQi \\cite{Qi88} has been systematically  investigated by Fujishige \\cite{Fujishige14,F05-book} and Murota \\cite{Murota03book} in the context of  discrete convex analysis (or the theory of submodular functions). The \nintegral formulation \\eqref{eq:disjoint-pair-Lovasz-def-integral}, however,   is \nmore convenient to obtain a closed formula of the equivalent continuous\noptimization problem for a combinatorial optimization problem. Moreover, the references and the present paper \nfocus on different aspects, with the exception of the submodularity theorem (i.e., $f$ is bisubmodular iff $f^L$ is convex).\n\n\\item $k$-way version:\nfor a function $f:\\mathcal{P}(V)^k\\to \\ensuremath{\\mathbb{R}}$, the {\\sl simple $k$-way Lov\\'asz extension} $f^L:\\ensuremath{\\mathbb{R}}^{kn}\\to \\ensuremath{\\mathbb{R}}$ is defined as\n\\begin{equation}\\label{eq:Lovasz-Form-1}\nf^L(\\vec x^1,\\cdots,\\vec x^k)=\\int_{\\min \\vec x}^{\\max \\vec x}f(V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))dt+ f(V,\\cdots,V)\\min\\vec x,\n\\end{equation}\nwhere $V^t(\\vec x^i)=\\{j\\in V:x^i_j>t\\}$, $\\min\\vec x=\\min\\limits_{i,j} x^i_j$ and $\\max\\vec x=\\max\\limits_{i,j} x^i_j$.  For $\\ensuremath{\\mathcal{A}}\\subset\\ensuremath{\\mathcal{P}}^k(V)$ and $f:\\ensuremath{\\mathcal{A}}\\to\\ensuremath{\\mathbb{R}}$, we take  ${\\mathcal D}_\\ensuremath{\\mathcal{A}}=\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^{kn}_{\\ge0}:(V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))\\in\\ensuremath{\\mathcal{A}},\\forall t\\in\\ensuremath{\\mathbb{R}}\\}$ as a feasible domain of the $k$-way Lov\\'asz extension $f^L$. \n\nBy the Lov\\'asz extension of  submodular functions on distributive\n  lattices \\cite{F05-book,Murota03book},  our $k$-way version\n  \\eqref{eq:Lovasz-Form-1} can be reduced to the classical version on\n  distributive lattices. Our main purposes and key results, however,  are\n  different from that approach. In fact, we mainly aim to deal with discrete\n  fractional programming by the $k$-way  Lov\\'asz extension, while those references concentrate on  submodularity and  convex optimization. \n\\end{enumerate}\n\n All these multi-way Lov\\'asz extensions satisfy the optimal identity Eq.~\\eqref{eq:D-to-C-formal}\n\n\\begin{introthm}[Theorem \\ref{thm:tilde-H-f} and Proposition \\ref{pro:fraction-f\/g}]\\label{thm:tilde-fg-equal}\nGiven two  functions $f,g:\\ensuremath{\\mathcal{A}}\\to [0,+\\infty)$, let $\\tilde{f}$ and $\\tilde{g}$ be two real\nfunctions on ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ satisfying $\\tilde{f}(\\vec1_{A_1,\\cdots,A_k})=f(A_1,\\cdots,A_k)$ and $\\tilde{g}(\\vec1_{A_1,\\cdots,A_k})=g(A_1,\\cdots,A_k)$. Then Eq.~\\eqref{eq:D-to-C-formal} holds if $\\tilde{f}$ and $\\tilde{g}$ further possess the properties (P1) or (P2) below. Correspondingly, if $\\tilde{f}$ and $\\tilde{g}$ fulfil (P1') or (P2),\nthere similarly holds\n$$\\max\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A_1,\\cdots,A_k)}{g(A_1,\\cdots,A_k)}=\\sup\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\widetilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.$$\nHere the optional additional conditions of $\\tilde{f}$ and $\\tilde{g}$ are:\n\n(P1) $\\tilde{f}\\ge f^L$ and $\\tilde{g}\\le g^L$.\\;\\;\\;  (P1') $\\tilde{f}\\le f^L$ and $\\tilde{g}\\ge g^L$.\n\n(P2)\n$\\tilde{f}=((f^\\alpha)^L)^{\\frac1\\alpha}$ and $\\tilde{g}=((g^\\alpha)^L)^{\\frac1\\alpha}$, where $\\alpha>0$.\n\nHere $f^L$ is either the original or the disjoint-pair or the $k$-way Lov\\'asz extension.\n\\end{introthm}\n\nTheorem \\ref{thm:tilde-fg-equal}\nshows that by the multi-way Lov\\'asz extension, the  combinatorial\noptimization in quotient form can be transformed to  fractional\nprogramming. And based on this fractional optimization, we propose an\neffective local convergence scheme, which relaxes the Dinkelbach-type\niterative scheme and mixes the inverse power method and the steepest descent\nmethod. Furthermore, many other  continuous iterations, such as\nKrasnoselski-Mann iteration, and the stochastic subgradient method, could be directly applied here. We refer the readers to \\cite{JostZhang} for another development on  equalities between  discrete and continuous optimization problems via various generalizations of Lov\\'asz extension. \n\n\nThe power of Theorem \\ref{thm:tilde-fg-equal} is embodied in many new examples and applications including  Cheeger-type problems, various\nisoperimetric constants and max $k$-cut problems (see Subsections \\ref{sec:max-k-cut}, \\ref{sec:boundary-graph-1-lap} and \\ref{sec:variantCheeger}).\nAnd moreover, we find that not only combinatorial optimization,\nbut also some combinatorial invariants like the independence number and the chromatic number, can\nbe transformed into a continuous representation by this scheme.\n\n\\begin{introthm}[Sections \\ref{sec:independent-number} and \\ref{sec:chromatic-number}]\n\\label{thm:graph-numbers}\nFor an unweighted and undirected simple graph $G=(V,E)$ with $\\#V=n$, its independence number can be represented as\n$$\\alpha(G)=\\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n\\setminus\\{\\vec 0\\}}\\frac{\\sum\\limits_{\\{i,j\\}\\in E}(|x_i-x_j|+|x_i+x_j|)- 2\\sum\\limits_{i\\in V}(\\deg_i-1)|x_i|}{2\\|\\vec x\\|_\\infty},$$\nwhere $\\deg_i=\\#\\{j\\in V:\\{j,i\\}\\in E\\}$, $i\\in V$, and its chromatic number is\n\\begin{equation*}\n\\gamma(G)= n^2-\\max\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^{n^2}\\setminus\\{\\vec 0\\}}\\sum\\limits_{k\\in V}\\frac{n\\sum\\limits_{\\{i,j\\}\\in E}(|x_{ik}-x_{jk}|+|x_{ik}+x_{jk}|)+2n\\|\\vec x^{k}\\|_{\\infty}-2n\\deg_k\\|\\vec x^{k}\\|_1- 2\\|\\vec x^{,k}\\|_{\\infty}}{2\\|\\vec x\\|_\\infty},\n\\end{equation*}\nwhere $\\vec x=(x_{ki})_{k,i\\in V}$, $\\vec x^{k}=(x_{k1},\\cdots,x_{kn})$ and $\\vec x^{,k}=(x_{1k},\\cdots,x_{nk})^T$.\nThe maximum matching number of $G$ can be expressed as\n$$\\max\\limits_{\\vec y\\in\\ensuremath{\\mathbb{R}}^E\\setminus\\{\\vec 0\\}}\\frac{\\|\\vec y\\|_1^2}{\\|\\vec y\\|_1^2-2\\sum_{e\\cap e'=\\varnothing}y_ey_{e'}}.$$\n\\end{introthm}\n\n \\vspace{0.15cm}\n\n\n\n\nThere are some equivalent continuous  reformulations of the maxcut problem and\nthe  independence number of a graph in the literature. However, a continuous  reformulation of the coloring number has not yet been proposed. The main reason seems to be the complexity of coloring a graph. Hence, it is very difficult to discover a continuous form of the coloring number by direct  observation. \n\n\\begin{introthm}[Theorem \\ref{thm:tilde-H-f}]\\label{thm:tilde-fg-equal-PQ}\nGiven   functions $f_1,\\cdots,f_n:\\ensuremath{\\mathcal{A}}\\to[0,+\\infty)$,and  $p$-homogeneous functions $P,Q:[0,+\\infty)^n\\to[0,+\\infty)$, we have\n$$\\max\\limits_{A\\in\\ensuremath{\\mathcal{A}}}\\frac{P(f_1(A),\\cdots,f_n(A))}{Q(f_1(A),\\cdots,f_n(A))}=\\sup\\limits_{x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{P(f_1^L(\\vec x),\\cdots,f_n^L(\\vec x))}{Q(f_1^L(\\vec x),\\cdots,f_n^L(\\vec x))}$$\nif $P^{\\frac1p}$ is \nsubadditive  and  $Q^{\\frac1p}$ is superadditive. One can replace  `max' by `min' if $P^{\\frac1p}$ is \nsuperadditive  and  $Q^{\\frac1p}$ is subadditive.\n\\end{introthm}\n\nTheorems \\ref{thm:tilde-fg-equal},  \\ref{thm:tilde-fg-equal-PQ} and \\ref{thm:tilde-H-f}  can be seen as natural and nontrivial generalizations of the related original works by   Hein's group \\cite{HS11}.\n\n\\vspace{0.1cm}\n\n\\textbf{Connections with  spectral graph theory}\n\nSpectral graph theory aims to derive properties of a (hyper-)graph from its eigenvalues and eigenvectors. Going beyond the linear case, nonlinear spectral graph theory is developed in terms of discrete geometric analysis and difference equations on (hyper-)graphs.\nEvery discrete eigenvalue problem can be formulated as a variational problem  for an\nobjective functional, a Rayleigh-type quotient.\nIn some cases, this\nfunctional is natural and easy to obtain,\nsince one may compare the discrete version with its original continuous analog in geometric analysis. However, in other\nsituations, there is no such analog. Fortunately, we find a unified framework based on multi-way Lov\\'asz extension to produce appropriate objective functions from a combinatorial problem (see Sections\n\\ref{sec:CC-transfer} and \\ref{sec:eigenvalue}).\n\nMore precisely, for a combinatorial problem with a  discrete objective function of the form $\\frac{f(A)}{g(A)}$, we might obtain some correspondences by studying the  \nset-valued eigenvalue problem\n$$\n \\nabla f^L (\\vec x)\\bigcap \\lambda\\nabla  g^L (\\vec x) \\ne\\varnothing\n$$\nwhich is simply called the eigenvalue problem of the function pair $(f^L,g^L)$.  \nHereafter we use $\\nabla$ to denote the (Clarke) sub-gradient operator acting on Lipschitz functions.\n\\begin{center}\n\\begin{tikzpicture}[node distance=6cm]\n\n\\node (graph) [startstop] {  combinatorial  quantities };\n\n\\node (spectrum) [startstop, right of=convex, xshift=2.6cm]  { eigenvalues and eigenvectors };\n\n\\draw [arrow](graph) --node[anchor=south] { \\small Spectral graph theory} (spectrum);\n\\draw [arrow](spectrum) --node[anchor=north] { } (graph);\n\\end{tikzpicture}\n\\end{center}\n\n\n\n\n\\vspace{0.16cm}\nWe shall consider the following three concepts:\n\\begin{itemize}\n\\item {\\sl Eigenvectors and eigenvalues}:\\;\\;\nThe set-valued eigenvalue problems above are usually written as $\\vec0\\in \\nabla f^L (\\vec x)-\\lambda\\nabla  g^L (\\vec x)$ by using the Minkowski summation of convex sets. We call $\\lambda$ an eigenvalue and $\\vec x$ an eigenvector associated to $\\lambda$.\n\n\\item {\\sl Critical points and critical values}:\\;\\;\nThe set of critical points\n$\\left\\{\\vec x\\left|0\\in\\nabla \\frac{f^L(\\vec x)}{g^L(\\vec x)}\\right.\\right\\}$\n and the corresponding critical values.\n\n\n\\item {\\sl Minimax critical values} (i.e., {\\sl variational eigenvalues in Rayleigh quotient form}):\\;\\;\nThe Lusternik-Schnirelman theory tells us that the min-max\nvalues\n\\begin{equation}\\label{eq:def-c_km}\n\\lambda_{m}=\\inf_{\\Psi\\in \\Gamma_m}\\sup\\limits_{\\vec x \\in \\Psi}\\frac{f^L(\\vec x)}{g^L(\\vec x)}\n\\end{equation}\nare critical values of $f^L(\\cdot)\/g^L(\\cdot)$. Here $\\Gamma_m$ is a class of certain topological objects at level $m$, e.g., the family of subsets with L-S category (or Krasnoselskii's $\\mathbb{Z}_2$-genus) not smaller than $m$.\n\\end{itemize}\n\n  There are the following relations between these three classes:\n$$\\{\\text{Eigenvalues in Rayleigh quotient}\\}\\subset\\{\\text{Critical values}\\}\\subset\\{\\text{Eigenvalues}\\}.$$\nFor linear spectral theory, the above three classes  coincide. However, for the non-smooth spectral theory derived by Lov\\'asz extension, we only have the inclusion relations.\n\n\nWe have the following result on the eigenvalue problem for the  disjoint-pair Lov\\'asz extension, while  for the results on the original Lov\\'asz extension, we refer to  Section \\ref{sec:eigenvalue} for details. \n\\begin{introthm}\n\\label{introthm:eigenvalue}\nGiven $f,g:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$, then every eigenvalue of $(f^L,g^L)$ has an eigenvector of the form $\\vec 1_A-\\vec1_B$. Moreover, we have the following claims:\n\\begin{itemize}\n\\item If $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ and $2g(A,B)=g(A,V\\setminus A)+g(V\\setminus B,B)$  for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$, then every eigenvalue of $(f^L,g^L)$ has an eigenvector of the form  $\\vec 1_A-\\vec1_{V\\setminus A}$.\n\\item If $g=\\mathrm{Const}$,  then for any $A\\subset V$, $\\vec 1_A-\\vec1_{V\\setminus A}$ is an eigenvector.\n\\item If $f(A,B)=\\hat{f}(A)+\\hat{f}(B)$ and $g(A,B)=\\hat{g}(A)+\\hat{g}(B)$  for some  symmetric function  $\\hat{f}:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ (i.e., $\\hat{f}(A)=\\hat{f}(V\\setminus A)$, $\\forall A$) and non-decreasing submodular function  $\\hat{g}:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}_+$,  then the second \neigenvalue $\\lambda_2$ of $(f^L,g^L)$ equals \n$$\\min\\limits_{\\vec x\\bot\\vec 1}\\frac{f^L(\\vec x)}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g^L(\\vec x-t\\vec 1)}=\\min\\limits_{A\\in\\mathcal{P}(V)\\setminus\\{\\varnothing,V\\}}\\frac{\\hat{f}(A)}{\\min\\{\\hat{g}(A),\\hat{g}(V\\setminus A)\\}}=\\min\\limits_{(A,B)\\in\\mathcal{P}_2(V)\\setminus\\{(\\varnothing,\\varnothing )\\}}\\max\\{\\frac{\\hat{f}(A)}{\\hat{g}(A)},\\frac{\\hat{f}(B)}{\\hat{g}(B)}\\}.$$\n\\end{itemize}\n\n\n\\end{introthm}\n\nThis generalizes  recent results on the\ngraph 1-Laplacian and Cheeger's constant \\cite{HeinBuhler2010,TVhyper-13,Chang16,CSZ15,CSZ17}. { And as a new application, we show  that the   min-cut problem and the   max-cut problem are equivalent to solving  the \nsmallest  nontrivial (i.e., the second) eigenvalue and the largest eigenvalue of a certain nonlinear eigenvalue problem (see Theorem \\ref{thm:mincut-maxcut-eigen}).\n}\n\n\\vspace{0.16cm}\n\n\\textbf{Applications to   frustration in signed network }\n\nAs a key measure for analysing signed networks, the frustration index on a signed graph  quantifies  how far a signature  is\nfrom being balanced (see Section \\ref{sec:frustration}). Computing the frustration index  is NP-hard, and few algorithms have been proposed  \\cite{ArefWilson19,ArefMasonWilson20}. \n\n\n\nConsidering a signed graph $(V,E_+\\cup E_-)$ with  $E_+$ (resp. $E_-$)\nthe set of positive (resp. negative)  edges,   based on the  disjoint-pair Lov\\'asz  extension, we obtain an equivalent continuous optimization of the frustration index (or the line index of balance \\cite{Harary59}):\n$$\\#E_-+\\min\\limits_{ x\\ne0}\\frac{\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|-\\sum_{\\{i,j\\}\\in E_-}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}.$$\nThis new reformulation can be computed via typical algorithms in continuous optimization. \n\nAlso, we propose the  eigenvalue problem\n\\begin{equation}\\label{eq:frustration-eigen}\n\\nabla\\left(\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|+\\sum_{\\{i,j\\}\\in E_-}|x_i+x_j|\\right)\\bigcap\\lambda \\nabla\\|\\vec x\\|_\\infty \\ne\\varnothing    \n\\end{equation}\nand we show an iterative scheme for searching the frustration index based on the smallest eigenvalue of the nonlinear eigenvalue problem \\eqref{eq:frustration-eigen}. See Section \\ref{sec:frustration} for details and  more results.  \n\n\\vspace{0.16cm}\n\n Since the  transformation of a combinatorial optimization to a\ncontinuous optimization or a nonsmooth  eigenvalue problem usually leads\n  to a quotient,\nthe task for  fractional programming then becomes to compute  an optimal value or an eigenvector. In Section \\ref{sec:algo}, we present a general  algorithm which is available to compute the resulting continuous reformulations arising in Theorems \\ref{thm:tilde-fg-equal}, \\ref{thm:graph-numbers}, \\ref{thm:tilde-fg-equal-PQ} and \\ref{introthm:eigenvalue}. \n\n In summary, we present a systematic study for  constructing  nonlinear\neigenvalue problems and equivalent continuous  reformulations for\ncombinatorial quantities, which capture the key properties of the original   combinatorial problems. This is helpful to increase understanding of certain combinatorial problems by the corresponding  eigenvalue problems and the  equivalent continuous  reformulations. The following picture summarizes the relations between the various concepts developed and studied in this paper.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[node distance=4.5cm]\n\n\\node (CQ) [startstop] {\\begin{tabular}{l}\nCombinatorial\\\\ Quantities\n\\end{tabular}};\n\n\\node (DO) [process, right of=CQ, xshift=-1cm]  { \\begin{tabular}{l}\n Discrete\\\\\n Optimization\n\\end{tabular}};\n\n\\node (CO) [startstop, right of=DO, yshift=0cm, xshift=2.6cm] {\\begin{tabular}{l}\n Continuous\\\\\n Optimization\n\\end{tabular} };\n\\node (DM) [io1, below of=CQ, xshift=0.1cm,yshift=2cm] {\\begin{tabular}{l}\n Discrete Morse theory\n\\end{tabular}\n };\n\\node (F\/G) [process, right of=DM, xshift=2.2cm] {\\begin{tabular}{l}\n(topological) Morse theory    \\\\\n  (metric) critical point theory\n\\end{tabular}\n};\n\\node (FG) [startstop, below of=F\/G, yshift=2.6cm,xshift=3.3cm] {\n\\begin{tabular}{r}\n(Nonlinear) Spectral theory\\\\\n\\end{tabular}};\n\n\\node (algorithm) [io1, right of=F\/G, xshift=2cm] {\\begin{tabular}{l}\n Continuous\\\\\n Programming\\\\\n\\& Algorithm\n\\end{tabular}};\n\\node (submodular) [io2, above of=DO, yshift=-2.6cm] {Submodularity};\n\\node (convexity) [io2, right of=submodular, xshift=5.2cm] {Convexity};\n\\draw [arrow](CQ) --node[anchor=south] { } (DO);\n\\draw [arrow](DO) --node[anchor=south] { } (CQ);\n\\draw [arrow](DO) --node[anchor=south]{Discrete-to-Continuous }node[anchor=north] {extension} (CO);\n\\draw [arrow](DM) -- node[anchor=south] {\\cite{JZ-prepare21}}node[anchor=north] { Part I } (F\/G);\n\\draw [arrow](FG) -- node[anchor=south] {} (algorithm);\n\\draw [arrow](F\/G) -- node[anchor=south] {  } (FG);\n\\draw [arrow](CO) -- node[anchor=south] {  } (algorithm);\n\\draw [arrow](CO) -- node[anchor=south] {  } (F\/G);\n\\draw [arrow](CO) -- node[anchor=south] {  } (FG);\n\\draw [arrow](submodular) -- node[anchor=south] {  Lov\\'asz extension } (convexity);\n\\draw [arrow](submodular) -- node[anchor=south] { } (DO);\n\\draw [arrow](convexity) -- node[anchor=south] { } (submodular);\n\\draw [arrow](convexity) -- node[anchor=south] { } (algorithm);\n\\end{tikzpicture}\n\\caption{\\label{fig:flowchart} The relationship between the aspects\nstudied in our work.}\n\\end{figure}\n\n\n\n\\begin{notification}\nSince this paper contains  many interacting parts and relevant results, some notions and concepts may have slightly distinct\nmeanings in different sections, but this will be stated at the beginning of each section.\n\\end{notification}\n\n\n\n\n\\section{A preliminary:  Lov\\'asz extension and submodular functions}\n\\label{sec:Lovasz}\n\n\n\nWhile most of the results on submodularity are known in the field of discrete\nconvex analysis, we present some details in a simple manner, which should be helpful to understand our main results in Section \\ref{sec:main}.\n\n\nWe first formalize some important results about the original Lov\\'asz extension.\n\n\n\\begin{defn}\nTwo vectors $\\vec x$ and $\\vec y$ are {\\sl comonotonic} if $(x_i-x_j)(y_i-y_j)\\ge0$, $\\forall i,j\\in \\{1,2,\\cdots,n\\}$.\n\nA function $F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ is  {\\sl comonotonic additive} if $F(\\vec x+\\vec y)=F(\\vec x)+F(\\vec y)$ for any comonotonic pair $\\vec x$ and $\\vec y$.\n\\end{defn}\nThe following proposition shows that a function is comonotonic additive if and only if it can be expressed as the Lov\\'asz extension of some function.\n\\begin{pro}\n\\label{pro:comonotonic-additivity}\n$F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ is the Lov\\'asz extension $F=f^L$ of some function $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F$ is comonotonic additive.\n\\end{pro}\n\nRecall the following known results:\n\n\\begin{theorem}[Lov\\'asz]\\label{thm:Lovasz}\nThe following conditions are equivalent: (1) $f$ is submodular; (2) $f^L$ is convex; (3) $f^L$ is submodular.\n\\end{theorem}\n\n\\begin{remark}\nThe fact that $f$ is submodular if and only if  $f^L$ is submodular is provided by Propositions 7.38 and 7.39 in \\cite{Murota03book}. We shall give a detailed proof for a generalized version of Theorem \\ref{thm:Lovasz} (see Theorem \\ref{thm:submodular-L-equivalent}).   \n\\end{remark}\n\n\n\\begin{theorem}\nMurota]\\label{thm:Chateauneuf-Cornet}\n$F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ is the Lov\\'asz extension $F=f^L$ of some submodular $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F$ is positively one-homogeneous, submodular and $F(\\vec x+t\\vec 1)=F(\\vec x)+tF(\\vec 1)$.\n\\end{theorem}\n\n\\begin{remark}\nTheorem \\ref{thm:Chateauneuf-Cornet} was   originally  proved by establishing a  one-to-one correspondence between positively  \nhomogeneous L-convex functions and submodular  functions  (see \n  Theorem 7.40 in Murota's book  \\cite{Murota03book}). An alternative proof is\n  given in   \\cite{CC18}.\n\\end{remark}\n\n\nWe shall  establish these results for  the disjoint-pair version and the $k$-way version of the Lov\\'asz extension.\n\n\\subsection{Disjoint-pair and $k$-way Lov\\'asz extensions}\n\nSince it is natural to set $f(\\varnothing,\\varnothing)=0$, one may write \\eqref{eq:disjoint-pair-Lovasz-def-integral} as\n\\begin{equation}\\label{eq:disjoint-pair-Lovasz-def-integral2}\nf^{L}(\\vec x)=\\int_0^{\\infty} f(V_+^t(\\vec x),V_-^t(\\vec x))dt,\n\\end{equation}\n\\begin{equation}\\label{eq:disjoint-pair-Lovasz-def}\nf^{L}(\\vec x)=\\sum_{i=0}^{n-1}(|x_{\\sigma(i+1)}|-|x_{\\sigma(i)}|)f(V_{\\sigma(i)}^+(\\vec x),V_{\\sigma(i)}^-(\\vec x)),\n\\end{equation}\nwhere $\\sigma:V\\cup\\{0\\}\\to V\\cup\\{0\\}$ is a bijection such that $|x_{\\sigma(1)}|\\le |x_{\\sigma(2)}| \\le \\cdots\\le |x_{\\sigma(n)}|$ and $\\sigma(0)=0$, where $x_0:=0$, and\n$$V_{\\sigma(i)}^\\pm(\\vec x):=\\{j\\in V:\\pm x_j> |x_{\\sigma(i)}|\\},\\;\\;\\;\\; i=0,1,\\cdots,n-1.$$\nWe regard $\\ensuremath{\\mathcal{P}}_2(V)=3^V$ as $\\{-1,0,1\\}^n$ by identifying the disjoint pair $(A,B)$ with the ternary (indicator) vector $\\vec 1_A-\\vec1_B$.\n\nOne may compare the original and the disjoint-pair Lov\\'asz extensions by writing \\eqref{eq:disjoint-pair-Lovasz-def-integral} as\n\\begin{equation}\\label{eq:disjoint-pair-form}\n\\int_{\\min_i |x_i|}^{\\max_i |x_i|} f(V_+^t(\\vec x),V_-^t(\\vec x))dt+\\min_i |x_i| f(V_+,V_-),\n\\end{equation}\nwhere $V_\\pm=\\{i\\in V:\\pm x_i>0\\}$. Note that \\eqref{eq:disjoint-pair-form} is very similar to \\eqref{eq:Lovaintegral}.\n\n\\begin{defn}\nGiven $V_i=\\{1,\\cdots,n_i\\}$, $i=1,\\cdots,k$, and a function $f:\\mathcal{P}(V_1)\\times \\cdots\\times \\mathcal{P}(V_k)\\to \\ensuremath{\\mathbb{R}}$, the  $k$-way Lov\\'asz extension $f^L: \\ensuremath{\\mathbb{R}}^{V_1}\\times\\cdots\\times \\ensuremath{\\mathbb{R}}^{V_k}\\to \\ensuremath{\\mathbb{R}}$ can be written as\n\\begin{align*}\nf^L(\\vec x^1,\\cdots,\\vec x^k)&=\\int_{\\min \\vec x}^{\\max \\vec x}f(V^t_1(\\vec x^1),\\cdots,V^t_k(\\vec x^k))dt+ f(V_1,\\cdots,V_k)\\min\\vec x\\\\&\n=\\int_{-\\infty}^0(f(V^t_1(\\vec x^1),\\cdots,V^t_k(\\vec x^k))-f(V_1,\\cdots,V_k))dt + \\int_0^{+\\infty}f(V^t_1(\\vec x^1),\\cdots,V^t_k(\\vec x^k)) d t\n\\end{align*}\nwhere $V^t_i(\\vec x^i)=\\{j\\in V_i:x^i_j>t\\}$, $\\min\\vec x=\\min\\limits_{i,j} x^i_j$ and $\\max\\vec x=\\max\\limits_{i,j} x^i_j$.\n\\end{defn}\n\n\\begin{defn}[$k$-way analog for disjoint-pair Lov\\'asz extension]\n\n   Given  $V_i=\\{1,\\cdots,n_i\\}$, $i=1,\\cdots,k$, and a function $f:\\mathcal{P}_2(V_1)\\times \\cdots\\times \\mathcal{P}_2(V_k)\\to \\ensuremath{\\mathbb{R}}$, define $f^L: \\ensuremath{\\mathbb{R}}^{V_1}\\times\\cdots\\times \\ensuremath{\\mathbb{R}}^{V_k}\\to \\ensuremath{\\mathbb{R}}$ by\n   $$f^L(\\vec x^1,\\cdots,\\vec x^k)=\n   \\int_0^{\\|\\vec x\\|_\\infty} f(V_{1,t}^+(\\vec x^1),V_{1,t}^-(\\vec x^1),\\cdots,V_{k,t}^+(\\vec x^k),V_{k,t}^-(\\vec x^k))dt\n   $$\n   where $V_{i,t}^\\pm(\\vec x^i)=\\{j\\in V_i:\\pm x^i_j>t\\}$, $\\|\\vec x\\|_\\infty=\\max\\limits_{i=1,\\cdots,k} \\|\\vec x^i\\|_\\infty$. We can replace $\\|\\vec x\\|_\\infty$ by $+\\infty$ if we set $f(\\varnothing,\\cdots,\\varnothing)=0$.\n\\end{defn}\n\\vspace{0.19cm}\n\nSome basic properties of the multi-way Lov\\'asz extension are shown below.\n\\begin{pro}\\label{pro:multi-way-property}\nFor the multi-way Lov\\'asz extension $f^L(\\vec x)$, we have\n\\begin{enumerate}[(a)]\n\\item  $f^L(\\cdot)$ is positively one-homogeneous, piecewise linear, and Lipschitz continuous.\n\\item $(\\lambda f)^L=\\lambda f^L$, $\\forall\\lambda\\in\\ensuremath{\\mathbb{R}}$.\n\n\\end{enumerate}\n\\end{pro}\n\n\\begin{pro}\\label{pro:setpair-property}\nFor the disjoint-pair Lov\\'asz extension $f^L(\\vec x)$, we have\n\\begin{enumerate}[(a)]\n\\item $f^L$ is Lipschitz continuous, and $|f^L(x)-f^L(y)|\\le 2\\max\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)}f(A,B) \\|x-y\\|_1$, $\\forall x,y\\in \\mathbb{R}^n$. Also,  $|f^L(x)-f^L(y)|\\le 2\\sum\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)}f(A,B)  \\|x-y\\|_\\infty$, $\\forall x,y\\in \\mathbb{R}^n$.\n\\item $f^L(-\\vec x)=\\pm f^L(\\vec x)$, $\\forall \\vec x\\in\\ensuremath{\\mathbb{R}}^V$ if and only if $f(A,B)=\\pm f(B,A)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$.\n\\item\\label{pro:pro-c} $f^L(\\vec x+\\vec y)=f^L(\\vec x)+f^L(\\vec y)$ whenever $V_\\pm(\\vec y)\\subset V_\\pm(\\widetilde{\\vec x})$, where $\\widetilde{\\vec x}$ has components $\\widetilde{ x}_i=\\begin{cases}x_i,&\\text{ if }|x_i|=\\|\\vec x\\|_\\infty,\\\\ 0,&\\text{ otherwise}.\\end{cases}$\n\\end{enumerate}\n\\end{pro}\n\\begin{proof} (a) and (b) are actually known results and their proofs are elementary.\n(c) can be derived from the definition \\eqref{eq:disjoint-pair-Lovasz-def}.\n\\end{proof}\n\n\\begin{defn\n\\label{def:associate-piece}\nTwo vectors $\\vec x$ and $\\vec y$ are said to be absolutely comonotonic \nif $x_iy_i\\ge0$, $\\forall i$, and $(|x_i|-|x_j|)(|y_i|-|y_j|)\\ge0$, $\\forall i,j$\n\\end{defn}\n\n\\begin{pro}\\label{pro:setpair-character}\nA continuous function $F$ is a disjoint-pair Lov\\'asz extension of some function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$, if and only if\n$F(\\vec x)+F(\\vec y)=F(\\vec x+\\vec  y)$ whenever $\\vec x$ and $\\vec y$ are absolutely comonotonic.\n\\end{pro}\n\n\\begin{proof} By the definition of the disjoint-pair Lov\\'asz extension (see \\eqref{eq:disjoint-pair-Lovasz-def}),\nwe know that $F$ is a disjoint-pair Lov\\'asz extension of some function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if\n$\\lambda F(\\vec x)+(1-\\lambda)F(\\vec y)=F(\\lambda\\vec x+(1-\\lambda)\\vec  y)$ for all absolutely comonotonic\nvectors $\\vec x$ and $\\vec y$, $\\forall \\lambda\\in[0,1]$. Therefore, we only need to prove the sufficiency part.\n\nFor $\\vec x\\in\\ensuremath{\\mathbb{R}}^V$, since $s\\vec x$ and $t\\vec x$ with $s,t\\ge 0$ are absolutely comonotonic,\n $F(s\\vec x)+F(t\\vec x)=F((s+t)\\vec x)$, which yields a Cauchy equation on the half-line. Thus the continuity assumption implies the linearity of $F$ on the ray $\\ensuremath{\\mathbb{R}}^+\\vec x$, which implies the property $F(t\\vec x)=tF(\\vec x)$, $\\forall t\\ge 0$, and hence $\\lambda F(\\vec x)+(1-\\lambda)F(\\vec y)=F(\\lambda\\vec x+(1-\\lambda)\\vec  y)$ for any absolutely comonotonic\n vectors $\\vec x$ and $\\vec y$, $\\forall \\lambda\\in[0,1]$. This completes the proof.\n\\end{proof}\n\nFor relations between the original and the disjoint-pair Lov\\'asz extensions, we further have\n\\begin{pro}\\label{pro:setpair-original}\n For $h:\\ensuremath{\\mathcal{P}}(V)\\to [0,+\\infty)$ with $h(\\varnothing)=0$, and $f:\\ensuremath{\\mathcal{P}}_2(V)\\to [0,+\\infty)$ with $f(\\varnothing,\\varnothing)=0$ \\footnote{In fact, if $h(\\varnothing)\\ne 0$ or $f(\\varnothing,\\varnothing)\\ne0$, one may change the value and it does not affect the related  Lov\\'asz extension.}, we have:\n\\begin{enumerate}[(a)]\n\\item If $f(A,B)=h(A)+h(V\\setminus B)-h(V)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then   $f^L=h^L$.\n\\item  If $f(A,B)=h(A)+h(B)$ and $h(A)=h(V\\setminus A)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then $f^L=h^L$.\n\\item  If $f(A,B)=h(A)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then $f^L(\\vec x)=h^L(\\vec x)$, $\\forall \\vec x\\in[0,\\infty)^V$.\n\\item If $f(A,B)=h(A\\cup B)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then  $f^L(\\vec x)=h^L(\\vec x^++\\vec x^-)$.\n\\item  If $f(A,B)=h(A)\\pm h(B)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then $f^L(\\vec x)=h^L(\\vec x^+)\\pm h^L(\\vec x^-)$.\n\\end{enumerate}\nHere $\\vec x^\\pm:=(\\pm \\vec x)\\vee \\vec0$.\n\\end{pro}\n\n\\begin{proof}\n\\begin{enumerate}[(a)]\n\\item Note that\n\\begin{align*}\nf^L(\\vec x)&=\\int_0^{\\|\\vec x\\|_\\infty}f(V^+_t(\\vec x),V^-_t(\\vec x))dt\n=\\int_0^{\\|\\vec x\\|_\\infty}(h(V_t(\\vec x))+h(V_{-t}(\\vec x))-h(V))dt\\\\\n&=\\int_{-\\|\\vec x\\|_\\infty}^{\\|\\vec x\\|_\\infty}h(V_t(\\vec x))dt- \\|\\vec x\\|_\\infty h(V)\n=\\int_{x_{\\sigma(1)}}^{x_{\\sigma(n)}}h(V_t(\\vec x))dt +x_{\\sigma(1)} h(V) = h^L(\\vec x),\n\\end{align*}\nwhere we use $\\|x\\|_\\infty=\\max\\{-x_{\\sigma(1)},x_{\\sigma(n)}\\}$ and $h(\\varnothing)=0$.\n\\item This is a direct consequence of (a) since  $h(V)=h(\\varnothing)=0$ and $h(B)=h(V\\setminus B)$.\n\\item For any $\\vec x\\in \\ensuremath{\\mathbb{R}}^V$ with $x_i\\ge0$, we note that $f^L(\\vec x)=\\int_0^{\\|\\vec x\\|_\\infty}h(V^+_t(\\vec x))dt=\\int_0^{\\max x_i}h(V_t(\\vec x))dt=\\int_{\\min x_i}^{\\max x_i}h(V_t(\\vec x))dt+\\min x_i h(V)=h^L(\\vec x)$.\n\\item\nSimilar to (c), one can check that $f^L(\\vec x)=h^L(\\vec x^++\\vec x^-)$.\n\\item It is straightforward.\n\\end{enumerate}\n\\end{proof}\n\nIn the sequel, we will not distinguish the original and the disjoint-pair Lov\\'asz extensions, since the reader can infer it\nfrom the domains ($\\ensuremath{\\mathcal{P}}(V)$ or $\\ensuremath{\\mathcal{P}}_2(V)$). Sometime we work on $\\ensuremath{\\mathcal{P}}(V)$ only, and in this situation, the disjoint-pair Lov\\'asz extension acts on the redefined  $f(A,B)=h(A\\cup B)$ as Proposition \\ref{pro:setpair-original} states.\n\nThe next result is useful for the application on graph coloring.\n\n\\begin{pro}\\label{pro:separable-summation}\nFor the  simple $k$-way Lov\\'asz extension of $f:\\ensuremath{\\mathcal{P}}(V_1)\\times\\cdots\\times\\ensuremath{\\mathcal{P}}(V_k)\\to \\ensuremath{\\mathbb{R}}$ with the separable summation form $f(A_1,\\cdots,A_k):=\\sum_{i=1}^kf_i(A_i)$, $\\forall (A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}(V)^k$, we have $f^L(\\vec x^1,\\cdots,\\vec x^k)=\\sum_{i=1}^kf_i^L(\\vec x^i)$, $\\forall (\\vec x^1,\\cdots,\\vec x^k)$.\n\nFor $f:\\ensuremath{\\mathcal{P}}_2(V_1)\\times\\cdots \\times\\ensuremath{\\mathcal{P}}_2(V_k)\\to \\ensuremath{\\mathbb{R}}$ with the form $f(A_1,B_1\\cdots,A_k,B_k):=\\sum_{i=1}^kf_i(A_i,B_i)$, $\\forall (A_1,B_1,\\cdots,A_k,B_k)\\in \\ensuremath{\\mathcal{P}}_2(V_1)\\times\\cdots \\times \\ensuremath{\\mathcal{P}}_2(V_k)$, there similarly holds $f^L(\\vec x^1,\\cdots,\\vec x^k)=\\sum_{i=1}^kf_i^L(\\vec x^i)$.\n\\end{pro}\n\n\\subsection{Submodularity and Convexity}\\label{sec:SubmodularityConvexity}\nIn this subsection, we give new analogs of Theorems \\ref{thm:Lovasz} and \\ref{thm:Chateauneuf-Cornet} for the disjoint-pair Lov\\'asz extension and the $k$-way Lov\\'asz extension. The major difference to existing results in the literature is that we work with the restricted or the enlarged domain of a function.\n\nLet's first recall the standard concepts of submodularity:\n\\begin{enumerate}[({S}1)]\n\\item A discrete function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$   is submodular if $f(A)+f(B)\\ge f(A\\cup B)+f(A\\cap B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$, where $\\ensuremath{\\mathcal{A}}\\subset\\mathcal{P}(V)$ is an algebra  (i.e., $\\ensuremath{\\mathcal{A}}$ is  closed under  union and intersection).\n\\item A continuous function $F:\\ensuremath{\\mathbb{R}}^n\\to \\ensuremath{\\mathbb{R}}$ is submodular if\n $F(\\vec x)+F(\\vec y)\\ge F(\\vec x\\vee \\vec y)+F(\\vec x\\wedge \\vec y)$, where $(\\vec x\\vee \\vec y)_i=\\max\\{x_i,y_i\\}$ and $(\\vec x\\wedge \\vec y)_i=\\min\\{x_i,y_i\\}$, $i=1,\\cdots,n$. For a sublattice ${\\mathcal D}\\subset\\ensuremath{\\mathbb{R}}^n$ that  is closed under $\\vee$ and $\\wedge$, one can define  submodularity in the same way.\n\\end{enumerate}\n\n\\begin{notification} The discussion about algebras of sets can be reduced to lattices.\nClassical submodular functions on a sublattice of the Boolean lattice\n$\\{0,1\\}^n$ and their continuous versions on $\\ensuremath{\\mathbb{R}}^n$ are presented in (S1) and\n(S2), respectively. Bisubmodular functions on a graded sub-poset (partially\nordered set)  of   $\\{-1,0,1\\}^n$ are defined in \\eqref{eq:2-submodular} below.\n\\end{notification}\n\n    Now, we\n    recall the concept of bisubmodularity and introduce its continuous version.\n    \\begin{enumerate}\n\\item[(BS1)] A discrete function $f:\\mathcal{P}_2(V)\\to \\ensuremath{\\mathbb{R}}$   is bisubmodular if $\\forall\\,(A,B),(C,D)\\in \\ensuremath{\\mathcal{P}}_2(V)$\n\\begin{equation}\\label{eq:2-submodular}\nf(A,B)+f(C,D)\\ge f((A\\cup C)\\setminus (B\\cup D),(B\\cup D)\\setminus(A\\cup C))+f(A\\cap C,B\\cap D).\n\\end{equation}\nOne can denote $A\\vee B=((A_1\\cup B_1)\\setminus (A_2\\cup B_2),(A_2\\cup B_2)\\setminus (A_1\\cup B_1))$ and $A\\wedge B=(A_1\\cap B_1,A_2\\cap B_2)$, where $A=(A_1,A_2)$, $B=(B_1,B_2)$. For a subset  $\\ensuremath{\\mathcal{A}}\\subset \\mathcal{P}_2(V)$ that is closed under $\\vee$ and $\\wedge$, the bisubmodularity of $f:\\ensuremath{\\mathcal{A}}\\to\\ensuremath{\\mathbb{R}}$ can be expressed as $f(A)+f(B)\\ge f(A\\vee B)+f(A\\wedge B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$.\n\\end{enumerate}\n\nIf we were to continue the definition of submodularity stated in (S2), we would obtain nothing new.  Hence, the proof of Theorem \\ref{thm:Chateauneuf-Cornet} cannot directly apply to our situation. To overcome this issue, we need to provide a matched definition of bisubmodularity for functions on $\\ensuremath{\\mathbb{R}}^n$, and an appropriate and careful modification of the translation linearity\ncondition.\n\n\\begin{enumerate}\n \\item[(BS2)] A continuous function $F:\\ensuremath{\\mathbb{R}}^n\\to \\ensuremath{\\mathbb{R}}$ is bisubmodular if\n $F(x)+F(y)\\ge F(x\\vee y)+F(x\\wedge y)$, where\n $$(x \\vee y)_i=\\begin{cases}\\max\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\ge0,\\\\\n  \\min\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\le0, \\\\\n  0,&\\text{ if } x_iy_i<0,\\end{cases}\\;\\;\\;\\;\\;\\;\\;(x \\wedge y)_i=\\begin{cases}\\min\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\ge0,\\\\\n  \\max\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\le0, \\\\\n  0,&\\text{ if } x_iy_i<0.\\end{cases}$$\n\\end{enumerate}\n\n\\begin{pro}\\label{pro:bisubmodular-continuous}\nA function $F:\\ensuremath{\\mathbb{R}}^V\\to \\ensuremath{\\mathbb{R}}$ is a disjoint-pair Lov\\'asz extension of a bisubmodular function if and only if $F$ is (continuously) bisubmodular (in the sense of (BS2)) and for any $\\vec x\\in\\ensuremath{\\mathbb{R}}^V ,\\,t\\ge0$,\n{ \\linespread{0.95} \\begin{enumerate}\n\\item[] $F(t\\vec x)=tF(\\vec x)$ (positive homogeneity);\n\\item[] $F(\\vec x+t\\vec 1_{V^+,V^-})\\ge F(\\vec x)+F(t\\vec 1_{V^+,V^-})$ for some\\footnote{This is some kind of `translation linearity' if we adopt the assumption $F(\\vec x+t\\vec 1_{V^+,V^-})= F(\\vec x)+F(t\\vec 1_{V^+,V^-})$. } $V^\\pm\\supset V^\\pm(\\vec x)$ with $V^+\\cup V^-=V$.\n\\end{enumerate} }\n\nHenceforth, $\\vec 1_{A,B}$ is defined as $\\vec 1_A-\\vec 1_B$ for simplicity.\n\\end{pro}\n\nThe proof is a modification of the previous  version on the original Lov\\'asz extension for  submodular functions. \n\n\\begin{proof}\nTake the discrete function $f$ defined as $f(A_1,A_2)=F(\\vec 1_{A_1,A_2})$. One can check the bisubmodularity of $f$ directly. Fix an $\\vec x\\in\\ensuremath{\\mathbb{R}}^n$ and let $\\sigma:V\\cup\\{0\\}\\to V\\cup\\{0\\}$ be a bijection such that $|x_{\\sigma(1)}|\\le |x_{\\sigma(2)}| \\le \\cdots\\le |x_{\\sigma(n)}|$ and $\\sigma(0)=0$, where $x_0:=0$, and\n$$V_{\\sigma(i)}^\\pm=V_{\\sigma(i)}^\\pm(\\vec x):=\\{j\\in V:\\pm x_j> |x_{\\sigma(i)}|\\},\\;\\;\\;\\; i=0,1,\\cdots,n-1.$$\nAlso, we denote $\\vec x_{V_{\\sigma(i)}^+,V_{\\sigma(i)}^-}=\\vec x * \\vec 1_{V_{\\sigma(i)}^+\\cup V_{\\sigma(i)}^-}$ (i.e., the restriction of $\\vec x$ onto $V_{\\sigma(i)}^+\\cup V_{\\sigma(i)}^-$,  with other components $0$), where $\\vec x*\\vec y:=(x_1y_1,\\cdots,x_ny_n)$.\n\nFor simplicity, in the following formulas, we identify $\\sigma(i)$ with $i$ for all $i=0,\\cdots,n$.\n\nIt follows from $|x_{i+1}|\\vec 1_{V_{i}^+,V_{i}^-}\\bigvee \\vec x_{V_{i+1}^+,V_{i+1}^-}=\\vec x_{V_{i}^+,V_{i}^-}$ and\n$$|x_{i+1}|\\vec 1_{V_{i}^+,V_{i}^-}\\bigwedge \\vec x_{V_{i+1}^+,V_{i+1}^-}= |x_{i+1}|\\vec 1_{V_{i+1}^+,V_{i+1}^-}$$ that\n\\begin{align*}\nf^{L}(\\vec x)&=\\sum_{i=0}^{n-1}(|x_{i+1}|-|x_{i}|)f(V_{i}^+,V_{i}^-)\n\\\\&=\\sum_{i=0}^{n-1}|x_{i+1}|\\left(f(V_{i}^+,V_{i}^-)-f(V_{i+1}^+,V_{i+1}^-)\\right)\n\\\\&=\\sum_{i=0}^{n-1}\\left\\{F\\left(|x_{i+1}|\\vec 1_{V_{i}^+,V_{i}^-}\\right)-F\\left(|x_{i+1}|\\vec 1_{V_{i+1}^+,V_{i+1}^-}\\right)\\right\\}\n\\\\&\\ge\\sum_{i=0}^{n-1}\\left\\{F\\left(\\vec x_{V_{i}^+,V_{i}^-}\\right)-F\\left(\\vec x_{V_{i+1}^+,V_{i+1}^-}\\right)\\right\\}=F(\\vec x).\n\\end{align*}\nOn the other hand,\n\\begin{align*}\nf^{L}(\\vec x)&=\\sum_{i=0}^{n-1}(|x_{i+1}|-|x_{i}|)f(V_{i}^+,V_{i}^-)\n=\\sum_{i=0}^{n-1}F\\left((|x_{i+1}|-|x_{i}|)\\vec 1_{V_{i}^+,V_{i}^-}\\right)\n\\\\&=\\sum_{i=0}^{n-2}\\left\\{F((|x_{i+1}|-|x_{i}|)\\vec 1_{V_{i}^+,V_{i}^-})-F((|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-})\\right\\}\n\\\\&\\;\\;\\;\\;\\;+\\left\\{\\sum_{i=0}^{n-2}F((|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-})\\right\\}+F\\left((|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}\\right)\n\\\\&\\le \\sum_{i=0}^{n-2}\\left\\{F\\left(\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-}\\right)-F\\left(\\vec x_{V_{i+1}^+,V_{i+1}^-}-|x_{i+1}|\\vec 1_{V^+,V^-}+(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}\\right)\\right\\}\n\\\\&\\;\\;\\;\\;\\;+\\left\\{\\sum_{i=0}^{n-2}(|x_{i+1}|-|x_{i}|)F(\\vec 1_{V^+,V^-})\\right\\}+F\\left((|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}\\right)\n\\\\&\\le \\sum_{i=0}^{n-2}\\left(F(\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-})-F(\\vec x_{V_{i+1}^+,V_{i+1}^-}-|x_{i+1}|\\vec 1_{V^+,V^-})\\right)+F\\left((|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}\\right)\n\\\\&=F(\\vec x)\n\\end{align*}\naccording to $(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}\\bigwedge (\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-})= (|x_{i+1}|-|x_{i}|)\\vec 1_{V_{i}^+,V_{i}^-}$ and $$(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}\\bigvee (\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-})=\\vec x_{V_{i+1}^+,V_{i+1}^-}-|x_{i+1}|\\vec 1_{V^+,V^-}+(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}$$\nfor $i=0,\\cdots,n-2$, as well as $\\vec x_{V_{n-1}^+,V_{n-1}^-}-|x_{n-1}|\\vec 1_{V^+,V^-}= (|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}$.  Therefore, we have  $F(\\vec x)=f^L(\\vec x)$. The proof is completed.\n\\end{proof}\n\n\n\\begin{pro}\\label{pro:setpair-character2}\nA continuous function $F$ is a disjoint-pair Lov\\'asz extension of some function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F(\\vec x\\vee c\\vec 1_{V^+,V^-})+F(\\vec x-\\vec x\\vee c\\vec 1_{V^+,V^-})=F(\\vec x)$ (or $F(\\vec x\\wedge c\\vec 1_{V^+,V^-})+F(\\vec x-\\vec x\\wedge c\\vec 1_{V^+,V^-})=F(\\vec x)$), for some  $V^\\pm\\supset V^\\pm(\\vec x)$  with  $V^+\\cup V^-=V$, $\\forall c\\ge0$ and $\\vec x\\in\\ensuremath{\\mathbb{R}}^n$.\n\\end{pro}\n\n\\begin{proof}\nWe only need to prove that the condition implies the absolutely comonotonic additivity of $F$,\nand then apply Proposition \\ref{pro:setpair-character}. Note that the property $F(\\vec x\\vee c\\vec 1)+F(\\vec x-\\vec x\\vee c\\vec 1)=F(\\vec x)$ implies a  summation form of $F$ which agrees with the form of the disjoint-pair Lov\\'asz extension. Then using the absolutely comonotonic additivity, we get the desired result.\n\\end{proof}\n\nThe $k$-way submodularity can be naturally defined as (S1) and (S2):\n\\begin{enumerate}\n\\item[({KS})] Given a tuple $V=(V_1,\\cdots,V_k)$ of finite sets and $\\ensuremath{\\mathcal{A}}\\subset\\{(A_1,\\cdots,A_k):A_i\\subset V_i,\\,i=1,\\cdots,k\\}$,\na discrete function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$   is $k$-way submodular if $f(A)+f(B)\\ge f(A\\vee B)+f(A\\wedge B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$, where $\\ensuremath{\\mathcal{A}}$ is a lattice under the corresponding  lattice operations join $\\vee$ and meet $\\wedge$ defined by $A\\vee B=(A_1\\cup B_1,\\cdots, A_k\\cup B_k)$ and $A\\wedge B=(A_1\\cap B_1,\\cdots, A_k\\cap B_k)$.\n\\end{enumerate}\n\n\n\\begin{theorem}\\label{thm:submodular-L-equivalent}\nUnder the assumptions and notations in (KS) above, ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ is also closed under $\\wedge$ and $\\vee$, with $\\wedge$ and $\\vee$ as in (S2).\n Moreover, the following statements are equivalent:\n \\begin{enumerate}[a)]\n \\item $f$ is $k$-way submodular on $\\ensuremath{\\mathcal{A}}$;\n\\item the $k$-way Lov\\'asz extension $f^L$  is convex on each convex subset of ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$;\n\\item the $k$-way Lov\\'asz extension $f^L$  is submodular on ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$.\n\\end{enumerate}\n\nIf one replaces (KS) and (S2) by (BS1) and (BS2) respectively for the bisubmodular setting, then all the above results hold analogously.\n\\end{theorem}\n\nThe proof is a slight  variation of the  original version by  Lov\\'asz,  and\nis provided for convenience. \n\n\\begin{proof}\nNote that $V^t(\\vec x)\\vee V^t(\\vec y)=V^t(\\vec x\\vee \\vec y)$ and $V^t(\\vec x)\\wedge V^t(\\vec y)=V^t(\\vec x\\wedge \\vec y)$, where $V^t(\\vec x):=(V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$.\nSince $\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}$ if and only if $V^t(\\vec x)\\in \\ensuremath{\\mathcal{A}}$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$, and $\\ensuremath{\\mathcal{A}}$ is a lattice, ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ must be\na lattice that is closed under the operations $\\wedge$ and $\\vee$.\nAccording to the $k$-way Lov\\'asz extension \\eqref{eq:Lovasz-Form-1},\nwe may write\n$$f^L(\\vec x)=\\int_{-N}^Nf(V^t(\\vec x))dt-Nf(V)\n$$\nwhere $N>\\|\\vec x\\|_\\infty$ is a sufficiently large number\\footnote{Here we set $f(\\varnothing,\\cdots,\\varnothing)=0$}.\nNote that $\\vec 1_A\\vee \\vec 1_B=\\vec 1_{A\\vee B}$ and $\\vec 1_A\\wedge \\vec 1_B=\\vec 1_{A\\wedge B}$.\nCombining the above results, we immediately get\n$$f(A)+f(B)\\ge f(A\\vee B)+ f(A\\wedge B) \\; \\Leftrightarrow \\; f^L(\\vec x)+f^L(\\vec y)\\ge f^L(\\vec x\\vee \\vec y)+f^L(\\vec x\\wedge \\vec y),$$\nwhich proves (a) $\\Leftrightarrow$ (c). Note that for  $\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}$, $f^L(\\vec x)=\\sum\\limits_{A\\in \\ensuremath{\\mathcal{C}}(\\vec x)}\\lambda_Af(A)$ for a unique chain $\\ensuremath{\\mathcal{C}}(\\vec x)\\subset\\ensuremath{\\mathcal{A}}$ that is determined by $\\vec x$ only, and\nthe extension $f^{\\mathrm{convex}}(\\vec x):=\\inf\\limits_{\\{\\lambda_A\\}_{A\\in\\ensuremath{\\mathcal{A}}}\\in\\Lambda(\\vec x)}\\sum\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\lambda_Af(A)$ is convex on each convex subset of ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$, where $\\Lambda(\\vec x):=\\{\\{\\lambda_A\\}_{A\\in\\ensuremath{\\mathcal{A}}}\\in\\ensuremath{\\mathbb{R}}^\\ensuremath{\\mathcal{A}}:\\sum\\limits_{A\\in\\ensuremath{\\mathcal{A}}}\\lambda_A\\vec 1_A=\\vec x,\\,\\lambda_A\\ge 0\\text{ whenever }A\\ne V\\}$. We only need to prove $f^L(\\vec x)=f^{\\mathrm{convex}}(\\vec x)$ if and only if $f$ is submodular. In fact, along a standard idea proposed in Lov\\'asz's original paper \\cite{Lovasz}, one could prove that for a (strictly) submodular function, the set $\\{A:\\lambda_A^*\\ne0\\}$ must be a chain, where $\\sum\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\lambda_A^*f(A)=f^{\\mathrm{convex}}(\\vec x)$ achieves the minimum over $\\Lambda(\\vec x)$, and  one can then easily check that it agrees with $f^L$. The converse can be proved\nin a standard way: $f(A)+f(B)=f^L(\\vec 1_A)+f^L(\\vec 1_B)\\ge 2f^L(\\frac{1}{2}(\\vec 1_A+\\vec 1_B))=f(\\vec 1_A+\\vec 1_B)=f(\\vec 1_{A\\vee B}+\\vec 1_{A\\wedge B})=f(\\vec 1_{A\\vee B})+f(\\vec 1_{A\\wedge B})=f(A\\vee B)+f(A\\wedge B)$. Now, the proof is completed.\n\nFor the  bisubmodular case, the above reasoning can be repeated  with minor differences.\n\\end{proof}\n\\label{pro:how-to-be-k-way-Lovasz-submodular}\n\n\n\n\\begin{remark}\nWe show some examples about how both convexity and continuous\nsubmodularity can be satisfied. In fact, it is easy to see that the $l^p$-norm $\\|\\vec x\\|_p$ is both convex and continuously\nsubmodular on $\\ensuremath{\\mathbb{R}}_+^n$, while the $l^1$-norm $\\|\\vec x\\|_1$ is convex and continuously\nsubmodular on the whole  $\\ensuremath{\\mathbb{R}}^n$. Besides, an elementary proof shows that a one-homogeneous continuously\nsubmodular function on $\\ensuremath{\\mathbb{R}}_+^2$ must be convex.\n\\end{remark}\n\n\n\n\n\n\n\\section{Main  results on  optimization and eigenvalue problems}\n\\label{sec:main}\n\nWe uncover the links between  combinatorial optimizations and continuous  programmings as well as eigenvalue\nproblems  in a general setting.\n\\subsection{Combinatorial and continuous optimization}\n\\label{sec:CC-transfer}\n\nAs we have told in the introduction, the application of the Lov\\'asz extension to non-submodular optimization meets with several difficulties, and in this section, we start attacking those. First, we set up some useful results.\n\n\\begin{notification}\\label{notification:fL}\nIn this section, $\\ensuremath{\\mathbb{R}}_{\\ge0}:=[0,\\infty)$ is the set of all non-negative numbers. We use $f^L$ to denote the multi-way Lov\\'asz extension which can be either the original or the disjoint-pair or the $k$-way Lov\\'asz extension.\n\\end{notification}\n\n\\begin{theorem}\\label{thm:tilde-H-f}\n Given set functions $f_1,\\cdots,f_n:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}_{\\ge0}$, and a zero-homogeneous function $H:\\ensuremath{\\mathbb{R}}^n_{\\ge0}\\setminus\\{\\vec 0\\}\\to\\ensuremath{\\mathbb{R}}\\cup\\{+\\infty\\}$\n with $H(\\vec a+\\vec b)\\ge\\min\\{H(\\vec a),H(\\vec b)\\}$, $\n \\forall \\vec a,\\vec b\\in\\ensuremath{\\mathbb{R}}^n_{\\ge0}\\setminus\\{\\vec 0\\}$, we have\n \\begin{equation}\\label{eq:H-minimum}\n \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))=\\inf\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)),\\end{equation}\nwhere $\\ensuremath{\\mathcal{A}}'=\\{A\\in\\ensuremath{\\mathcal{A}}: (f_1(A),\\cdots,f_n(A))\\in\\mathrm{Dom}(H)\\}$, ${\\mathcal D}'=\\{\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathbb{R}}_{\\ge0}^V:\\,(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x))\\in\\mathrm{Dom}(H)\\}$ and $\\mathrm{Dom}(H)=\\{\\vec a\\in \\ensuremath{\\mathbb{R}}^n _{\\ge0}\\setminus\\{\\vec 0\\}: H(\\vec a)\\in\\ensuremath{\\mathbb{R}}\\}$\n\\end{theorem}\n\n\\begin{proof}\nBy the property of $H$, $\\;\\forall t_i\\ge0\\,,n\\in\\mathbb{N}^+,\\, a_{i,j}\\ge0,i=1,\\cdots,m,\\,j=1,\\cdots,n$,\n\\begin{align*}H\\left(\\sum_{i=1}^m t_i a_{i,1},\\cdots,\\sum_{i=1}^m t_i a_{i,n}\\right)&=H\\left(\\sum_{i=1}^m t_i \\vec a^i\\right)\\ge \\min_{i=1,\\cdots,m} H(t_i\\vec a^i) \\\\ &= \\min_{i=1,\\cdots,m} H(\\vec a^i)\n= \\min_{i=1,\\cdots,m} H(a_{i,1},\\cdots,a_{i,n}).\n\\end{align*}\nTherefore, in  the case of the original Lov\\'asz extension, for any $\\vec x\\in{\\mathcal D}'$,\n\\begin{align}\n&H\\left(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)\\right)  \\label{eq:psi-H}\n\\\\ =\\, & H\\left(\\int_{\\min \\vec x}^{\\max  \\vec x}f_1(V^t(\\vec x))dt+ f_1(V(\\vec x))\\min \\vec x,\\cdots,\\int_{\\min \\vec x}^{\\max \\vec x}f_n(V^t(\\vec x))dt+ f_n(V(\\vec x))\\min \\vec x\\right)\\notag\n\\\\ =\\, & H\\left(\\sum_{i=1}^m (t_i-t_{i-1}) f_1(V^{t_{i-1}}(\\vec x)),\\cdots,\\sum_{i=1}^m (t_i-t_{i-1}) f_n(V^{t_{i-1}}(\\vec x)) \\right)\\notag\n\\\\ \\ge\\, & \\min_{i=1,\\cdots,m} H\\left(f_1(V^{t_{i-1}}(\\vec x)),\\cdots,f_n(V^{t_{i-1}}(\\vec x)) \\right)\\notag\n\\\\ \\ge\\, &  \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))\\label{eq:min-H}\n\\\\ =\\, &  \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1^L(\\vec1_A),\\cdots,f_n^L(\\vec1_A))\\notag\n\\\\ \\ge\\, &  \\inf\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)).\\label{eq:inf-H}\n\\end{align}\nCombining \\eqref{eq:psi-H} with \\eqref{eq:min-H}, we have $\\inf\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x))\\ge \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))$, and then together with \\eqref{eq:min-H} and \\eqref{eq:inf-H}, we get the reverse inequality. Hence, \\eqref{eq:H-minimum} is proved for the original Lov\\'asz extension $f^L$. For multi-way settings, the proof is similar.\n\\end{proof}\n\n\\begin{remark}\\label{remark:thm-H} Duality: If one replaces $H(\\vec a+\\vec b)\\ge\\min\\{H(\\vec a),H(\\vec b)\\}$ by $H(\\vec a+\\vec b)\\le\\max\\{H(\\vec a),H(\\vec b)\\}$, then\n   \\begin{equation}\\label{eq:H-maximum}\n   \\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))=\\sup\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)).\\end{equation}\n   The proof of the  identity \\eqref{eq:H-maximum} is similar to that of \\eqref{eq:H-minimum}, and thus we omit it.\n   \\end{remark}\n   \\begin{remark}\nA function $H:[0,+\\infty)^n\\to \\overline{\\ensuremath{\\mathbb{R}}}$ has the (MIN) property\nif\n$$H\\left(\\sum_{i=1}^m t_i \\vec a^i\\right)\\ge \\min_{i=1,\\cdots,m} H(\\vec a^i),\\;\\forall t_i>0\\,,m\\in\\mathbb{N}^+,\\,\\vec a^i\\in [0,+\\infty)^n.$$\nThe (MAX) property is formulated analogously.\n\nWe can verify that the (MIN) property is equivalent to the\nzero-homogeneity and $H(\\vec x+\\vec y)\\ge\\min\\{H(\\vec x),H(\\vec y)\\}$. A similar correspondence holds for the (MAX) property.\n\\end{remark}\n\n\\begin{remark}Theorem \\ref{thm:tilde-H-f} shows that if $H$ has the (MIN)  or (MAX) property, then the corresponding combinatorial optimization is equivalent to a continuous optimization by means of  \n the multi-way Lov\\'asz extension.\n Here are some examples:\n \n Given $c,c_i\\ge 0$ with $\\sum_i c_i>0$, let $H(f_1,\\cdots,f_n)=\\frac{c_1f_1+\\cdots+c_nf_n-c\\sqrt{f_1^2+\\cdots+f_n^2}}{f_1+\\cdots+f_n}$. Then $H$ satisfies the (MIN)  property, and by Theorem \\ref{thm:tilde-H-f}, we have\n $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{\\sum_i c_if_i(A)-c\\sqrt{\\sum_if_i^2(A)}}{\\sum_i f_i(A)}=\\inf\\limits_{\\psi\\in {\\mathcal D}'}\\frac{\\sum_i c_if_i^L(\\psi)-c\\sqrt{\\sum_i (f_i^L(\\psi))^2}}{\\sum_i f_i^L(\\psi)}.$$\n\nTaking $H(f_1,\\cdots,f_n)=\\frac{(c_1f_1^p+\\cdots+c_nf_n^p)^{\\frac1p}}{f_1+\\cdots+f_n}$ for some $p> 1$, then $H$ satisfies the (MAX)  property, and by Theorem \\ref{thm:tilde-H-f}, there holds $$\\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{(\\sum_i c_if_i(A)^p)^{\\frac1p}}{\\sum_i f_i(A)}=\\sup\\limits_{\\psi\\in {\\mathcal D}'}\\frac{(\\sum_i c_if_i^L(\\psi)^p)^{\\frac1p}}{\\sum_i f_i^L(\\psi)}.$$\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:tilde-fg-equal-PQ}]\nWithout loss of generality, we may assume that $P(f_1,\\cdots,f_n)$ is one-homogeneous and subaddtive, while $Q(f_1,\\cdots,f_n)$ is one-homogeneous and superadditive on $(f_1,\\cdots,f_n)\\in\\ensuremath{\\mathbb{R}}_{\\ge0}^n$. \n\nThen $H(f_1,\\cdots,f_n)=\\frac{P(f_1,\\cdots,f_n)}{Q(f_1,\\cdots,f_n)}$ is zero-homogeneous on $[0,+\\infty)^n$,   and \n$$H(\\vec f+\\vec g)=\\frac{P(\\vec f+\\vec g)}{Q(\\vec f+\\vec g)}\\le \\frac{P(\\vec f)+P(\\vec g)}{Q(\\vec f)+Q(\\vec g)}\\le \\max\\{\\frac{P(\\vec f)}{Q(\\vec f)},\\frac{P(\\vec g)}{Q(\\vec g)}\\}=\\max\\{H(\\vec f),H(\\vec g)\\}$$\nwhere $\\vec f=(f_1,\\cdots,f_n)$ and $\\vec g=(g_1,\\cdots,g_n)$.\n\n\n\nThen the proof is completed by Theorem \\ref{thm:tilde-H-f} \n (and Remark \\ref{remark:thm-H}).\n\\end{proof}\n\n\n\\begin{example}\\label{exam:maxcut1}\nGiven a finite graph $(V,E)$, for $\\{i,j\\}\\in E$, let $f_{\\{i,j\\}}(A)=1$ if $A\\cap\\{i,j\\}=\\{i\\}$ or $\\{j\\}$, and $f_{\\{i,j\\}}(A)=0$ otherwise. Let $g(A)=|A|$ for $A\\subset V$. It is clear that $\\frac{\\left(\\sum_{\\{i,j\\}\\in E}f_{\\{i,j\\}}^p\\right)^{\\frac1p}}{g}$ satisfies the condition of  Theorem \\ref{thm:tilde-fg-equal-PQ}. Thus, we  derive that\n$$\\max\\limits_{A\\ne\\varnothing}\\frac{|\\partial A|^{\\frac1p}}{|A|}=\\max\\limits_{A\\ne\\varnothing}\\frac{(\\sum_{\\{i,j\\}\\in E}f_{\\{i,j\\}}^p(A))^{\\frac1p}}{g(A)} =\\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{\\sum_{i\\in V}x_i}=\\max\\limits_{x\\ne 0}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{\\sum_{i\\in V}|x_i|}.$$\nSimilarly, we have\n$$\\max\\limits_{A\\ne\\varnothing}|\\partial\nA|^{\\frac1p}=\\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{(\\sum_{\\{i,j\\}\\in\n    E}|x_i-x_j|^p)^{\\frac1p}}{\\max\\limits_{i\\in V}x_i}=\\max\\limits_{x\\ne\n  0}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{2\\|\\vec\n  x\\|_\\infty},$$\nwhich gives a  continuous representation of the Max-Cut problem. \nThe last equality holds due to the following reason: letting $F(\\vec x)=(\\sum_{\\{i,j\\}\\in\n    E}|x_i-x_j|^p)^{\\frac1p}$, we can check that  \n$\\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{F(\\vec x)}{\\max_{i\\in V}x_i}$ achieves its maximum at some characteristic vector $\\vec 1_A$, and then $\\vec 1_A-\\vec 1_{V\\setminus A}$ is a maximizer of $\\frac{F(\\vec x)}{2\\|\\vec\n  x\\|_\\infty}$ on $\\ensuremath{\\mathbb{R}}^V\\setminus\\vec0$. \n  \n \n  Similarly,    $\\max\\limits_{x\\ne\n  0}\\frac{F(\\vec x)}{2\\|\\vec\n  x\\|_\\infty}$ achieves its maximum at $\\vec 1_A-\\vec 1_{V\\setminus A}$ for some $A$, and then $\\vec1_A$ indicates a maximizer of $\\frac{F(\\vec x)}{\\max_{i\\in V}x_i}$ on the first orthant $\\ensuremath{\\mathbb{R}}^V_+$. We need the factor 2  because $F(\\vec 1_A-\\vec 1_{V\\setminus A})=2F(\\vec 1_A)$. \n  \nIt should be noted that the two equivalent continuous reformulations are derived by the original and disjoint-pair Lov\\'asz extensions in the following two ways: $$\\max\\limits_{A\\ne\\varnothing}|\\partial\nA|^{\\frac1p}=\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}}\\frac{(\\sum_{\\{i,j\\}\\in E}f_{\\{i,j\\}}^p(A))^{\\frac1p}}{1} = \\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{(\\sum_{\\{i,j\\}\\in\n    E}|x_i-x_j|^p)^{\\frac1p}}{\\max_{i\\in V}x_i}$$\n    where we use  $f^L_{\\{i,j\\}}(\\vec x)=|x_i-x_j|$ and $1^L=\\max_{i\\in V}x_i$; \n      $$\\max\\limits_{A\\ne\\varnothing}|\\partial\nA|^{\\frac1p}=\\max\\limits_{(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{ (\\varnothing,\\varnothing)\\}}\\frac{\\left(\\sum_{\\{i,j\\}\\in E}(f_{\\{i,j\\}}(A)+f_{\\{i,j\\}}(B))^p\\right)^{\\frac1p}}{2} =\\max\\limits_{x\\ne\n  0}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{2\\|\\vec\n  x\\|_\\infty}$$\n  where we use the fact that the disjoint-pair Lov\\'asz extension of $(A,B)\\mapsto f_{\\{i,j\\}}(A)+f_{\\{i,j\\}}(B)$ is $|x_i-x_j|$ and the disjoint-pair Lov\\'asz extension of $(A,B)\\mapsto 1$ is $\\|\\vec x\\|_\\infty$.\n\\end{example}\n\n\n\\begin{example}\\label{exam:maxcut2}\nThere are many other equalities that can be obtained by  Theorem \\ref{thm:tilde-fg-equal-PQ}, such as:\n$$\\min\\limits_{A\\ne\\varnothing,V}\\frac{|\\partial A|}{|A|^{\\frac1p}}=\\min\\limits_{x\\in\\ensuremath{\\mathbb{R}}^V:\\,\\min x=0}\\frac{\\sum_{\\{i,j\\}\\in E}|x_i-x_j|}{(\\sum_{i\\in V}x_i^p)^{\\frac1p}}$$\nand \n$$\\max\\limits_{(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}}\\frac{2|E(A,B)|^{\\frac1p}}{\\vol(A\\cup B)}= \\max\\limits_{x\\ne 0}\\frac{(\\sum_{\\{i,j\\}\\in E}(|x_i|+|x_j|-|x_i+x_j|)^p)^{\\frac1p}}{\\sum_{i\\in V}\\deg_i|x_i|}$$\nwhenever $p\\ge 1$. Here, $\\vol A =\\sum_{i\\in A} \\deg_i$.\\\\ \nThe last equality shows a variant of the dual Cheeger constant. A slight modification gives \n$$\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}2|\\partial A|^{\\frac1p}=\\max\\limits_{(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)}2|E(A,B)|^{\\frac1p}=\\max\\limits_{x\\ne 0}\\frac{\\left(\\sum_{\\{i,j\\}\\in E}(|x_i|+|x_j|-|x_i+x_j|)^p\\right)^{\\frac1p}}{\\|\\vec x\\|_\\infty}$$\nshowing a new continuous formulation of the Maxcut problem.\n\\end{example}\n\nTaking $n=2$ and $H(f_1,f_2)=\\frac{f_1}{f_2}$ in Theorem \\ref{thm:tilde-H-f}, then such an $H$ satisfies both (MIN) and (MAX) properties. So, we get\n$$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{f_1(A)}{f_2(A)}=\\inf\\limits_{\\psi\\in {\\mathcal D}'}\\frac{f_1^L(\\psi)}{f_2^L(\\psi)},\\;\\;\\;\\text{ and } \\;\\; \\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{f_1(A)}{f_2(A)}=\\sup\\limits_{\\psi\\in {\\mathcal D}'}\\frac{f_1^L(\\psi)}{f_2^L(\\psi)}.$$\nIn fact, we can get more:\n\\end{remark}\n\n\n\\begin{pro}\\label{pro:fraction-f\/g}\n Given two  functions $f,g:\\ensuremath{\\mathcal{A}}\\to [0,+\\infty)$, let $\\tilde{f},\\tilde{g}:{\\mathcal D}_\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$ satisfy $\\tilde{f}\\ge f^L$, $\\tilde{g}\\le g^L$, $\\tilde{f}(\\vec1_{A})=f(A)$ and $\\tilde{g}(\\vec1_{A})=g(A)$. Then\n$$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}=\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.$$\n\nIf we replace the condition $\\tilde{f}\\ge f^L$ and $\\tilde{g}\\le g^L$  by $\\tilde{f}\\le f^L$ and $\\tilde{g}\\ge g^L$, then $$\\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}=\\sup\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.$$\n\nFor any $\\alpha\\ne0$,  then $\\tilde{f}=((f^\\alpha)^L)^{\\frac1\\alpha}$ and\n$\\tilde{g}=((g^\\alpha)^L)^{\\frac1\\alpha}$ satisfy the above two\nidentities. \n\\end{pro}\n\n\\begin{proof}\nIt is obvious that\n $$\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}\\le\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)} \\frac{\\widetilde{f}(\\vec1_{A})}{\\widetilde{g}(\\vec1_{A})} = \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}.$$\n On the other hand, for any $\\psi\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})$, $g^L(\\psi)\\ge \\tilde{g}(\\psi)>0$. Hence, there exists $t\\in (\\min \\widetilde{\\beta}\\psi-1,\\max \\widetilde{\\beta}\\psi+1)$ satisfying $g(V^t(\\psi))>0$. Here $\\widetilde{\\beta}\\psi=\\psi$ (resp., $|\\psi|$), if $f^L$ represents either the original or the $k$-way Lovasz extension of $f$ (resp., either the disjoint-pair or the $k$-way disjoint-pair Lovasz extension). So, the set $W(\\psi):=\\{t\\in\\ensuremath{\\mathbb{R}}: g(V^t(\\psi))>0\\}$ is nonempty. Since $\\{V^t(\\psi):t\\in W(\\psi)\\}$ is finite, there exists $t_0\\in W(\\psi)$ such that $\\frac{f(V^{t_0} (\\psi))}{g(V^{t_0} (\\psi))}=\\min\\limits_{t\\in W(\\psi)}\\frac{f(V^{t} (\\psi))}{g(V^{t} (\\psi))}$. Accordingly, $f(V^{t} (\\psi))\\ge \\frac{f(V^{t_0} (\\psi))}{g(V^{t_0} (\\psi))}g(V^{t} (\\psi))$ for any $t\\in W(\\psi)$, and thus\n  $$f(V^{t} (\\psi))\\ge Cg(V^{t} (\\psi)),\\;\\;\\;\\text{ with }\\;\\;C=\\min\\limits_{t\\in W(\\psi)}\\frac{f(V^{t} (\\psi))}{g(V^{t} (\\psi))}\\ge0,$$\n  holds for any $t\\in\\ensuremath{\\mathbb{R}}$ (because $g(V^{t} (\\psi))=0$ for $t\\in\\ensuremath{\\mathbb{R}}\\setminus W(\\psi)$ which means that the above inequality automatically holds).\n  Consequently,\n \\begin{align*}&\\tilde{f}(\\psi)\\ge f^L(\\psi)\n\\\\ =\\,& \\int_{\\min \\widetilde{\\beta}\\psi}^{\\max  \\widetilde{\\beta}\\psi}f(V^t(\\psi))dt+ f(V(\\psi))\\min \\widetilde{\\beta}\\psi\n\\\\ \\ge\\,& C \\int_{\\min \\widetilde{\\beta}\\psi}^{\\max  \\widetilde{\\beta}\\psi}g(V^t(\\psi))dt+ g(V(\\psi))\\min \\widetilde{\\beta}\\psi.\n\\\\ =\\,&C g^L(\\psi)\\ge C\\tilde{g}(\\psi).\n \\end{align*}\n It follows that\n $$\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}\\ge C\\ge \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}$$\n and thus the proof is completed. The dual case is similar.\n\nFor $\\alpha>0$, we can simply suppose $\\ensuremath{\\mathrm{supp}}(g)=\\ensuremath{\\mathcal{A}}$. Then \n \\begin{align*}\n \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f(A)}{g(A)}=\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{ (f^\\alpha)^{\\frac1\\alpha}(A)}{(g^\\alpha)^{\\frac1\\alpha}(A)}=\\left(\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f^\\alpha(A)}{ g^\\alpha(A)}\\right)^{\\frac1\\alpha}\n = \\left(\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{( f^\\alpha)^L(\\psi)}{(g^\\alpha)^L(\\psi)}\\right)^{\\frac1\\alpha}=\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{(( f^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}{(( g^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}.\n \\end{align*}\n \nFor $\\alpha<0$, we may suppose without loss of generality that $g(A)>0$ and $f(A)>0$ for any $A\\in\\ensuremath{\\mathcal{A}}$. Then, in this case,  \n \\begin{align*}\n \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f(A)}{g(A)}=\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{ (f^\\alpha)^{\\frac1\\alpha}(A)}{(g^\\alpha)^{\\frac1\\alpha}(A)}=\\left(\\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f^\\alpha(A)}{ g^\\alpha(A)}\\right)^{\\frac1\\alpha}\n= \\left(\\sup\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{( f^\\alpha)^L(\\psi)}{(g^\\alpha)^L(\\psi)}\\right)^{\\frac1\\alpha}=\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{(( f^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}{(( g^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}.\n \\end{align*}\n \n This completes the proof.\n\\end{proof}\n\nIt is worth noting that in Proposition  \\ref{pro:fraction-f\/g}, $\\ensuremath{\\mathcal{A}}$ can be a family of some set-tuples, and $f^L$ is the  multi-way Lov\\'asz extension of the corresponding  $f$. We point out that we can replace the Lov\\'asz extension $f^L$ by any other extension $f^E$ with the property that $f^E\/g^E$ achieves its minimum and maximum at some $0$-$1$ vector $\\vec 1_A$ for some $A\\in\\ensuremath{\\mathcal{A}}$. \n Similarly, we have:\n\n\\begin{pro}\\label{pro:maxconvex}\nLet $f,g:\\ensuremath{\\mathcal{A}}\\to [0,+\\infty)$ be two set functions and $f:=f_1-f_2$ and $g:=g_1-g_2$ be decompositions of differences of submodular functions.\n\nLet $\\widetilde{f}_2,\\widetilde{g}_1$ be the restriction of  positively one-homogeneous convex functions onto ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$, with $f_1(A)= \\widetilde{f}_1(\\vec1_{A})$ and $g_2(A)= \\widetilde{g}_2(\\vec1_{A})$.\nDefine $\\widetilde{f}=f_1^L-\\widetilde{f}_2$ and $\\widetilde{g}=\\widetilde{g}_1-g_2^L$. Then,\n$$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}=\\min\\limits_{\\vec x\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\widetilde{g})}\\frac{\\widetilde{f}(\\vec x)}{\\widetilde{g}(\\vec x)}.$$\n\\end{pro}\n\n\\begin{remark}\nHirai et al introduce the generalized Lov\\'asz extension of $f:\\mathcal{L}\\to\\overline{\\ensuremath{\\mathbb{R}}}$ on a graded poset $\\mathcal{L}$ (see \\cite{Hirai18,HH17-arxiv}). Since $f^L(\\vec x)=\\sum_i\\lambda_i f(\\vec p_i)$ for $\\vec x=\\sum_i\\lambda_i \\vec p_i$ lying in the orthoscheme complex $K(\\mathcal{L})$, the same results as stated in Theorem \\ref{thm:tilde-H-f} and Proposition \\ref{pro:fraction-f\/g} hold for such a generalized Lov\\'asz extension $f^L$.   Propositions \\ref{pro:fraction-f\/g} and \\ref{pro:maxconvex} are also generalizations of Theorem 3.1 in \\cite{HS11} and Theorem 1 (b) in \\cite{BRSH13}.\n\\end{remark}\n\nAlthough the continuous representations translate the original problems into  equivalently difficult\noptimization problems,\nwe should point out that the continuous reformulations ensure that many fast algorithms in continuous programming  can be applied directly to certain  combinatorial optimization problems.  For example, the fractional form of the equivalent continuous optimizations shown in  Theorem \\ref{thm:tilde-fg-equal} as well as Propositions \\ref{pro:fraction-f\/g} and \\ref{pro:maxconvex}  implies that we can directly adopt the Dinkelbach iteration in Fractional Programming \\cite{SI83} to solve them. In addition, since the equivalent continuous formulation is Lipschitz, we can also adopt the \nstochastic subgradient method  \\cite{Davis19-FoCM} to solve certain discrete optimization problems directly.\n\nTables~\\ref{tab:L-one} and \\ref{tab:L-two} and Propositions \\ref{pro:discrete:one-to-two}, \\ref{pro:discrete:one-to-k} and \\ref{pro:discrete:two-to-k} present a general correspondence between set or set-pair functions and their Lov\\'asz extensions. We shall make use of several of those in Section \\ref{sec:examples-Applications}. Note that the first four lines in Table \\ref{tab:L-one} for the original Lov\\'asz extension, and the first five lines in Table \\ref{tab:L-two} for the disjoint-pair Lov\\'asz extension are known (see \\cite{HS11,CSZ18}).\n\n\\begin{table}\n\\centering\n\\caption{\\small Original Lov\\'asz extension of some objective functions.}\n\\begin{tabular}{|l|l|}\n               \\hline\n               Set function $f(A)=$ & Lov\\'asz extension $f^L(\\vec x)=$  \\\\\n               \\hline\n                 $\\#E(A,V\\setminus A)$ & $\\sum\\limits_{\\{i,j\\}\\in E}|x_i-x_j|$\\\\\n                 \\hline\n               $C$&$C\\max_i x_i$\\\\\n               \\hline\n               $\\vol(A)$ & $\\sum_i \\deg_i x_i$\\\\\n               \\hline\n               $\\min\\{\\vol(A),\\vol(V\\setminus A)\\}$& $\\min\\limits_{t\\in \\mathbb{R}}\\|\\vec x-t \\vec 1\\|_1$\\\\\n               \\hline\n               $\\#A\\cdot\\#(V\\setminus A)$ & $\\sum\\limits_{i,j\\in V}|x_i-x_j|$ \\\\\n               \\hline\n               $\\#V(E(A,V\\setminus A))$ & $\\sum\\limits_{i=1}^n(\\max\\limits_{j\\in N(i)}x_j-\\min\\limits_{j\\in N(i)}x_j)$\n               \\\\\n               \\hline\n            \\end{tabular}\n             \\label{tab:L-one}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{\\small Disjoint-pair Lov\\'asz extension of several objective functions.}\n\\begin{tabular}{|l|l|}\n               \\hline\n               Objective function $f(A,B)=$ & Disjoint-pair Lov\\'asz extension $f^L(\\vec x)=$  \\\\\n               \\hline\n                 $\\#E(A,V\\setminus A)+\\#E(B,V\\setminus B)$& $\\sum\\limits_{\\{i,j\\}\\in E}|x_i-x_j|$\\\\\n                 \\hline\n               $\\#E(A,B)$&$\\frac12\\left(\\sum\\limits_{i\\in V}\\deg_i|x_i|- \\sum\\limits_{\\{i,j\\}\\in E}|x_i+x_j|\\right)$\\\\\n               \\hline\n               $C$&$C\\|\\vec x\\|_\\infty$\\\\\n               \\hline\n               $\\vol(A)+\\vol(B)$&$\\sum\\limits_{i\\in V}\\deg_i|x_i|$\\\\\n               \\hline\n               $\\min\\{\\vol(A),\\vol(V\\setminus A)\\}+\\min\\{\\vol(B),\\vol(V\\setminus B)\\}$&$\\min\\limits_{\\alpha\\in \\mathbb{R}}\\|(x_1,\\cdots,x_n)-\\alpha \\vec 1\\|$\\\\\n               \\hline\n              $ \\# E(A\\cup B,A\\cup B)$ & $\\sum_{i\\sim j}\\min\\{|x_i|,|x_j|\\}$ \\\\\n              \\hline\n               $\\# (A\\cup B)\\cdot\\# E(A\\cup B,A\\cup B)$ & $\\sum_{k\\in V,i\\sim j}\\min\\{|x_k|,|x_i|,|x_j|\\}$\\\\ \\hline\n$\\#(A\\cup B)\\cdot\\#(V\\setminus (A\\cup B))$ & $\\sum_{i>j}||x_i|-|x_j||$ \\\\ \\hline\n            \\end{tabular}\n             \\label{tab:L-two}\n\\end{table}\n\n\n\\begin{pro}\\label{pro:discrete:one-to-two}\nSuppose $f,g:\\ensuremath{\\mathcal{P}}(V)\\to [0,+\\infty)$ are two set functions with $g(A)>0$ for any $A\\in \\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$.\nThen $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}}\\frac{f(A)}{g(A)}=\\min\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}(V)^2\\setminus\\{(\\varnothing,\\varnothing)\\}}\\frac{f(A)+f(B)}{g(A)+g(B)}=\\min\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}}\\frac{f(A)+f(B)}{g(A)+g(B)},$$\nwhere the right identity needs additional assumptions like\n$f(\\varnothing)=g(\\varnothing)=0$\\footnote{This setting is natural, as the\n  Lov\\'asz extension doesn't use the datum on $\\varnothing$.} or  that $f$\n  and $g$ are symmetric.\\footnote{A function $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is {\\sl symmetric} if $f(A)=f(V\\setminus A)$, $\\forall A\\subset V$.}\nReplacing $f(B)$ and $g(B)$ by $f(V\\setminus B)$ and $g(V\\setminus B)$, all the above identities hold without any additional assumption. Clearly, replacing `min' by `max', all statements still hold.\n\\end{pro}\n\n\\begin{pro}\\label{pro:discrete:one-to-k}\nSuppose $f,g:\\ensuremath{\\mathcal{P}}(V)\\to [0,+\\infty)$ are two set functions with $g(A)>0$ for any $A\\in \\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$.\nThen $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{P}}(V)}\\frac{f(A)}{g(A)}=\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}(V)^k}\\frac{\\sum_{i=1}^kf(A_i)}{\\sum_{i=1}^kg(A_i)}=\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}(V)^k}\\sqrt[k]{\\frac{\\prod_{i=1}^kf(A_i)}{\\prod_{i=1}^kg(A_i)}}=\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}_k(V)}\\frac{\\sum_{i=1}^kf(A_i)}{\\sum_{i=1}^kg(A_i)},$$\nwhere the last identity needs additional assumptions like $f(\\varnothing)=g(\\varnothing)=0$.\n\\end{pro}\n\n\\begin{pro}\\label{pro:discrete:two-to-k}\nSuppose $f,g:\\ensuremath{\\mathcal{P}}_2(V)\\to [0,+\\infty)$ are two set functions with $g(A,B)>0$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$.\nThen $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{P}}_2(V)}\\frac{f(A,B)}{g(A,B)}=\\min\\limits_{(A_1,B_1,\\cdots,A_k,B_k)\\in \\ensuremath{\\mathcal{P}}_2(V)^k}\\frac{\\sum_{i=1}^kf(A_i,B_i)}{\\sum_{i=1}^kg(A_i,B_i)}=\\min\\limits_{(A_1,B_1,\\cdots,A_k,B_k)\\in \\ensuremath{\\mathcal{P}}_{2k}(V)}\\frac{\\sum_{i=1}^kf(A_i,B_i)}{\\sum_{i=1}^kg(A_i,B_i)},$$\nwhere the last identity needs additional assumptions like $f(\\varnothing,\\varnothing)=g(\\varnothing,\\varnothing)=0$\\footnote{This setting is natural, as the  disjoint-pair Lov\\'asz extension doesn't use the information on $(\\varnothing,\\varnothing)$.}.\n\\end{pro}\n\nTogether with Propositions \\ref{pro:setpair-original} and \\ref{pro:discrete:one-to-two}, one may directly transfer the data from  Table \\ref{tab:L-one} to Table \\ref{tab:L-two}. Similarly, by employing Propositions \\ref{pro:separable-summation}, \\ref{pro:discrete:one-to-k} and \\ref{pro:discrete:two-to-k}, the $k$-way Lov\\'asz extension of some special functions can be transformed to the  original and the disjoint-pair versions.\n\n\\begin{pro}\\label{pro:Lovasz-f-pre}\nFor any $a<b$, and for any $f$, \n$$ \\min\\limits_{x\\in [a,b]^n}f^L(\\vec x)= \\min\\limits_{x\\in \\{a,b\\}^n}f^L(\\vec x)=af(V)+(b-a)\\min\\limits_{A\\subset V}f(A).$$\nClearly, we can replace all `min' by `max'.\n\\end{pro}\n\n\\begin{proof}\nSince $f^L$ is linear on each piece $\\triangle_\\sigma\\cap [a,b]^n:=\\{\\vec x\\in\n\\ensuremath{\\mathbb{R}}^n:a\\le x_i\\le b,x_{\\sigma(1)}\\le \\cdots\\le x_{\\sigma(n)}\\}$, where\n$\\sigma:\\{1,\\cdots,n\\}\\to \\{1,\\cdots,n\\}$ is a permutation, the maximum and\nminimum of $f^L$ on $\\triangle_\\sigma\\cap [a,b]^n$ can be reached at some\nvertices of the simplex  $\\triangle_\\sigma\\cap [a,b]^n$. Note that the\nvertices of $\\triangle_\\sigma\\cap [a,b]^n$ are included in $\\{a,b\\}^n$ (i.e.,\nthe vertices of the hypercube $ [a,b]^n$). Thus, the maximum and minimum of\n$f^L$ on $ [a,b]^n$ can be attained at some points in  $\\{a,b\\}^n$. By the definition of Lov\\'asz extension, it is easy to check that $f^L(b\\vec1_B+a\\vec1_{V\\setminus B})=af(V)+(b-a)f(B)$.  The proof is completed.\n\\end{proof}\n\n\n\n\\subsection{Eigenvalue problems for Lov\\'asz extension}\n\\label{sec:eigenvalue}\n\nIn this section, we give the proof of Theorem \\ref{introthm:eigenvalue} by\nestablishing some  properties on the nonlinear eigenvalue problem of the\nfunction pair $(f^L,g^L)$.\n\\begin{defn}\n $\\lambda \\in \\ensuremath{\\mathbb{R}}$ is called an eigenvalue, and $\\vec x \\in\n\\ensuremath{\\mathbb{R}}^n\\setminus\\{\\vec0\\}$ a corresponding eigenvector of the\nfunction pair $(f^L,g^L)$ if\n$$\\vec0\\in \\nabla f^L (\\vec x)- \\lambda\\nabla  g^L (\\vec x).$$\nWe then also call $(\\lambda, \\vec x)$ an eigenpair. \n    \\end{defn}\n\n\n\n\\begin{pro}\\label{pro:Lovasz-eigen-pre}\nIn the setting of the original Lov\\'asz extension, for any eigenvalue $\\lambda$ of $(f^L,g^L)$, there exists $A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$  such that $\\lambda=f(A)\/g(A)$ and $\\vec 1_A$ is a corresponding eigenvector.  Indeed, every eigenvalue has an  eigenvector in   $\\{a,b\\}^n$, for given distinct real numbers $a$ and $b$. \nMoreover, we have:\n\\begin{itemize}\n    \\item if $g(V)\\ne0$, then $\\lambda=f(V)\/g(V)$ is the only possible eigenvalue of $(f^L,g^L)$; \n    \\item if $g(V)=0$ and $(f^L,g^L)$ has at least one eigenvalue, then $f(V)=0$ and in this case, $(f^L,g^L)$ may have many distinct eigenvalues.\n\\end{itemize}\n\\end{pro}\n\\begin{proof}\n We need the following  basic statement.\n\n\\textbf{Argument}. For a piecewise linear function $F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ with finite pieces, if $F$ is linear on a convex subset  $\\Omega$, then $\\nabla F(\\vec x)\\subset \\nabla F(\\vec y)$ for any relative interior point $\\vec x$ of $\\Omega$ and any relative boundary point $\\vec y$ in $ \\overline{\\Omega}$. \n\n\\vspace{0.16cm}\n\nSuppose that $(\\lambda,\\vec x)$ is an eigenpair of $(f^L,g^L)$. Since $f^L$ is one-homogeneous and  linear along the direction $\\vec 1$, we can assume without loss of generality that  $\\vec x$ lies in the interior of the simplex $\\triangle$ with vertices $\\vec 1_{A_1},\\cdots,\\vec1_{A_k}$, where $A_1,\\cdots,A_k$ are upper level sets of $\\vec x$.  Applying  the above argument to the piecewise linear functions $f^L$ and $g^L$, we immediately get $\\vec0\\in\\nabla f^L(\\vec x)-\\lambda \\nabla g^L(\\vec x)\\subset \\nabla f^L(\\vec 1_{A_i})-\\lambda \\nabla g^L(\\vec 1_{A_i})$ for any $i=1,\\cdots,k$, meaning that each $\\vec 1_{A_i}$ is an eigenvector of  $(f^L,g^L)$. \n\nMoreover, by the definition of Lov\\'asz extension, we can check that $\\forall \\vec v\\in\\nabla f^L(\\vec x)$, $\\sum_{i\\in V} v_i=f(V)$. Therefore, $\\nabla f^L(\\vec x)\\cdot\\vec1=f(V)$, $\\forall \\vec x$. Hence, for any eigenpair $(\\lambda,\\vec x)$, $f(V)-\\lambda g(V)=(\\nabla f^L(\\vec x)-\\lambda\\nabla g^L(\\vec x) )\\cdot\\vec1=0$, implying that $\\lambda=f(V)\/g(V)$ if $g(V)\\ne 0$; and $f(V)=0$ if $g(V)= 0$.\n\nFor the case of $f(V)=g(V)= 0$, we may take a look at the example\n$f(A)=|\\partial A|$ and $g(A)=\\min\\{\\vol(A),\\vol(V\\setminus A)\\}$ defined on a\nsimple graph $G=(V,E)$. Then the eigenvalue problem of $(f^L,g^L)$ reduces to\nthe problem for the  graph 1-Laplacian. And it is known that the 1-Laplacian may have many different  eigenvalues \\cite{CSZ17}.\n\\end{proof}\n\n\\begin{pro}\\label{pro:Lovasz-eigen}\nIn the setting of the  disjoint-pair  Lov\\'asz extension, for any eigenvalue $\\lambda$ of $(f^L,g^L)$, there exists $(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$  such that $\\lambda=f(A,B)\/g(A,B)$ and $\\vec 1_A-\\vec1_B$ is a corresponding eigenvector.  If we further assume that $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ and $2g(A,B)=g(A,V\\setminus A)+g(V\\setminus B,B)$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$, then for any eigenvalue $\\lambda$ of $(f^L,g^L)$, there exists $A\\subset V$  such that $\\lambda=f(A,V\\setminus A)\/g(A,V\\setminus A)$ and $\\vec 1_A-\\vec1_{V\\setminus A}$ is a corresponding eigenvector.\n\\end{pro}\n\n\\begin{proof}\n The general result on  the disjoint-pair  Lov\\'asz extension is similar to \n Proposition \\ref{pro:Lovasz-eigen-pre}, and thus we omit the proof. \n \nLet's focus on the special case that $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ and $2g(A,B)=g(A,V\\setminus A)+g(V\\setminus B,B)$  for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$. We shall prove that in this case, \ntypical eigenvectors can be taken from  $\\{-1,1\\}^n$. \n\n\\vspace{0.16cm}\n\n\\textbf{Claim}. For any $A_1\\subset \\cdots\\subset A_k$ with $(A_1,A_k)\\ne (\\varnothing,V)$, $f^L$ is linear on  $\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{V\\setminus A_i}:i=1,\\cdots,k\\}$. \n\n\\textbf{Proof}. The absolute  comonotonicity of the disjoint-pair Lov\\'asz extension implies that $f^L$ is linear on $\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{B_i}:i=1,\\cdots,k\\}$ whenever $A_1\\subset \\cdots\\subset A_k$ and $B_1\\subset \\cdots\\subset B_k$ and $A_k\\cap B_k=\\varnothing$. Thus, for any $\\sigma,\\tau:\\{1,\\cdots,k\\}\\to \\{1,\\cdots,k\\}$  with $\\sigma(1)\\le \\cdots\\le \\sigma(k)\\le\\tau(k)\\le\\cdots\\le\\tau(1)$, $f^L$ is linear on $\\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\}$. Note that\n\\begin{align*}\n\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{V\\setminus A_i}:i=1,\\cdots,k\\}&=\\bigcup_{\\sigma,\\tau} \\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\} \n\\\\&=\\bigcup_{l=1}^k\\bigcup_{\\sigma(1)=1,\\sigma(k)=\\tau(k)=l,\\tau(1)=k}\\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\}\n\\end{align*}\nwhere in the second line we can  further assume that each $\\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\}$ is a $(k-1)$-dim simplex, and there are exactly $\\sum_{l=1}^k{k-1\\choose l-1}=2^{k-1}$ simplexes of dimension $(k-1)$.  \nSince\n\\begin{align*}\n2f^L(\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}})\n&=2f(A_{\\sigma(i)},V\\setminus A_{\\tau(i)})\n\\\\&=f(A_{\\sigma(i)},V\\setminus A_{\\sigma(i)})+f(A_{\\tau(i)},V\\setminus A_{\\tau(i)})\n\\\\&=\nf^L(\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\sigma(i)}})+f^L(\\vec1_{A_{\\tau(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}})\n\\end{align*}\nand $\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\sigma(i)}}+\\vec1_{A_{\\tau(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}=2(\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}})$, $f^L$ must be linear on the segment $ \\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\sigma(i)}},\\vec1_{A_{\\tau(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}\\}$. Therefore, one can check that $f^L$ is linear on the simplex  $\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{V\\setminus A_i}:i=1,\\cdots,k\\}$. \n\n\\vspace{0.16cm}\n\nGiven an eigenvalue $\\lambda$, let $\\vec 1_A-\\vec 1_B$ be a corresponding eigenvector for some $(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$. By the above claim and argument, it can be verified that  both $\\vec1_A-\\vec 1_{V\\setminus A}$ and  $\\vec 1_{V\\setminus B}-\\vec1_B$ are eigenvectors w.r.t. $\\lambda$.  \n\\end{proof}\n\n\\begin{remark}\nIn Proposition \\ref{pro:Lovasz-eigen}, the  condition $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$ is natural and easy to satisfy. Below, we provide two examples satisfying the condition. \n\nExample 1. $f(A,B)=\\hat{f}(A)+\\hat{f}(B)$ for some $\\hat{f}:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ with $\\hat{f}(A)=\\hat{f}(V\\setminus A)$ for any $A\\subset V$. \n In this case,  $f^L(\\vec x)=\\hat{f}^L(\\vec x)$. \n\nExample 2. $f(A,B)=1$ whenever $A\\cup B\\ne\\varnothing$. In this case,  $f^L(\\vec x)=\\|\\vec x\\|_\\infty$.  \n\nOne may observe  that in the above examples, $f$ is symmetric, i.e.,\n$f(A,B)=f(B,A)$, but it is not a sufficient condition for Proposition\n\\ref{pro:Lovasz-eigen}. In fact, for symmetric functions  $f$ and $g$ on\n$\\ensuremath{\\mathcal{P}}_2(V)$,  not every  eigenvalue has an eigenvector possessing the form\nof $\\vec 1_A-\\vec1_{V\\setminus A}$. In fact, taking $f(A,B)=\\#E(A,V\\setminus\nA)+\\#E(B,V\\setminus B)$ and $g(A,B)=\\vol(A\\cup B)$, we have $f^L(\\vec\nx)=\\sum_{\\{i,j\\}\\in E}|x_i-x_j|$  and $g^L(\\vec x)=\\sum_{i\\in\n  V}\\deg(i)|x_i|$. Letting $V=\\{1,2,3\\}$ and $E=\\{\\{1,2\\},\\{2,3\\},\\{1,3\\}\\}$,\nit is known in \\cite{Chang16} that $\\vec 1_A-\\vec1_{V\\setminus A}$ cannot be\nan eigenvector w.r.t. the largest eigenvalue of the  1-Laplacian, $\\forall A\\subset V$.\n\\end{remark}\n\n We conclude the following  result, which asserts that every\nvector\nin $\\{-1,1\\}^n$ is an eigenvector of $(f^L,\\|\\cdot\\|_\\infty)$ if $f^L$ is nonnegative,  where $f^L$ indicates the disjoint-pair Lov\\'asz extension of $f$. \n\n\\begin{pro}\\label{pro:set-pair-infty-norm}\nLet $f:\\ensuremath{\\mathcal{P}}_2(V)\\to[0,+\\infty)$ be nonnegative. Then, for any $A\\subset V$,  $\\vec 1_A-\\vec 1_{V\\setminus A}$ is an eigenvector of  $(f^L,\\|\\cdot\\|_\\infty)$.\n\\end{pro}\n\n\\begin{proof}\nBy the definition of the eigenvalue problem of $(f^L,\\|\\cdot\\|_\\infty)$, we only need to prove that $\\nabla f^L(\\vec x)\\cap ((-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})\\cup \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}) \\ne \\varnothing$ for any $\\vec x\\in\\{-1,1\\}^n$, where $\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}:=\\{\\vec y\\in\\ensuremath{\\mathbb{R}}^n:y_ix_i\\ge 0,\\forall i\\}=\\mathrm{cone}(\\nabla \\|\\vec x\\|_\\infty)$. We first assume that $f$ is positive-definite, i.e., $f(A,B)>0$ whenever $(A,B)\\ne(\\varnothing,\\varnothing)$, and we shall apply the following argument about polar cones to this case. \n\n\\vspace{0.16cm}\n\n\\textbf{Argument}. Let $C$ and $\\Omega$ be two convex cones in $\\ensuremath{\\mathbb{R}}^n$ such that $\\Omega\\cap ((-C)\\cup C)=\\{\\vec0\\}$. Then $\\Omega^*\\cap C^*\\ne\\{\\vec 0\\}$ and $\\Omega^*\\cap (-C^*)\\ne\\{\\vec 0\\}$, where $C^*$ indicates the polar cone of $C$. \n\nProof: Indeed, $\\Omega^*\\cap C^*=(\\Omega\\cup C)^*\\supset (\\Omega+ C)^*$, where $\\Omega+ C$ is the Minkowski summation of $C$ and $\\Omega$. If $\\Omega+ C=\\ensuremath{\\mathbb{R}}^n$, then for any $-\\vec c\\in (-C)\\setminus\\{\\vec0\\}$, there exist $\\vec a\\in\\Omega\\setminus\\{\\vec0\\}$ and $\\vec c'\\in C\\setminus\\{\\vec0\\}$ such that $\\vec a+\\vec c'=-\\vec c$. This implies $\\vec a=-\\vec c'-\\vec c\\in -C\\setminus\\{\\vec0\\}$, which contradicts  the condition that $\\Omega\\cap(-C)=\\{\\vec0\\}$. Therefore, the convex cone $\\Omega+ C$ is not the whole space $\\ensuremath{\\mathbb{R}}^n$, which implies that $(\\Omega+ C)^*\\ne\\{\\vec0\\}$. Consequently, $\\Omega^*\\cap C^*\\ne\\{\\vec 0\\}$ and similarly, $\\Omega^*\\cap (-C^*)\\ne\\{\\vec 0\\}$. The proof is completed.\n\n\\vspace{0.16cm}\n\nSuppose on the contrary, that $\\nabla f^L(\\vec x)\\cap ((-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})\\cup \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}) =\\varnothing$ for some $\\vec x\\in\\{-1,1\\}^n$. \nFixing such an $\\vec x$, then $\\mathrm{cone}(\\nabla f^L(\\vec x))\\cap ((-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})\\cup \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}) =\\{\\vec0\\}$, and by the above  argument, we have  $\\mathrm{cone}^*(\\nabla f^L(\\vec x))\\cap\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}=\\mathrm{cone}^*(\\nabla f^L(\\vec x))\\cap(-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})^* \\ne\\{\\vec0\\}$. \n\nHowever, since $f$ is positive-definite, it is known \nthat  $\\mathrm{cone}^*(\\nabla f^L(\\vec x))\\subset T_x(\\{\\vec y:f^L(\\vec y)\\le f^L(\\vec x)\\})$, meaning that $ T_x(\\{\\vec y:f^L(\\vec y)\\le f^L(\\vec x)\\})\\cap \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}\\ne\\{\\vec0\\}$, where $T_x$ represents the tangent cone at $\\vec x$. Now, suppose $\\vec x=\\vec 1_{A_n}-\\vec 1_{B_n}$ with $A_n\\sqcup B_n=V$. Every  permutation  $\\sigma:\\{1,\\cdots,n\\}\\to \\{1,\\cdots,n\\}$ determines a  sequence $\\{(A_i,B_i):i=1,\\cdots,n\\}\\subset \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$ by the iterative construction:   $A_1\\cup B_1=\\{\\sigma(1)\\}$ and  $A_{i+1}\\cup B_{i+1}=A_i\\cup B_i\\cup \\{\\sigma(i+1)\\}$, $i=1,\\cdots,n-1$. \n\nSince $f(A_i,B_i)>0$, $f^L(\\vec x^i)=1$ where $\\vec x^i:=(\\vec1_{A_i}-\\vec1_{B_i})\/f(A_i,B_i)$, $i=1,\\cdots,n$. Also, $T_{x^n}\\{\\vec y:f^L(\\vec y)\\le 1\\}=T_x(\\{\\vec y:f^L(\\vec y)\\le f^L(\\vec x)\\})$.\nWithout loss of generality, we may assume that $\\vec x=\\vec x^n$.\n\nThe definition of $f^L$ yields that $\\mathrm{conv}(\\vec0,\\vec x^1,\\cdots,\\vec x^n)\\subset \\{\\vec y:f^L(\\vec y)\\le 1\\}$.  We denote by $\\triangle_\\sigma=\\mathrm{conv}(\\vec0,\\vec x^1,\\cdots,\\vec x^n)$ since the  construction of $\\vec x^1,\\cdots,\\vec x^n$ depends on the permutation  $\\sigma$. \nFor any $\\vec y=\\sum_{i=1}^nt_i\\vec x^i\\in\\mathrm{conv}(\\vec0,\\vec x^1,\\cdots,\\vec x^n)\\setminus \\{\\vec x\\}$, \n$(\\vec y-\\vec x)_{\\sigma(n)}x_{\\sigma(n)}=-(1-t_n)x_{\\sigma(n)}^2<0$, and thus  $\\vec y-\\vec x\\not\\in \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}$.\nHence, $T_x(\\triangle_\\sigma)\\cap \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}=\\{\\vec0\\}$. It follows from the fact  $T_x(\\{\\vec y:f^L(\\vec y)\\le 1\\})=\\bigcup_{\\sigma}T_x(\\triangle_\\sigma)$ that $T_x(\\{\\vec y:f^L(\\vec y)\\le 1\\})\\cap \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}=\\{\\vec0\\}$. This is a contradiction. \n\n\\vspace{0.16cm}\n\nNow we turn to the general case that $f\\ge 0$. Take a sequence $\\{f_n\\}_{n\\ge 1}$ of  positive-definite functions on $\\ensuremath{\\mathcal{P}}_2(V)$ such that $f_n\\to f$ as $n$ tends to $+\\infty$. Then it can be verified that for any $\\vec v_n\\in \\nabla f_n^L(\\vec x)$, all limit  points of $\\{\\vec v_n\\}_{n\\ge 1}$ belong to $\\nabla f^L(\\vec x)$. Now, there exist $\\vec u_n\\in \\nabla \\|\\vec x\\|_{\\infty}$ and $\\lambda_n=f^L_n(\\vec x)\/\\|\\vec x\\|_{\\infty}>0$ such that $\\lambda_n\\vec u_n\\in \\nabla f^L_n(\\vec x)$. Then for any limit point $\\vec u$ of $\\{\\vec u_n\\}_{n\\ge 1}$, $\\vec u\\in  \\nabla \\|\\vec x\\|_{\\infty}$ and $\\lambda\\vec u\\in \\nabla f^L(\\vec x)$ where $\\lambda=\\lim\\limits_{n\\to+\\infty}\\lambda_n$. Therefore, $(\\lambda,\\vec x)$ is an eigenpair of $(f^L,\\|\\cdot\\|_\\infty)$.\n\nThe proof is completed. \n\\end{proof}\n\nBy Propositions \\ref{pro:Lovasz-eigen} and  \\ref{pro:set-pair-infty-norm}, we have \n\\begin{cor}\nIf $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then the set of  eigenvalues of $(f^L,\\|\\cdot\\|_\\infty)$ coincides with $\\{f(A,V\\setminus A):A\\subset V\\}$, and every vector in $\\{-1,1\\}^n$ is an eigenvector.\n\\end{cor}\n\n\\begin{remark}\nThe proof of  Proposition  \\ref{pro:set-pair-infty-norm} heavily depends on\nthe property  that $N_v(X)=-T_v(X)$ for any vertex $v$ of the  hypercube  $X:=\\{\\vec x:\\|\\vec x\\|_\\infty\\le1\\}$.  Characterizing  the class of  polytopes satisfying   $N_v=-T_v$ for any vertex $v$  remains an open problem, where $N_v$ is the normal cone at $v$ and $T_v$ is the tangent  cone at $v$. \n\\end{remark}\n\nMotivated by Propositions \\ref{pro:Lovasz-eigen} and  \\ref{pro:set-pair-infty-norm}, we suggest a combinatorial eigenvalue problem for $(f,g)$ as follows:\n\nGiven $A\\subset V$ and a permutation  $\\sigma:\\{1,\\cdots,n\\}\\to \\{1,\\cdots,n\\}$, there exists a unique sequence $\\{(A_i^\\sigma,B_i^\\sigma):i=1,\\cdots,n\\}\\subset \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$   satisfying $A_1^\\sigma\\subset \\cdots \\subset A_n^\\sigma=A$, $B_1^\\sigma\\subset \\cdots \\subset B_n^\\sigma=V\\setminus A$, $A_1^\\sigma\\cup B_1^\\sigma=\\{\\sigma(1)\\}$ and  $A_{i+1}^\\sigma\\cup B_{i+1}^\\sigma=A_i^\\sigma\\cup B_i^\\sigma\\cup \\{\\sigma(i+1)\\}$, $i=1,\\cdots,n-1$.\nLet $\\vec u^{A,\\sigma}\\in\\ensuremath{\\mathbb{R}}^n$ be defined by\n$$u^{A,\\sigma}_i=\\begin{cases} f(A_{\\sigma^{-1}(i)}^\\sigma,B_{\\sigma^{-1}(i)}^\\sigma)-f(A_{\\sigma^{-1}(i)-1}^\\sigma,B_{\\sigma^{-1}(i)-1}^\\sigma)\n,&\\text{ if }i\\in A,\\\\\nf(A_{\\sigma^{-1}(i)-1}^\\sigma,B_{\\sigma^{-1}(i)-1}^\\sigma)-f(A_{\\sigma^{-1}(i)}^\\sigma,B_{\\sigma^{-1}(i)}^\\sigma)\n,&\\text{ if }i\\not\\in A.\\end{cases}\n$$\nDenote by $S(f,A)=\\{\\vec u^{A,\\sigma}:\\sigma\\in S_n\\}$ and\n$$\\nabla f(A,B):=\\mathrm{conv}\\left(\\bigcup\\limits_{\\tilde{A}:\\,A\\subset \\tilde{A}\\subset V\\setminus B}S(f,\\tilde{A})\\right),\\;\\;\\forall (A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)$$\nwhere $S_n$ is the permutation group over $\\{1,\\cdots,n\\}$.\n\n\\begin{defn}[Combinatorial eigenvalue problem]\nGiven $f,g:\\ensuremath{\\mathcal{P}}_2(V)\\to \\ensuremath{\\mathbb{R}}$, the combinatorial eigenvalue problem of $(f,g)$ is to find $\\lambda\\in\\ensuremath{\\mathbb{R}}$ and $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$ such that $\\nabla f(A,B)\\cap \\lambda \\nabla g(A,B)\\ne\\varnothing$.\n\\end{defn}\n\nSince it can be verified that $\\nabla f^L(\\vec1_A-\\vec1_B)=\\nabla f(A,B)$, Proposition \\ref{pro:Lovasz-eigen} (or Theorem \\ref{introthm:eigenvalue}) implies that the combinatorial eigenvalue problem for $(f,g)$ is equivalent to the nonlinear eigenvalue problem of $(f^L,g^L)$.\n\n{ By Propositions \\ref{pro:Lovasz-eigen-pre} and \\ref{pro:Lovasz-eigen}, for a pair of functions $f$ and $g$ on $\\ensuremath{\\mathcal{P}}(V)$ (resp.,  $\\ensuremath{\\mathcal{P}}_2(V)$), every eigenvalue of the function pair   $(f^L,g^L)$ generated by Lov\\'asz extension has an eigenvector of the form $\\vec1_A$ (resp., $\\vec1_A-\\vec1_B$) for some $A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$ (resp., $(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$). We may call such a set $A$ (resp., $(A,B)$) an  eigen-set of $(f,g)$. And, we are interested in the eigen-sets and the corresponding eigenvalues, which  encode the \nkey information about the data structure generated by the function pair $(f,g)$.     }  \n\nNext, we study the second eigenvalue of the function pair $(f^L,g^L)$, which\nis closely related to a combinatorial Cheeger-type constant  of the form \n$$\\ensuremath{\\mathrm{Ch}}(f,g):=\\min\\limits_{A\\in\\mathcal{P}(V)\\setminus\\{\\varnothing,V\\}}\\frac{f(A)}{\\min\\{g(A),g(V\\setminus A)\\}}$$\nwhere $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is symmetric, i.e., $f(A)=f(V\\setminus A)$, $\\forall A$, and $g:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}_+$ is  submodular and non-decreasing. \n\n\\begin{pro}\\label{eq:Cheeger-identity}\nLet $f_s,g_s:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ be defined by $f_s(A,B)=f(A)+f(B)$ and $g_s(A,B)=g(A)+g(B)$. Then \n$$\\ensuremath{\\mathrm{Ch}}(f,g)=\\text{the second eigenvalue of the function pair }(f_s^L,g_s^L)\\;(\\text{or equivalently }(f_s,g_s)).$$\n\\end{pro}\n\nWe need the following  auxiliary proposition.\n\\begin{pro}\\label{pro:original-vs-disjoint-pair}Suppose that $g:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}_+$ is non-decreasing, i.e., $g(A)\\le g(B)$ whenever $A\\subset B$. \nLet $G:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be the disjoint-pair Lov\\'asz extension of the function $(A,B)\\mapsto g(A)+g(B)$.   Then the  Lov\\'asz extension of the function $A\\mapsto\\min\\{g(A),g(V\\setminus A)\\}$ is $\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}G(\\vec x-t\\vec1)$.\n\\end{pro}\n\\begin{proof}\nDenote by $g_{m}(A)=\\min\\{g(A),g(V\\setminus A)\\}$ and $g_s(A,B)=g(A)+g(B)$, where $g_{m}^L$ is the original Lov\\'asz extension of  $g_{m}$, and $g_s^{L}$ is the disjoint-pair Lov\\'asz extension of $g_s$. \nSince $g$ is non-decreasing, $g(V^t(\\vec x))$ must be non-increasing on $t\\in \\ensuremath{\\mathbb{R}}$, i.e., $g(V^{t_1}(\\vec x))\\ge g(V^{t_2}(\\vec x))$ whenever $t_1\\le t_2$. Hence, there exists $t_0\\in\\ensuremath{\\mathbb{R}}$ such that $g(V^t(\\vec x))\\ge g(V\\setminus V^t(\\vec x))$, $\\forall t\\le t_0$; and   $g(V^t(\\vec x))\\le g(V\\setminus V^t(\\vec x))$, $\\forall t\\ge t_0$.\nThen \\begin{align*}\ng_{m}^L(\\vec x)&= \\int_{\\min \\vec x}^{\\max \\vec x}g_{m}(V^t(\\vec x))dt+\\min\\vec x g_{m}(V)\n\\\\&=\\int_{\\min \\vec x}^{t_0}g(V\\setminus V^t(\\vec x))dt+\\int_{t_0}^{\\max\\vec x}g(V^t(\\vec x))dt\n\\\\&=\\int_{\\min (\\vec x-t_0\\vec1)}^0g(V\\setminus V^t(\\vec x-t_0\\vec1))dt+\\int_{0}^{\\max(\\vec x-t_0\\vec1)}g(V^t(\\vec x-t_0\\vec1))dt\n\\\\&=\\int_{-\\|\\vec x-t_0\\vec1\\|_\\infty}^0g(V\\setminus V^t(\\vec x-t_0\\vec1))dt+\\int_{0}^{\\|\\vec x-t_0\\vec1\\|_\\infty}g(V^t(\\vec x-t_0\\vec1))dt\n\\\\&=\\int_{0}^{\\|\\vec x-t_0\\vec1\\|_\\infty}g(V^t(\\vec x-t_0\\vec1))+g(V\\setminus V^{-t}(\\vec x-t_0\\vec1))dt\n\\\\&=g_{s}^{L}(\\vec x-t_0\\vec1)=\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^{L}(\\vec x-t\\vec1).\n\\end{align*}\nThe proof is completed. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{eq:Cheeger-identity}]\nSince $f$ is symmetric, by Proposition \\ref{pro:setpair-original}, $f_s^L(\\vec x)=f^L(\\vec x)=f_m^L(\\vec x)$. \n\nSince $g$ is positive,  submodular and non-decreasing, it is not difficult to check that\n$g_s$ is bisubmodular.  Thus, by the equivalence of submodularity and convexity, $g_s^L$ is a convex function. \nTherefore, we have\n$$\\min\\limits_{\\vec x\\bot\\vec 1}\\frac{f_s^L(\\vec x)}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^L(\\vec x-t\\vec 1)}=\\min\\limits_{\\text{nonconstant }\\vec x\\in\\ensuremath{\\mathbb{R}}_+^n}\\frac{f_s^L(\\vec x)}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^L(\\vec x-t\\vec 1)}=\\min\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}_+^n:\\min\\vec x=0}\\frac{f_m^L(\\vec x)}{g_m^L(\\vec x)}=\\min\\limits_{A\\ne\\varnothing,V}\\frac{f_m(A)}{g_m(A)}=\\ensuremath{\\mathrm{Ch}}(f,g),$$\nwhere the first equality is based on the fact that $\\vec x\\mapsto f_s^L(\\vec x)=f^L(\\vec x)$ and $\\vec x\\mapsto\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^L(\\vec x-t\\vec 1)$ are translation invariant along $\\vec1$, the second equality is derived by Proposition \\ref{pro:original-vs-disjoint-pair}, and the third one follows from Theorem \\ref{thm:tilde-fg-equal}.\n\\end{proof}\nFinally, we prove that for any $A,B\\ne \\varnothing$ with $A\\cap B=\\varnothing$,  $$\\max\\{\\frac{f(A)}{g(A)},\\frac{f(B)}{g(B)}\\}\\ge\\min\\{\\frac{f(A)}{\\min\\{g(A),g(V\\setminus A)\\}},\\frac{f(B)}{\\min\\{g(B),g(V\\setminus B)\\}}\\}.$$\nSuppose the contrary, and keep $f(A)=f(V\\setminus A)$ in mind. Then, we have $g(A)>g(V\\setminus A)$ and $g(B)>g(V\\setminus B)$, implying $g(A)+g(B)>g(V\\setminus A)+g(V\\setminus B)$.\nSince $A\\subset V\\setminus B$ and $g$ is non-decreasing, one has $g(A)\\le g(V\\setminus B)$. Similarly, $g(B)\\le g(V\\setminus A)$, which leads to a contradiction. \n\nCombining all the results and discussions in this section, we complete the proof of Theorem \\ref{introthm:eigenvalue}. \n\n\n\\subsection{A relaxation of a Dinkelbach-type scheme}\n \\label{sec:algo}\n\nWe would like to establish an iteration framework for finding minimum and maximum eigenvalues.  These extremal eigenvalues play significant roles in optimization theory. They can be found via the  so-called Dinkelbach iterative scheme \\cite{D67}. This will provide  a good starting point for an appropriate\niterative algorithm for the resulting fractional programming. Actually, the equivalent continuous optimization has a fractional form, but such kind of fractions have been hardly touched in the field of fractional programming \\cite{SI83}, where optimizing the ratio of a concave function to a convex one is usually considered. For convenience, we shall work in  a normed space $X$ in this subsection.\n\nFor a convex function $F:X\\to \\mathbb{R}$,  its sub-gradient (or sub-derivative) $\\nabla F(\\vec x)$ is defined as the collection of $\\vec u\\in X^*$ satisfying  $F(\\vec y)-F(\\vec x)\\ge \\langle \\vec u,\\vec y-\\vec x\\rangle,\\;\\forall \\vec y\\in X$, where $X^*$ is the dual of $X$ and $\\langle \\vec u,\\vec y-\\vec x\\rangle$ is the action of $\\vec u$ on $\\vec y-\\vec x$.\nThe concept of a sub-gradient has been extended to Lipschitz functions. This is called the  Clarke derivative \\cite{Clarke}:\n $$\\nabla F(\\vec x)=\\left\\{\\vec u\\in X^*\\left|\\limsup_{\\vec y\\to \\vec x, t\\to 0^+}\\frac{F(\\vec y+t\\vec h)-F(\\vec y)}{t}\\ge \\langle \\vec u,\\vec h\\rangle,\\forall \\vec h\\in X\\right.\\right\\}.$$\n And it can even be  generalized to the class of lower semi-continuous functions \\cite{DM94,D10}.\n\n\\begin{theorem}[Global convergence of a Dinkelbach-type  scheme \\cite{D67}] \\label{thm:global convergence}\nLet $S$ be a compact set and let $F,G:S\\to\\mathbb{R}$ be two continuous functions with $G(\\vec x)>0$, $\\forall \\vec x\\in S$. Then the sequence $\\{r^k\\}$ generated by the two-step iterative scheme\n\\begin{numcases}{}\n\\vec x^{k+1}=\\argopti\\limits_{\\vec  x\\in S} \\{F(\\vec x)-r^k G(\\vec x)\\}, \\label{iter0-1}\n\\\\\nr^{k+1}=\\frac{F( \\vec x^{k+1})}{G(\\vec x^{k+1})},\n\\label{iter0}\n\\end{numcases}\n from any initial point $\\vec  x^0\\in S$, converges monotonically to a global optimum of $F(\\cdot)\/G(\\cdot)$,\nwhere `opti' is `min' or `max'.\n\\end{theorem}\n\n\\begin{cor}\nIf $F\/G$ is a zero-homogeneous continuous function, then the iterative scheme \\eqref{iter0-1}\\eqref{iter0} from any initial point $\\vec  x^0$ converges monotonically to a global optimum on the cone spanned by $S$ (i.e., $\\{t\\vec x: t>0, \\vec x\\in S\\}$).\n\\end{cor}\n\nWe note that Theorem \\ref{thm:global convergence} generalizes Theorem 3.1 in \\cite{CSZ15} and Theorem 2 in \\cite{CSZ18}.  Since it is a Dinkelbach-type iterative algorithm in the field of fractional programming, we omit the proof\n\nMany minimization problems in the field of fractional programming possess the form\n$$\n\\min\\,\\frac{\\text{convex }F}{\\text{concave }G},\n$$\nwhich is not necessary for a  convex programming problem. The original Dinkelbach iterative scheme turns the ratio form to the inner problem \\eqref{iter0-1} with the form lik\n$$\n\\min \\; (\\text{convex }F-\\text{concave }G),\n$$\nwhich is indeed a convex programming problem. However, most of our examples are in the form\n$$\n\\min\\frac{\\text{convex }F}{\\text{convex }G},\n$$\ni.e., both the numerator and the denominator of the fractional object function are convex.\nSince the difference of two convex functions may not be convex, the inner\nproblem \\eqref{iter0-1} is no longer a convex optimization problem and hence might be very difficult to solve.\n\n\nIn other practical applications, we may encounter optimization problems of the form\n$$\n\\min\\frac{\\text{convex }F_1 - \\text{convex }F_2}{\\text{convex }G_1- \\text{convex }G_2}.\n$$\nThis is NP-hard in general. Fortunately, we can construct an effective relaxation of \\eqref{iter0-1}.\n\n\nThe starting point of the relaxation step is the following  classical fact:\n\\begin{pro}\\label{pro:difference-two-submodular}\nFor any function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$, there are two submodular functions $f_1$ and $f_2$ on $\\ensuremath{\\mathcal{A}}$ such that $f=f_1-f_2$.\n\\end{pro}\n\nAlthough this is  an old result,  for readers' convenience, we present a short\nproof below. \n\n\\begin{proof}\nTaking $g$ to be a strict submodular function and letting $$\\delta=\\min\\limits_{A,A'\\in \\ensuremath{\\mathcal{A}}}\\left(g(A)+g(A')-g(A\\vee A')-g(A\\wedge A')\\right)>0.$$ Set $f_2=Cg$ and $f_1=f+f_2$ for a sufficiently large $C>0$. It is clear that $f_2$ is strict submodular and $f_1$ is submodular. So, $f=f_1-f_2$, which completes the proof.\n\\end{proof}\n\nThanks to Proposition \\ref{pro:difference-two-submodular}, any discrete function can be expressed as the difference of two submodular functions. Since the Lov\\'asz extension of a submodular function is convex, every Lov\\'asz extension function is the difference of two convex functions. \n\nThen, for the fractional programming derived by Theorem \\ref{thm:tilde-fg-equal} (or Propositions  \\ref{pro:fraction-f\/g} and \\ref{pro:maxconvex}),  both the numerator\nand denominator can be rewritten as the differences of two convex functions. This  implies  that a simple and efficient\nalgorithm can be obtained via further relaxing the Dinkelbach iteration by\ntechniques in DC Programming \\cite{HT99}. It should be noted that the following recent works (especially the papers by Hein's group  \\cite{HeinBuhler2010,HS11,TVhyper-13,TMH18}) motivated us to investigate more on this direction:\n\\begin{enumerate}\n\\item  The efficient generalization of the inverse power method proposed by Hein et al  \\cite{HeinBuhler2010} and the extended steepest descent \nmethod by Bresson et al  \\cite{BLUB12}  deal with fractional programming  in the same spirit. For more relevant papers by Hein's group, we refer to \\cite{HS11} for the RatioDCA method, and  \\cite{TMH18} for the generalized RatioDCA technique.\n\\item In \\cite{MaeharaMurota15,MMM18}, the authors address difference convex programming (DC programming)\nfor discrete convex functions, in which  an algorithm and a convergence result similar to Theorem \\ref{th:gsd} are presented. \n\n\\item A simple iterative algorithm based on the continuous reformulation by the disjoint-pair Lov\\'asz extension provides the best cut values for maxcut on a G-set among all existing continuous algorithms \\cite{SZZmaxcut}. \n\\end{enumerate}\n In view of these recent  developments, and in order to enlarge the scope of fractional programming and RatioDCA method, it is helpful to study this\n  aspect by general formulations (see also Remark \\ref{remark:very-general-RatioDCA} for the most general form).   \n  Thus,  \nwe begin to establish a method based on convex programming for solving $\\mathtt{i}\n\\frac{F(\\vec x)}{G(\\vec x)}$ with $F=F_1-F_2$ and $G=G_1-G_2$ being two nonnegative functions, where $F_1,F_2,G_1,G_2$ are four  nonnegative convex functions on $X$. Let $\\{H_{\\vec y}(\\vec x):\\vec y\\in X\\}$ be a family of convex differentiable functions on $X$ with $H_{\\vec y}(\\vec x)\\ge H_{\\vec y}(\\vec y)$, $\\forall \\vec x\\in X$. Consider the following three-step iterative scheme\n\\begin{subequations}\n\\label{iter1}\n\\begin{numcases}{}\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\mathbb{B}} \\{F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\}, \\label{eq:twostep_x2}\n\\\\\nr^{k+1}=F( \\vec x^{k+1})\/G( \\vec x^{k+1}),\n\\label{eq:twostep_r2}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\n\\label{eq:twostep_s2}\n\\end{numcases}\n\\end{subequations}\nwhere $\\mathbb{B}$ is a convex body containing $\\vec 0$ as its inner point.  The following slight modification  \n\\begin{subequations}\n\\label{iter2}\n\\begin{numcases}{}\n\\vec  y^{k+1}\\in \\argmin\\limits_{\\vec x\\in X} \\{F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\}, \\label{eq:2twostep_x2}\n\\\\\nr^{k+1}=F( \\vec y^{k+1})\/G( \\vec y^{k+1}),~~ \\vec x^{k+1}=\\partial \\mathbb{B}\\cap\\{t\\vec y^{k+1}:t\\ge 0\\} \n\\label{eq:2twostep_r2}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\n\\label{eq:2twostep_s2}\n\\end{numcases}\n\\end{subequations}\n is available when $F\/G$ is zero-homogeneous and \\eqref{eq:2twostep_x2} has a\n solution. In  \\eqref{eq:2twostep_r2}, $\\vec x^{k+1}$ indicates  the\n normalization of $\\vec y^{k+1}$ w.r.t. the convex body $\\mathbb{B}$; in  particular, $\\vec x^{k+1}:=\\vec y^{k+1}\/\\|\\vec y^{k+1}\\|_2$ if we let  $\\mathbb{B}$ be the unit ball.     \n These schemes  mixing the inverse power (IP) method and steepest descent (SD) method can be well used in computing special eigenpairs of $(F,G)$. Note that the inner problem \\eqref{eq:twostep_x2} (resp.  \\eqref{eq:2twostep_x2}) is a convex optimization and thus many algorithms in convex programming are applicable. We should note that the above schemes provide a generalization of the RatioDCA  technique in \\cite{HS11}.\n\n\\begin{theorem}[Local convergence for the  mixed IP-SD scheme]\\label{th:gsd}\nThe sequence $\\{r^k\\}$ generated by the iterative scheme \\eqref{iter1} (resp. \\eqref{iter2}) from any initial point $\\vec x^0\\in \\mathrm{supp}(G)\\cap \\mathbb{B}$ (resp. $\\vec x^0\\in \\mathrm{supp}(G)$) converges monotonically, where $\\mathrm{supp}(G)$ is the support of $G$.\n\nNext we further assume that $X$ is of finite dimension. If one of the following  additional  conditions  holds, then  $\\lim_{k\\to+\\infty} r^k=r^*$ is an eigenvalue of the function pair $(F,G)$ \nin the sense that it fulfills $\\vec0\\in \\nabla F_1(\\vec x^*)-\\nabla F_2(\\vec x^*)-r^*\\left(\\nabla G_1(\\vec x^*)-\\nabla G_2(\\vec x^*)\\right)$,  where $\\vec x^*$ is a  cluster point of $\\{\\vec x^k\\}$.\n\n\\begin{itemize}\n\\item[Case 1.] For the  scheme \\eqref{iter1}, $F_2$ and $G_1$ are one-homogeneous, and  $F_1$ and $G_2$ are $p$-homogeneous with $p\\ge 1$, and $H_{\\vec x}=\\text{const}$, $\\forall \\vec x\\in\\mathbb{B}$. \n\\item[Case 2.1.]  For the  scheme \\eqref{iter2}, $F_1$, $F_2$, $G_1$ and $G_2$ are $p$-homogeneous with  $p>1$.\n\\item[Case 2.2.]  For the  scheme \\eqref{iter2}, $F_1$, $F_2$, $G_1$ and $G_2$\n  are one-homogeneous, and $H_{\\vec x}(\\vec x)$ is a continuous\n  function of $\\vec x\\in \\mathbb{B}$ and  $\\forall M>0$,  $\\exists C>0$ such that   $H_{\\vec x}(\\vec y)>M \\|\\vec y\\|_2$  whenever $\\vec x\\in\\mathbb{B}$ and  $\\|\\vec y\\|_2\\ge C$. \n\\end{itemize}\n\n\\end{theorem}\n\n\nTheorem \\ref{th:gsd} partially generalizes  Theorem 3.4 in  \\cite{CSZ15},\nand it is indeed an extension of both the IP and the SD method \\cite{BLUB12,CP11,M16,HeinBuhler2010}. \n\n\n\\begin{proof}[Proof of Theorem \\ref{th:gsd}]\n\nIt will be helpful to divide this proof into several parts and steps:\n\n\\begin{enumerate}\n\\item[Step 1.] We may assume $G(\\vec x^k)>0$ for any $k$.\nIn fact, the initial point $\\vec x^0$ satisfies $G(\\vec x^0)> 0$. We will show $F(\\vec x^1)=0$ if $G(\\vec x^1)=0$ and thus the iteration should be terminated at $\\vec x^1$. This tells us that we may assume $G(\\vec x^k)>0$ for all $k$ before the termination of the iteration.\n\nNote that\n\\begin{align*}&F_1(\\vec x^1)+r^0 G_2(\\vec x^1) -(\\langle \\vec u^0,\\vec x^1\\rangle+r^0 \\langle \\vec v^0,\\vec x^1\\rangle) + H_{\\vec x^0}(\\vec x^1)\\\\ \\le~& F_1(\\vec x^0)+r^0 G_2(\\vec x^0) -(\\langle \\vec u^0,\\vec x^0\\rangle+r^0 \\langle \\vec v^0,\\vec x^0\\rangle) + H_{\\vec x^0}(\\vec x^0),\n\\end{align*}\nwhich implies\n\\begin{align*}&F_1(\\vec x^1)-F_1(\\vec x^0)+r^0 (G_2(\\vec x^1)-G_2(\\vec x^0)) + H_{\\vec x^0}(\\vec x^1)-H_{\\vec x^0}(\\vec x^0)\\\\ \\le~& \\langle \\vec u^0,\\vec x^1-\\vec x^0\\rangle+r^0 \\langle \\vec v^0,\\vec x^1-\\vec x^0\\rangle\\le F_2(\\vec x^1)-F_2(\\vec x^0) +r^0 (G_1(\\vec x^1)-G_1(\\vec x^0)),\n\\end{align*}\ni.e.,\n\\begin{align}F(\\vec x^1)-F(\\vec x^0)+  H_{\\vec x^0}(\\vec x^1)-H_{\\vec x^0}(\\vec x^0)&\\le r^0 (G(\\vec x^1)-G(\\vec x^0))\\label{eq:important-inequality}\\\\&=-r^0G(\\vec x^0)=-F(\\vec x^0).\\notag\n\\end{align}\nSince the equality holds, we have $F(\\vec x^1)=0$, $H_{\\vec x^0}(\\vec x^1)=H_{\\vec x^0}(\\vec x^0)$, $\\langle \\vec u^0,\\vec x^1-\\vec x^0\\rangle=F_2(\\vec x^1)-F_2(\\vec x^0)$ and $\\langle \\vec v^0,\\vec x^1-\\vec x^0\\rangle=G_1(\\vec x^1)-G_1(\\vec x^0)$. So this step is finished.\n\n\n\n\\item[Step 2.] $\\{r^k\\}_{k=1}^\\infty$ is monotonically decreasing and hence convergent.\n\nSimilar to \\eqref{eq:important-inequality} in Step 1, we can arrive at\n$$F(\\vec x^{k+1})-F(\\vec x^k)+  H_{\\vec x^k}(\\vec x^{k+1})-H_{\\vec x^k}(\\vec x^k) \\le r^k (G(\\vec x^{k+1})-G(\\vec x^k)),$$\nwhich leads to\n$$F(\\vec x^{k+1})\\le r^k G(\\vec x^{k+1}).$$\nSince $G(\\vec x^{k+1})$ is assumed to be positive,\n$r^{k+1}=F(\\vec x^{k+1})\/G(\\vec x^{k+1})\\le r^k$.\n Thus, there exists $r^*\\in [r_{\\min},r^0]$ such that $\\lim\\limits_{k\\to+\\infty}r^k=r^*$.\n\\end{enumerate}\n\nIn the sequel, we assume that the  dimension of $X$ is finite.\n\n\\begin{enumerate}\n\\item[Step 3.]   $\\{\\vec x^k\\}$, $\\{\\vec u^k\\}$ and $\\{\\vec v^k\\}$ are sequentially compact.\n\nIn this setting, $\\mathbb{B}$ must be compact. In consequence, there exist $k_i$, $r^*$, $\\vec x^*$,  $\\vec x^{**}$, $\\vec u^*$ and $\\vec v^*$ such that $\\vec x^{k_i}\\to \\vec x^*$, $\\vec x^{k_i+1}\\to \\vec x^{**}$, $\\vec u^{k_i}\\to \\vec u^*$ and $\\vec v^{k_i}\\to \\vec v^*$, as $i\\to +\\infty$.\n\n Clearly, the statements in Steps 1, 2 and 3 are also available for the  scheme \\eqref{iter2}.\n\n\\item[Step 4.] For the  scheme \\eqref{iter1},   $\\vec x^*$ is a minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$.  For the  scheme  \\eqref{iter2}, under the additional assumptions introduced in  Case 2.1 or Case  2.2, $\\vec x^*$ is a minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $X$. \n\nLet $g(r,\\vec y,\\vec u,\\vec v)=\\min\\limits_{  \\vec x\\in \\mathbb{B}} \\{F_1(\\vec x)+r G_2(\\vec x) -(\\langle \\vec u,\\vec x\\rangle+r \\langle \\vec v,\\vec x\\rangle) + H_{\\vec y}(\\vec x)\\}$. It is standard to verify that $g(r,\\vec y,\\vec u,\\vec v)$ is continuous on $\\mathbb{R}^{1}\\times X\\times X^*\\times X^*$ according to the compactness of $\\mathbb{B}$.\n\nSince $g(r^{k_i},\\vec x^{k_i},\\vec u^{k_i},\\vec v^{k_i})=r^{k_i+1}$, taking $i\\to+\\infty$, one obtains $g(r^*,\\vec x^*,\\vec u^*,\\vec v^*)=r^*$.\n\nBy Step 3, $\\vec x^{**}$ attains the minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$. Suppose the contrary, that $\\vec x^*$ is not a minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$. Then\n\\begin{align*}&F_1(\\vec x^{**})+r^* G_2(\\vec x^{**}) -(\\langle \\vec u^*,\\vec x^{**}\\rangle+r^* \\langle \\vec v^*,\\vec x^{**}\\rangle) + H_{\\vec x^*}(\\vec x^{**})\\\\ <~& F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle+r^* \\langle \\vec v^*,\\vec x^*\\rangle) + H_{\\vec x^*}(\\vec x^*),\n\\end{align*}\nand thus $F(\\vec x^{**})<r^* G(\\vec x^{**})$ (similar to Step 1), which implies $G(\\vec x^{**})>0$ and $F(\\vec x^{**})\/ G(\\vec x^{**})<r^*$. This is a contradiction. Consequently, $  \\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$.\n\n On the scheme  \\eqref{iter2}, we refer to   Cases 2.1 and 2.2 below for details.  \n\n\nNext, we will\nverify that $(r^*,\\vec x^*)$ is an eigenpair  under certain  additional conditions. \n\n\\item[Case 1.]  On  the scheme \\eqref{iter1}, $F_2$ and $G_1$ are one-homogeneous, and  $F_1$ and $G_2$ are $p$-homogeneous with $p\\ge 1$, and $H_{\\vec x}=Const$, $\\forall \\vec x\\in\\mathbb{B}$. \n\nSince $H_{\\vec x^k}=Const$,  the above claim shows that $\\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle +r^* \\langle \\vec v^*,\\vec x\\rangle)$ on $\\mathbb{B}$. Also, since $F_2$ and $G_1$ are one-homogeneous, the Euler identity on homogeneous functions gives $F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle +r^* \\langle \\vec v^*,\\vec x^*\\rangle)=F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(F_2(\\vec x^*)+r^* G_1(\\vec x^*))=F(\\vec x^*)-r^* G(\\vec x^*)=0$. Thus, $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)\\ge 0$, $\\forall \\vec x\\in \\mathbb{B}$, and the equality holds when $\\vec x=\\vec x^*$.  \n\n\nSince $\\mathbb{B}$ contains $0$ as its inner point, we have $\\{\\alpha \\vec x: \\vec x\\in \\mathbb{B},\\alpha\\ge 1\\}=X$.\n Keeping  $\\alpha\\ge 1$ and $p\\ge 1$ in mind, for any $\\alpha\\ge 1$ and $\\vec x\\in \\mathbb{B}$,\n \\begin{align*}\n &F_1(\\alpha \\vec x)+r^* G_2(\\alpha \\vec x) -(\\langle \\vec u^*,\\alpha \\vec x\\rangle+r^* \\langle \\vec v^*,\\alpha \\vec x\\rangle)\n  \\\\ =~& \\alpha\\left(F_1( \\vec x)+r^* G_2( \\vec x) -(\\langle \\vec u^*, \\vec x\\rangle+r^* \\langle \\vec v^*, \\vec x\\rangle)\\right)+(\\alpha^p-\\alpha)(F_1( \\vec x)+r^* G_2( \\vec x))\n  \\\\ (\\text{by Step 4})~\\ge~& (\\alpha^p-\\alpha)(F_1( \\vec x)+r^* G_2( \\vec x)) \\ge 0.\n\n\n \\end{align*}\n\nConsequently, $\\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)$ on $X$, and thus\n\\begin{align*}\n\\vec 0 &\\in \\nabla|_{  \\vec x=  \\vec x^*} \\left(F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) \\right)\n\\\\&=\\nabla F_1(\\vec x^*)+r^* \\nabla G_2(\\vec x^*) -\\vec u^*-r^*\\vec v^*\n\\\\& \\subset \\nabla F_1(\\vec x^*)-\\nabla F_2(\\vec x^*)+r^*\\nabla G_2(\\vec x^*)-r^*\\nabla G_1(\\vec x^*).\n\\end{align*}\n\n\\item[Case 2.1.] On the  scheme \\eqref{iter2}, $F_1$, $F_2$, $G_1$ and $G_2$ are $p$-homogeneous with  $p>1$. \n\nDenote by $B:X\\to[0,+\\infty)$ the unique convex and one-homogeneous function satisfying $B(\\partial\\mathbb{B})=1$. Then the normalization of $\\vec x$ in  \\eqref{eq:2twostep_r2} can be expressed as $\\vec x\/B(\\vec x)$.\n\nThe compactness of $\\{\\vec x:B(\\vec x)\\le 1\\}$ and the upper semi-continuity and compactness of subderivatives imply that $\\bigcup_{\\vec x:B(\\vec x)\\le 1}\\nabla F_2(\\vec x)$ and $\\bigcup_{\\vec x:B(\\vec x)\\le 1}\\nabla G_1(\\vec x)$ are bounded sets. So, we have a uniform constant $C_1>0$ such that $\\|\\vec u\\|_2+r^*\\|\\vec v\\|_2\\le C_1$, $\\forall \\vec u\\in \\nabla F_2(\\vec x)$, $ \\vec v\\in \\nabla G_1(\\vec x)$, $\\forall \\vec x\\in\\mathbb{B}$. Let $C_2>0$ be such that  $\\|\\vec x\\|_2\\le C_2B(\\vec x)$, and  $C_3=\\min\\limits_{B(\\vec x)=1} F_1(\\vec x)>0$ (here we assume without loss of generality that $F_1(\\vec x)>0$ whenever $\\vec x\\ne \\vec 0$). For any $\\vec x$ with $B(\\vec x)\\ge \\max\\{2,(2C_1C_2\/C_3)^{\\frac{1}{p-1}}\\}$, and for any $\\vec x^*\\in \\mathbb{B}$, $ \\vec u^*\\in \\nabla F_2(\\vec x^*)$, $ \\vec v^*\\in \\nabla G_1(\\vec x^*)$, \n\\begin{align*}\n&F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^*\\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)\n\\\\ =~&   B(\\vec x)^p F_1(\\frac{\\vec x}{B(\\vec x)})+r^*B(\\vec x)^p G_2(\\frac{\\vec x}{B(\\vec x)}) -(\\|\\vec x\\|_2\\langle \\vec u^*,\\frac{\\vec x}{\\|\\vec x\\|_2}\\rangle+r^*\\|\\vec x\\|_2\\langle \\vec v^*,\\frac{\\vec x}{\\|\\vec x\\|_2}\\rangle) + H_{\\vec x^*}(\\vec x)\n\\\\ \\ge~& B(\\vec x)^pF_1(\\frac{\\vec x}{B(\\vec x)})-\\|\\vec x\\|_2(\\|\\vec u^*\\|_2+r^*\\|\\vec v^*\\|_2 ) + H_{\\vec x^*}(\\vec x^*)\n\\\\ \\ge~& B(\\vec x)^pC_3-C_2C_1B(\\vec x) + H_{\\vec x^*}(\\vec x^*)=B(\\vec x)(B(\\vec x)^{p-1}C_3-C_2C_1) + H_{\\vec x^*}(\\vec x^*)\n> H_{\\vec x^*}(\\vec x^*)\\\\ >~& -(p-1)(F_2(\\vec x^*)+r^* G_1(\\vec x^*))+ H_{\\vec x^*}(\\vec x^*)\n\\\\ =~& F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle+r^*\\langle \\vec v^*,\\vec x^*\\rangle) + H_{\\vec x^*}(\\vec x^*)\n\\end{align*}\nwhich means that the  minimizers of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^*\\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ exist and they always lie in the bounded set $\\{\\vec x:B(\\vec x)< \\max\\{2,(2C_1C_2\/C_3)^{\\frac{1}{p-1}}\\}\\}$. Since $B(\\vec x^k)=1$, $\\{\\vec y^k\\}$ must be a bounded sequence.  There exists  $\\{k_i\\}\\subset \\{k\\}$ such that $\\vec x^{k_i} \\to \\vec x^*$, $\\vec y^{k_i+1}\\to \\vec y^{**}$, $\\vec x^{k_i+1} \\to \\vec x^{**}$ for some $\\vec x^*$, $\\vec y^{**}$ and $\\vec x^{**}=\\vec y^{**}\/B(\\vec y^{**})$. Similar to Step 4 and  Case 1,  $\\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)+H_{\\vec x^*}(\\vec x)$ on $X$, and thus\n\\begin{align*}\n\\vec 0 &\\in \\nabla|_{  \\vec x=  \\vec x^*} \\left(F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)+H_{\\vec x^*}(\\vec x) \\right)\n\\\\&=\\nabla F_1(\\vec x^*)+r^* \\nabla G_2(\\vec x^*) -\\vec u^*-r^*\\vec v^*\n\\subset \\nabla F_1(\\vec x^*)-\\nabla F_2(\\vec x^*)+r^*\\nabla G_2(\\vec x^*)-r^*\\nabla G_1(\\vec x^*)\n\\end{align*}\n\n\\item[Case 2.2.] On the scheme \\eqref{iter2},  $F_1$, $F_2$, $G_1$ and $G_2$ are one-homogeneous; $H_x(\\vec x)$ is continuous of $\\vec x\\in \\mathbb{B}$ and for any $M>0$, there exists $C>0$ such that $H_{\\vec x}(\\vec y)>M\\cdot B(\\vec y)$ whenever $\\vec x\\in\\mathbb{B}$ and  $B(\\vec y)\\ge C$. \n\nTaking $M=C_1C_2+2$ in which the constants  $C_1$ and $C_2$ are introduced in Case 2.1, there exists $C>\\max\\{\\max\\limits_{x\\in \\mathbb{B}}H_x(\\vec x),1\\}$ such that $H_{\\vec x^*}(\\vec x)\\ge M\\cdot B(\\vec x)$ whenever $\\vec x^*\\in \\mathbb{B}$ and $B(\\vec x)\\ge C$.  \n\nSimilar to Case 2.1, for any $\\vec x^*\\in \\mathbb{B}$,  $\\vec x\\in X$ with $B(\\vec x)\\ge C$, and $\\forall \\vec u^*\\in \\nabla F_2(\\vec x^*)$, $ \\vec v^*\\in \\nabla G_1(\\vec x^*)$, \n\\begin{align*}\n&F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^*\\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)\n\\\\>~& B(\\vec x)(C_3-C_2C_1) + (C_1C_2+2)\\cdot B(\\vec x)\n\\ge 2 B(\\vec x) >H_{\\vec x^*}(\\vec x^*)\n\\\\ =~& F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle+r^*\\langle \\vec v^*,\\vec x^*\\rangle) + H_{\\vec x^*}(\\vec x^*).\n\\end{align*}\nThe remaining part can refer to  Case 2.1.\n   \\end{enumerate}\n  \n\\end{proof}\n\n\\begin{remark}\\label{remark:very-general-RatioDCA}\nAs some direct extensions of the so-called {\\sl generalized RatioDCA} in \\cite{TMH18}, we have the following modified schemes:\n\\begin{subequations}\n\\label{iter1-}\n\\begin{numcases}{}\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\mathbb{B}} F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k\\ge0, \\label{eq:twostep_x2-}\n\\\\\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\mathbb{B}} G_1(\\vec x)-\\langle \\vec w^k,\\vec x\\rangle-\\frac{1}{r^k}( F_1(\\vec x) - \\langle \\vec u^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k<0, \\label{eq:twostep_x22-}\n\\\\\nr^{k+1}=F( \\vec x^{k+1})\/G( \\vec x^{k+1}),\n\\label{eq:twostep_r2-}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\\;\\vec w^{k+1}\\in\\nabla G_2( \\vec x^{k+1})\n\\label{eq:twostep_s2-}\n\\end{numcases}\n\\end{subequations}\nand\n\\begin{subequations}\n\\label{iter2-}\n\\begin{numcases}{}\n\\vec  y^{k+1}\\in \\argmin\\limits_{\\vec x\\in X} F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k\\ge0, \\label{eq:2twostep_x2-}\n\\\\\n\\vec  y^{k+1}\\in \\argmin\\limits_{\\vec x\\in X} G_1(\\vec x)-\\langle \\vec w^k,\\vec x\\rangle-\\frac{1}{r^k}( F_1(\\vec x) - \\langle \\vec u^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k<0, \\label{eq:2twostep_x22-}\n\\\\\nr^{k+1}=F( \\vec y^{k+1})\/G( \\vec y^{k+1}),~~ \\vec x^{k+1}=\\partial \\mathbb{B}\\cap\\{t\\vec y^{k+1}:t\\ge 0\\} \n\\label{eq:2twostep_r2-}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\n\\label{eq:2twostep_s2-}\n\\end{numcases}\n\\end{subequations}\nin which the previous assumption $F_1-F_2\\ge 0$ in \\eqref{iter1} and \\eqref{iter2}  has been removed. For these modifications, a convergence property like Theorem \\ref{th:gsd} still holds.\n\n\\end{remark}\n\n\n\n\\begin{remark}\nTheorem \\ref{th:gsd} shows the local convergence of a general relaxation of  Dinkelbach's algorithm  in the spirit of DC programming.\n The DC programming  consists in  minimizing $F-G$ where $F$ and $G$ are convex functions. \nAs described in \\cite{MaeharaMurota15,MMM18}, both the original DC algorithm and its discrete version can be written as the simple iteration: $\\vec u^k\\in \\nabla G(\\vec x^k)$, $\\vec x^{k+1}\\in \\nabla F^\\star(\\vec u^k)$, where $F^\\star$ is the Fenchel conjugate of $F$. It is known that such an iteration is equivalent to the following scheme  \\begin{subequations}\n\\label{F-Giter1}\n\\begin{numcases}{}\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x} F(\\vec x) -\\langle \\vec u^k,\\vec x\\rangle, \\label{eq:F-Gtwostep_x2}\n\\\\ \\vec u^{k+1}\\in\\nabla G( \\vec x^{k+1}).\n\\label{eq:F-Gtwostep_s2}\n\\end{numcases}\n\\end{subequations}\nMoreover,\na slight variation of the above scheme by adding a normalization step\n\\begin{subequations}\n\\label{FGiter1}\n\\begin{numcases}{}\n  \\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^n} F(\\vec x) -\\langle \\vec u^k,\\vec x\\rangle, \\label{eq:FGtwostep_x2}\n\\\\ \\vec x^{k+1}=  \\hat{\\vec x}^{k+1}\/G(\\hat{\\vec x}^{k+1})^{\\frac1p}\n\\\\ \\vec u^{k+1}\\in\\nabla G( \\vec x^{k+1}).\n\\label{eq:FGtwostep_s2}\n\\end{numcases}\n\\end{subequations}\ncan be used to solve the fractional programming $\\min F\/G$, where $F$ and $G$ are convex and $p$-homogeneous with $p>1$. This scheme is nothing but\nAlgorithm 2 in \\cite{HeinBuhler2010}.\nIn fact, we can say more about it. \n\\end{remark}\n\\begin{pro}\nLet $F$ and $G$ be convex,   $p$-homogeneous and positive-definite functions on $\\ensuremath{\\mathbb{R}}^n$, where $p>1$.  Then, for any  initial point $\\vec x^0$, the sequence of the pairs $\\{(r^k,\\vec x^k)\\}_{k\\ge1}$ produced by the following scheme\n\\begin{subequations}\n\\label{iter1FG}\n\\begin{numcases}{}\n\\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n} F(\\vec x) -a_k\\langle \\vec u^k,\\vec x\\rangle, \\label{eq:twostep_x2FG}\n\\\\\n\\vec x^{k+1}= b_{k+1}  \n\\hat{\\vec x}^{k+1}\\; (\\mathrm{scaling}),\\;\\; r^{k+1}=F( \\vec x^{k+1})\/G( \\vec x^{k+1}),\n\\label{eq:twostep_r2FG}\n\\\\\n \\vec u^{k+1}\\in\\nabla G( \\vec x^{k+1}),\n\\label{eq:twostep_s2FG}\n\\end{numcases}\n\\end{subequations}\nconverges to an eigenpair $(r^*,\\vec x^*)$ of $(F,G)$ in the sense that $\\lim\\limits_{k\\to+\\infty}r^k= r^*$ and $\\vec x^*$ is a limit point of $\\{\\vec x^k\\}_{k\\ge1}$,  whenever $a_k,b_k>0$ as well as  both $\\{a_k\\}_{k\\ge 1}$ and  $\\{b_k\\hat{\\vec x}^k\\}_{k\\ge 1}$\n are  bounded away from $0$ and $\\infty$.\n\\end{pro}\n\nThe proof is very similar to the original proof of Theorem 3.1 in  \\cite{HeinBuhler2010},  with an additional trick  like the proof of Case 2.1 in Theorem \\ref{th:gsd}. It can be regarded as a supplement of both Theorem 3.1 in  \\cite{HeinBuhler2010} and Theorem \\ref{th:gsd}. It is also interesting that the scheme is stable under   perturbations of $a_k$ and $b_k$. Besides, it can be seen that the resulting eigenvalue $r^*$ should be  independent of the choice of $a_k$ and $b_k$. In fact, $r^*$ only depends on the initial data and the choice of subgradient $\\vec u^k$. \nThe assumption that  $F$ is positive-definite can be removed in some sense. Indeed, if $r^k\\le0$ for some $k$, we can modify \\eqref{eq:twostep_x2FG} as $\\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n} F(\\vec x) -r^kG(\\vec x)$ or $\\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n} G(\\vec x) -\\frac{1}{r^k}F(\\vec x)$ when $r^k<0$. Then $\\{r^k\\}$  converges to the global minimum of $F\/G$.\n\n\nAnother solver for the continuous optimization $\\min\\frac{F(\\vec x)}{G(\\vec x)}$ is\n the stochastic subgradient method:\n\\begin{equation*}\n\\vec x^{k+1}=\\vec x^k-\\alpha_k(\\vec y^k+\\vec\\xi^k),\\;\\;\\;\\vec y^k\\in\\nabla\\frac{F(\\vec x^k)}{G(\\vec x^k)},\n\\end{equation*}\nwhere $\\{\\alpha_k\\}_{k\\ge1}$ is a step-size sequence and $\\{\\vec\\xi^k\\}_{k\\ge1}$ is now a sequence of random variables (the ``noise'') on some probability space. Theorem 4.2 in \\cite{Davis19-FoCM} shows that under some natural assumptions, almost surely, every limit point of the stochastic subgradient iterates $\\{\\vec x^k\\}_{k\\ge1}$ is critical for $F\/G$, and the function values $\\{\\frac{F}{G}(\\vec x^k)\\}_{k\\ge1}$ converge.\n\n\\section{Examples and Applications}\n\\label{sec:examples-Applications}\n\n\n\\subsection{Submodular vertex cover and multiway partition problems}\nAs a first immediate application of Theorem \\ref{thm:tilde-fg-equal}, we obtain an easy way to rediscover the famous identity by Lov\\'asz, and the two typical submodular optimizations -- submodular vertex cover and multiway partition problems.\n\\begin{example}\nThe identity $\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}f(A)=\\min\\limits_{\\vec x\\in[0,1]^V}f^L(\\vec x)$ discovered by Lov\\'asz in his original paper \\cite{Lovasz} can be obtained by our result. In fact, $$\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}f(A)=\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}\\frac{f(A)}{1}=\\min\\limits_{\\vec x\\in [0,\\infty)^V}\\frac{f^L(\\vec x)}{\\max\\limits_{i\\in V}x_i}=\\min\\limits_{\\vec x\\in [0,1]^V}\\frac{f^L(\\vec x)}{\\max\\limits_{i\\in V}x_i}=\\min\\limits_{\\vec x\\in [0,1]^V,\\max\\limits_i x_i=1}f^L(\\vec x).$$\nChecking this is easy:\n if $f\\ge 0$, then $\\min\\limits_{\\vec x\\in [0,1]^V,\\max\\limits_i x_i=1}f^L(\\vec x)=0$; if $f(A)<0$ for some $A\\subset V$, then $\\min\\limits_{\\vec x\\in [0,1]^V,\\max\\limits_i x_i=1}f^L(\\vec x)=\\min\\limits_{\\vec x\\in [0,1]^V}f^L(\\vec x)$.\n\\end{example}\n\n\\paragraph{Vertex cover number }\nA vertex cover (or node cover) of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. The vertex cover number is the minimal cardinality of a  vertex cover. Similarly, the independence number of a graph is the maximal number of vertices not connected by edges. The sum of the vertex cover number and the independence number is  the  cardinality of the vertex set.\n\nBy a variation of the Motzkin-Straus theorem and Theorem \\ref{thm:graph-numbers}, the vertex cover number thus has at least two equivalent continuous representations similar to the independence number.\n\n\\paragraph{Submodular vertex cover problem}\nGiven a graph $G=(V,E)$, and a submodular function $f:\\ensuremath{\\mathcal{P}}(V)\\to[0,\\infty)$,  find a vertex cover $S\\subset V$ minimizing $f(S)$.\n\nBy Theorem \\ref{thm:tilde-fg-equal},\n$$\\min\\{f(S):S\\subset V,\\,S\\text{ is a vertex cover}\\}=\\min\\limits_{\\vec x\\in{\\mathcal D}}\\frac{f^L(\\vec x)}{\\|\\vec x\\|_\\infty}=\\min\\limits_{\\vec x\\in \\widetilde{{\\mathcal D}}}f^L(\\vec x)$$\nwhere ${\\mathcal D}=\\{\\vec x\\in[0,\\infty)^V:V^t(\\vec x)\\text{ vertex cover},\\,\\forall t\\ge0\\}=\\{\\vec x\\in[0,\\infty)^V:x_i+x_j>0,\\forall\\{i,j\\}\\in E,\\,\\{i:x_i=\\max_j x_j\\}\\text{ vertex cover}\\}$,  and $\\widetilde{{\\mathcal D}}=\\{\\vec x\\in{\\mathcal D}: \\|\\vec x\\|_\\infty=1\\}=\\{\\vec x\\ge\\vec 0:x_i+x_j\\ge 1,\\forall\\{i,j\\}\\in E,\\,\\{i:x_i=\\max_j x_j\\}\\text{ vertex cover}\\}$. Note that\n$$\\mathrm{conv}(\\widetilde{{\\mathcal D}})=\\{\\vec x:x_i+x_j\\ge 1,\\forall\\{i,j\\}\\in E,\\,x_i\\ge 0,\\forall i\\in V\\}.$$\n\nTherefore, $\\min\\limits_{\\vec x\\in \\mathrm{conv}(\\widetilde{{\\mathcal D}})}f^L(\\vec x)\\le \\min\\{f(S):\\text{ vertex cover }S\\subset V\\}$, which rediscovers the convex programming relaxation.\n\n\\paragraph{Submodular multiway partition problem}\nThis problem  is about to minimize $\\sum_{i=1}^k f(V_i)$ subject to $V=V_1\\cup\\cdots\\cup V_k$, $V_i\\cap V_j=\\varnothing$, $i\\ne j$, $v_i\\in V_i$, $i=1,\\cdots,k$, where $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is a submodular function.\n\n Letting $\\ensuremath{\\mathcal{A}}=\\{\\text{ partition }(A_1,\\cdots,A_k)\\text{ of }V:A_i\\ni a_i,\\,i=1,\\cdots,k\\}$, by Theorem \\ref{thm:tilde-fg-equal},\n $$\\min\\limits_{(A_1,\\cdots,A_k)\\in\\ensuremath{\\mathcal{A}}}\\sum_{i=1}^k f(A_i)=\\inf\\limits_{\\vec x\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{\\sum_{i=1}^k f^L(\\vec x^i)}{\\|\\vec x\\|_\\infty}=\\inf\\limits_{\\vec x\\in {\\mathcal D}'}\\sum_{i=1}^k f^L(\\vec x^i),$$\n  where ${\\mathcal D}_\\ensuremath{\\mathcal{A}}=\\{\\vec x\\in [0,\\infty)^{kn}: (V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))\\text{ is a partition}, V^t(\\vec x^i)\\ni a_i,\\forall t\\ge 0\\}=\\{\\vec x\\in [0,\\infty)^{kn}: \\vec x^i=t1_{A_i},A_i\\ni a_i,\\forall t\\ge 0\\}$, and ${\\mathcal D}'=\\{(\\vec x^1,\\cdots,\\vec x^k):\\vec x^i\\in [0,\\infty)^V,\\,\\vec x^i=\\vec 1_{A_i},A_i\\ni a_i\\}$. Note that $$\\mathrm{conv}({\\mathcal D}')=\\{(\\vec x^1,\\cdots,\\vec x^k):\\sum_{v\\in V} x^i_v=1,x^i_{a_i}=1,x^i_v\\ge0\\}.$$ So one rediscovers the corresponding convex programming relaxation $\\min\\limits_{\\vec x\\in \\mathrm{conv}({\\mathcal D}')}\\sum_{i=1}^k f^L(\\vec x^i)$.\n  \n\\subsection{Min-cut and Max-cut}\n\nGiven an undirected weighted   graph $(V,E,w)$, the  min-cut problem  \n$$\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|:=\\min\\limits_{S\\ne\\varnothing,V}|E(S,V\\setminus S)|=\\min\\limits_{S\\ne\\varnothing,V}\\sum\\limits_{i\\in S,j\\in V\\setminus S}w_{ij}$$\n and the \nmax-cut problem  $$\\max\\limits_{S\\ne\\varnothing,V}|\\partial S|:=\\max\\limits_{S\\ne\\varnothing,V}|E(S,V\\setminus S)|=\\max\\limits_{S\\ne\\varnothing,V}\\sum\\limits_{i\\in S,j\\in V\\setminus S}w_{ij}$$\nhave been  investigated systematically.\n\n\n\n\\begin{theorem}\\label{thm:mincut-maxcut-eigen}Let $(V,E,w)$ be a weighted  undirected graph. \nThen, we have the equivalent continuous optimization formulations for the min-cut and max-cut problems:\n$$\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|=\\min\\limits_{\\min _ix_i+\\max_i x_i=0}\\frac{\\sum_{ij\\in E}w_{ij}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}=\\tilde{\\lambda}_2,$$\n$$\\max\\limits_{S\\ne\\varnothing,V}|\\partial S|= \\max\\limits_{\\vec x\\ne \\vec0}\\frac{\\sum_{ij\\in E}w_{ij}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}=\\tilde{\\lambda}_{\\max},$$\nwhere $\\tilde{\\lambda}_2$ and $\\tilde{\\lambda}_{\\max}$ are the second \n(i.e., the smallest  nontrivial)  eigenvalue and the largest eigenvalue of the nonlinear eigenvalue problem:\n\\begin{equation}\\label{eq:mincut-maxcut-eigen}\n\\vec0\\in \\nabla\\sum_{ij\\in E}w_{ij}|x_i-x_j|-\\lambda\\nabla 2\\|\\vec x\\|_\\infty.    \n\\end{equation}\\end{theorem}\n\n\\begin{proof}\nWe only prove the min-cut case. It is clear that\n$$ \\min\\limits_{S\\ne\\varnothing,V}|\\partial S|= \\min\\limits_{A,B\\ne\\varnothing,A\\cap B=\\varnothing}\\frac{ |\\partial A|+|\\partial B|}{2}$$\nLet $\\ensuremath{\\mathcal{A}}=\\{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V):A,B\\ne\\varnothing\\}$. Then ${\\mathcal D}_\\ensuremath{\\mathcal{A}}=\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^n: \\max_i x_i=-\\min_i x_i>0\\}$, and by Theorem  \\ref{thm:tilde-fg-equal}, $$\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|=\\min\\limits_{(A,B)\\in\\ensuremath{\\mathcal{A}}}\\frac{ |\\partial A|+|\\partial B|}{2}=\\min\\limits_{\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{\\sum_{ij\\in E}w_{ij}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}. $$ \n In addition,  according to Theorem \\ref{introthm:eigenvalue}, the set of the eigenvalues of $(f^L,g^L)$ coincides with \n $$\\left\\{\\frac{f^L(\\vec 1_A-\\vec 1_{V\\setminus A})}{g^L(\\vec 1_A-\\vec 1_{V\\setminus A})}:A\\subset V\\right\\}=\\left\\{\\frac{f(A,V\\setminus A)}{g(A,V\\setminus A)}:A\\subset V\\right\\}=\\left\\{\\frac{|\\partial A|+|\\partial (V\\setminus A)|}{2}:A\\subset V\\right\\}=\\{|\\partial A|:A\\subset V\\},$$\n where $f(A,B)=|\\partial A|+|\\partial B|$ and $g(A,B)=2$. In consequence, $\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|$ is the second \neigenvalue  of $(f^L,g^L)$. The proof is completed.\n\\end{proof}\n\nEq.~\\eqref{eq:mincut-maxcut-eigen} shows the first nonlinear eigenvalue problem which possesses two nontrivial eigenvalues that are equivalent to two important graph optimization problems, respectively. \n\nIn addition, by our results, we  present a lot of equivalent continuous optimizations for the maxcut problem (see Examples \\ref{exam:maxcut1} and \\ref{exam:maxcut2}):\n\\begin{align*}\n\\max\\limits_{S\\subset V}|\\partial S|&=\\max\\limits_{x\\ne\n  0}\\frac{ \\sum_{\\{i,j\\}\\in E}w_{ij}(|x_i|+|x_j|-|x_i+x_j|)^p }{(2\\|\\vec\n  x\\|_\\infty)^p}\n\\\\&=\\max\\limits_{x\\ne\n  0}\\frac{ \\sum_{\\{i,j\\}\\in E}w_{ij}|x_i-x_j|^p }{(2\\|\\vec\n  x\\|_\\infty)^p}=\\max\\limits_{\\|\\vec\n  x\\|_\\infty\\le\\frac12 }  \\sum_{\\{i,j\\}\\in E}w_{ij}|x_i-x_j|^p  \n\\end{align*}\nfor any $p\\ge1$.  \n\n\\subsection{Max $k$-cut problem}\n\\label{sec:max-k-cut}\nThe max $k$-cut problem is to determine a graph $k$-cut by solving\n\\begin{equation}\\label{eq:maxk}\n\\mathrm{MaxC}_k(G)=\\max_{\\text{partition }(A_1,A_2,\\ldots,A_k)\\text{ of }V}\\sum_{i\\ne j}|E(A_i,A_j)|=\\max_{(A_1,A_2,\\ldots,A_k)\\in \\mathcal{C}_{k}(V)}\\sum_{i=1}^k|\\partial A_i|,\n\\end{equation}\nwhere $\\mathcal{C}_{k}(V)=\\{(A_1,\\ldots,A_k)\\big|A_i\\cap A_j = \\varnothing, \\bigcup_{i=1}^{k} A_i= V \\}$,  and $\\partial A_i:=E(A_i,V\\setminus A_i)$. We may write \\eqref{eq:maxk} as\n$$ \\mathrm{MaxC}_k(G)=\\max_{(A_1,A_2,\\ldots,A_{k-1})\\in \\mathcal{P}_{k-1}(V)}\\sum_{i=1}^{k-1}|\\partial A_i|+|\\partial (A_1\\cup\\cdots\\cup A_{k-1})|.$$\nTaking $f_k(A_1,\\cdots,A_k)=\\sum_{i=1}^{k}|\\partial A_i|+|\\partial (A_1\\cup\\cdots\\cup A_{k})|$, the $k$-way Lov\\'asz extension is $$f^L_k(\\vec x^1,\\cdots,\\vec x^k)=\\sum_{i=1}^k\\sum_{i\\sim j}|x^k_i-x^k_j|+\\sum_{j\\sim j'}\\left|\\max\\limits_{i=1,\\cdots,k} x^i_j-\\max\\limits_{i=1,\\cdots,k} x^i_{j'}\\right|.$$\nApplying  Theorem \\ref{thm:tilde-fg-equal}, we have\n$$ \\mathrm{MaxC}_{k+1}(G)=\\max\\limits_{\\vec x^i\\in\\ensuremath{\\mathbb{R}}^n_{\\ge0}\\setminus\\{\\vec0\\},\\,\\ensuremath{\\mathrm{supp}}(\\vec x^i)\\cap \\ensuremath{\\mathrm{supp}}(\\vec x^j)=\\varnothing}\\frac{\\sum_{i=1}^k\\sum_{i\\sim j}|x^k_i-x^k_j|+\\sum_{j\\sim j'}\\left|\\max\\limits_{i=1,\\cdots,k} x^i_j-\\max\\limits_{i=1,\\cdots,k} x^i_{j'}\\right|}{\\max\\limits_{i,j}x^i_j}$$\n\\subsection{Relative isoperimetric constants on a subgraph with boundary}\n\\label{sec:boundary-graph-1-lap}\nGiven a finite graph $G=(V,E)$ and a subgraph, we\nconsider the Dirichlet and Neumann eigenvalue problems for the corresponding 1-Laplacian. For $A\\subset V$, put $\\overline{A}=A\\cup \\delta A$, where $\\delta A$ is the set of points in  $A^c$ that are adjacent to some points in $A$ (see Fig.~\\ref{fig:AdeltaA}).\n\n\\begin{figure}[!h]\\centering\n\\begin{tikzpicture}[scale=0.69\n\\draw (0,0) to (1,0);\n\\draw (0,0) to (0,1);\n\\draw (1,0) to (2,0);\n\\draw (1,0) to (1,1);\n\\draw (1,1) to (0,1);\n\\draw (0,2) to (0,1);\n\\draw (0,2) to (1,2);\n\\draw (2,0) to (2,1);\n\\draw (2,2) to (1,2);\n\\draw (2,2) to (2,1);\n\\draw (1,1) to (1,2);\n\\draw (1,1) to (2,1);\n\\draw (0,0) to (-0.9,0);\n\\draw (0,0) to (0,-0.9);\n\\draw (1,0) to (1,-0.9);\n\\draw (2,0) to (2,-0.9);\n\\draw (0,2) to (-0.9,2);\n\\draw (0,1) to (-0.9,1);\n\\draw[densely dotted] (-2,2) to (-1.1,2);\n\\draw[densely dotted] (-2,1) to (-1.1,1);\n\\draw[densely dotted] (-2,0) to (-1.1,0);\n\\draw[densely dotted] (4,2) to (3.1,2);\n\\draw[densely dotted] (4,1) to (3.1,1);\n\\draw[densely dotted] (4,0) to (3.1,0);\n\\draw[densely dotted] (2,-2) to (2,-1.1);\n\\draw[densely dotted] (1,-2) to (1,-1.1);\n\\draw[densely dotted] (0,-2) to (0,-1.1);\n\\draw[densely dotted] (2,4) to (2,3.1);\n\\draw[densely dotted] (1,4) to (1,3.1);\n\\draw[densely dotted] (0,4) to (0,3.1);\n\\draw[densely dotted] (-0.1,3) to (-1,3);\n\\draw[densely dotted] (-0.1,-1) to (-1,-1);\n\\draw[densely dotted] (2.1,3) to (3,3);\n\\draw[densely dotted] (2.1,-1) to (3,-1);\n\\draw[densely dotted] (3,-0.1) to (3,-1);\n\\draw[densely dotted] (-1,-0.1) to (-1,-1);\n\\draw[densely dotted] (3,2.1) to (3,3);\n\\draw[densely dotted] (-1,2.1) to (-1,3);\n\\draw[dashed] (-1,1.9) to (-1,1.1);\n\\draw[dashed] (-1,0.9) to (-1,0.1);\n\\draw[dashed] (3,1.9) to (3,1.1);\n\\draw[dashed] (3,0.9) to (3,0.1);\n\\draw[dashed] (1.9,-1) to (1.1,-1);\n\\draw[dashed] (0.9,-1) to (0.1,-1);\n\\draw[dashed] (1.9,3) to (1.1,3);\n\\draw[dashed] (0.9,3) to (0.1,3);\n\\node (00) at (0,0) {$\\bullet$};\n\\node (10) at (1,0) {$\\bullet$};\n\\node (11) at (1,1) {$\\bullet$};\n\\node (01) at (0,1) {$\\bullet$};\n\\node (02) at (0,2) {$\\bullet$};\n\\node (20) at (2,0) {$\\bullet$};\n\\node (12) at (1,2) {$\\bullet$};\n\\node (21) at (2,1) {$\\bullet$};\n\\node (22) at (2,2) {$\\bullet$};\n\\node (03) at (0,3) {$\\circ$};\n\\node (30) at (3,0) {$\\circ$};\n\\node (13) at (1,3) {$\\circ$};\n\\node (31) at (3,1) {$\\circ$};\n\\node (23) at (2,3) {$\\circ$};\n\\node (32) at (3,2) {$\\circ$};\n\\node (01') at (0,-1) {$\\circ$};\n\\node (1'0) at (-1,0) {$\\circ$};\n\\node (11') at (1,-1) {$\\circ$};\n\\node (1'1) at (-1,1) {$\\circ$};\n\\node (21') at (2,-1) {$\\circ$};\n\\node (1'2) at (-1,2) {$\\circ$};\n\\draw (2.9,0) to (2,0);\n\\draw (0,2.9) to (0,2);\n\\draw (2.9,1) to (2,1);\n\\draw (1,2.9) to (1,2);\n\\draw (2.9,2) to (2,2);\n\\draw (2,2.9) to (2,2);\n\\end{tikzpicture}\n\\caption{\\label{fig:AdeltaA} In this graph, let  $A$ be the set of solid\n  points, $\\delta A$ the set of hollow points. We only consider the edges\n   for which one vertex is in $A$ and the other in $\\overline{A}$ (solid lines). We will ignore the dashed lines in $\\delta A$, and the dotted lines outside $\\overline{A}$.}\n\\end{figure}\n\nGiven $S\\subset \\overline{A}$, denote the boundary of $S$ relative to $A$ by\n$$\\partial_A S=\\{(u,v)\\in E:u\\in S\\cap A,v\\in \\delta A\\setminus S\\text{ or }u\\in S, v\\in A\\setminus S\\}.$$\nIf $S\\subset A$, then $\\partial_A S=\\{(u,v)\\in E:u\\in S, v\\in \\overline{A}\\setminus S\\}$.~\n\nThe Cheeger (cut) constant of the subgraph $A$ of $G$ is defined as\n$$h(A)=\\min_{S\\subset \\overline{A}}\\frac{|\\partial_A S|}{\\min\\{\\vol(A\\cap S),\\vol(A\\setminus S)\\}}.$$\nA set pair $(S,\\overline{A}\\setminus S)$ that achieves the Cheeger constant is called a Cheeger cut.\n\nThe Cheeger isoperimetric constant\\footnote{Some authors call it the Dirichlet isoperimetric constant.} of $A$ is defined as\n$$h_1(A)=\\min_{S\\subset  A}\\frac{|\\partial_A S|}{\\vol(S)},$$\nwhere a set $S$ achieving the Cheeger isoperimetric constant is called a Cheeger set. In the sequel, we fix  $A\\subset V$,  and we write $h(G)$ and $h_1(G)$ instead of $h(A)$ and $h_1(A)$, respectively.\n\nAccording to our generalized Lov\\'asz extension, we have\n\\begin{equation}\\label{eq:Dirichlet-Cheeger-h_1}\nh_1(G\n=\\inf_{\\vec x\\in \\mathbb{R}^n\\setminus\\{0\\}}\\frac{\\sum_{i\\sim j} |x_i-x_j|+\\sum_{i\\in A} p_i|x_i|}{\\sum_{i\\in A}d_i|x_i|}\n\\end{equation}\nand\n$$h(G\n=\\inf_{\\vec x\\in \\mathbb{R}^n\\setminus\\{0\\}}\\frac{\\sum_{i\\sim j,i,j\\in A}|x_i-x_j|+\\sum_{i\\sim j,i\\in A,j\\in\\delta A}|x_i-x_j|}{\\inf_{c\\in \\mathbb{R}}\\sum_{i\\in A}d_i|x_i-c|}.$$\n\nNote that the term on the right hand side of \\eqref{eq:Dirichlet-Cheeger-h_1} can be written as\n$$\n\\inf\\limits_{ \\vec x|_{V\\setminus S}=0,\\, \\vec x\\ne 0}\\mathcal{R}_1(x)\n$$\nwhich is called the {\\sl Dirichlet $1$-Poincare constant} (see \\cite{OSY19}) over $S$,\nwhere $$\\mathcal{R}_1(\\vec x):=\\frac{\\sum\\limits_{\\{i,j\\}\\in E} |x_i-x_j|}{\\sum_i d_i|x_i|}$$\nis called the $1$-Rayleigh quotient of $\\vec x$.\n\nWe can consider the corresponding spectral problems.\n\\begin{itemize}\n\\item Dirichlet eigenvalue problem:\n$$\\begin{cases}\n\\Delta_1 \\vec x\\cap \\mu D \\Sgn \\vec x\\ne\\varnothing,& \\text{ in } A\\\\\n\\vec x = 0,&\\text{ on }  \\delta A\n\\end{cases}$$  where $D$ is the diagonal matrix of the vertex degrees,  \nthat is,\n$$\\begin{cases}\n(\\Delta_1 \\vec x)_i-\\mu d_i\\Sgn x_i\\ni 0,&i\\in A\\\\\nx_i=0,&i\\in\\delta A\n\\end{cases}$$~\nwhose component form is: $\\exists$~$c_i\\in \\Sgn(x_i)$,~$z_{ij}\\in \\Sgn(x_i-x_j)$ satisfying $z_{ji}=-z_{ij}$ and\n$$\\sum_{j\\sim i} z_{ij}+ p_ic_i\\in \\mu d_i\\Sgn(x_i),~i\\in A,$$\nin which $p_i$ is the number of neighbors of $i$ in $\\delta A$.\n\\item Neumann eigenvalue problem: There exists $c_i\\in \\Sgn(x_i)$,~$z_{ij}\\in \\Sgn(x_i-x_j)$ with $z_{ji}=-z_{ij}$ such that\n$$\n\\begin{cases}\n\\sum_{j\\sim i,j\\in \\overline{A}}z_{ij}-\\mu d_i c_i=0,&i\\in A\\\\\n\\sum_{j\\sim i,j\\in A}z_{ij}=0,&i\\in\\delta A.\n\\end{cases}\n$$\n\\end{itemize}\nFor a graph $G$ with boundary, we use $\\Delta_1^D(G)$ and $\\Delta_1^N(G)$ to denote the Dirichlet 1-Laplacian and the Neumann 1-Laplacian, respectively. Then\n\\begin{pro}\n$$h_1(G)=\\lambda_1(\\Delta_1^D(G))\\;\\text{ and }\\;h(G)=\\lambda_2(\\Delta_1^N(G)).$$\n\\end{pro}\n\\begin{figure}\\centering\n\\begin{tikzpicture}[scale=0.69\n\\draw (0,0) to (2,0);\\draw (0,0) to (-0.9,0);\n\\draw (0,0) to (1,1);\n\\draw (0,0) to (1,-1);\n\\draw (1,1) to (1,-1);\\draw (1,1) to (1,1.9);\n\\draw (1,1) to (2,0);\n\\draw (1,-1) to (2,0);\\draw (1,-1) to (1,-1.9);\n\\draw (3,0) to (2,0);\\draw (3,0) to (3,0.9);\n\\draw (3,0) to (4,0);\\draw (3,0) to (3,-0.9);\n\\draw (6,0) to (4,0);\n\\draw (5,1) to (4,0);\n\\draw (5,-1) to (4,0);\n\\draw (5,1) to (6,0);\n\\draw (5,-1) to (6,0);\n\\draw (5,-1) to (5,1);\n\\draw (6,0) to (7,0);\n\\draw (8,0) to (7,0);\n\\draw (8,0) to (10,0);\n\\draw (8,0) to (9,1);\n\\draw (8,0) to (9,-1);\n\\draw (10,0) to (9,1);\n\\draw (10,0) to (9,-1);\n\\draw (9,1) to (9,-1);\n\\draw (10,0) to (10.9,0);\n\\draw (5,1) to (5,1.9);\n\\draw (5,1) to (5,-1.9);\n\\draw (9,1) to (9,1.9);\n\\draw (9,1) to (9,-1.9);\n\\draw (7,0) to (7,0.9);\n\\draw (7,0) to (7,-0.9);\n\\node (00) at (0,0) {$\\bullet$};\n\\node (11) at (1,1) {$\\bullet$};\n\\node (1-1) at (1,-1) {$\\bullet$};\n\\node (20) at (2,0) {$\\bullet$};\n\\node (30) at (3,0) {$\\bullet$};\n\\node (40) at (4,0) {$\\bullet$};\n\\node (60) at (6,0) {$\\bullet$};\n\\node (51) at (5,1) {$\\bullet$};\n\\node (5-1) at (5,-1) {$\\bullet$};\n\\node (70) at (7,0) {$\\bullet$};\n\\node (80) at (8,0) {$\\bullet$};\n\\node (100) at (10,0) {$\\bullet$};\n\\node (91) at (9,1) {$\\bullet$};\n\\node (9-1) at (9,-1) {$\\bullet$};\n\\node (-10) at (-1,0) {$\\circ$};\n\\node (110) at (11,0) {$\\circ$};\n\\node (12) at (1,2) {$\\circ$};\n\\node (1-2) at (1,-2) {$\\circ$};\n\\node (31) at (3,1) {$\\circ$};\n\\node (31) at (3,-1) {$\\circ$};\n\\node (71) at (7,1) {$\\circ$};\n\\node (7-1) at (7,-1) {$\\circ$};\n\\node (52) at (5,2) {$\\circ$};\n\\node (5-2) at (5,-2) {$\\circ$};\n\\node (92) at (9,2) {$\\circ$};\n\\node (9-2) at (9,-2) {$\\circ$};\n\\end{tikzpicture}\n\\caption{\\label{fig:k-nodal-domain} In this example, there are $3$ nodal domains of an eigenvector corresponding to the first Dirichlet eigenvalue of the graph 1-Laplacian. Each nodal domain is the vertex set of the $4$-order complete subgraph shown in the figure. }\n\\end{figure}\n\nFor a connected graph, the first eigenvector of $\\Delta_1^N(G)$ is constant\nand it has only one nodal domain while the first eigenvector of $\\Delta_1^D(G)$\n may have any number of nodal domains.\n \\begin{pro}\n For any $k\\in \\mathbb{N}^+$, there exists a connected graph $G$ with boundary such that its Dirichlet 1-Laplacian $\\Delta_1^D(G)$ has a first eigenvector (corresponding to $\\lambda_1(\\Delta_1^D(G))$) with exactly $k$ nodal domains; and its Neumann 1-Laplacian $\\Delta_1^N(G)$ possesses a second eigenvector (corresponding to $\\lambda_2(\\Delta_1^N(G))$) with exactly $k$ nodal domains.\n \\end{pro}\n\n\n \\begin{figure}\\centering\n \\begin{tikzpicture}[scale=0.8]\n\\node (1) at (0,1) {$\\bullet$};\n\\node (2) at (1,0) {$\\bullet$};\n\\node (3) at (1,1) {$\\bullet$};\n\\node (4) at (2,2) {$\\bullet$};\n\\node (5) at (3,1) {$\\bullet$};\n\\node (6) at (4,1) {$\\bullet$};\n\\node (7) at (3,0) {$\\bullet$};\n\\node (8) at (1,3) {$\\bullet$};\n\\node (9) at (1,4) {$\\bullet$};\n\\node (10) at (0,3) {$\\bullet$};\n\\node (11) at (3,3) {$\\bullet$};\n\\node (12) at (3,4) {$\\bullet$};\n\\node (13) at (4,3) {$\\bullet$};\n\\draw (0,1) to (1,1);\n\\draw (1,0) to (1,1);\n\\draw (1,1) to (2,2);\n\\draw (2,2) to (3,1);\n\\draw (3,1) to (4,1);\n\\draw (3,1) to (3,0);\n\\draw (2,2) to (1,3);\n\\draw (1,3) to (1,4);\n\\draw (1,3) to (0,3);\n\\draw (2,2) to (3,3);\n\\draw (3,3) to (3,4);\n\\draw (3,3) to (4,3);\n\\end{tikzpicture}\n\\caption{\\label{fig:second-eigen-k-nodal-domain} In this example, there are $4$ nodal domains of an eigenvector corresponding to the second Neumann eigenvalue of the graph 1-Laplacian. Each nodal domain is the vertex set of the $3$-order subgraph after removing the center vertex and its edges.}\n\\end{figure}\n\n\n\\subsection{Independence number}\\label{sec:independent-number}\nThe independence number $\\alpha(G)$ of an unweighted and undirected simple graph $G$ is  the largest cardinality of a subset of vertices in $G$, no two of which are adjacent. It can be seen as an optimization problem $\\max\\limits_{S\\subset V \\text{ s.t. }E(S)=\\varnothing}\\#S$. However, such a graph optimization is not global, and the feasible domain seems to be very complicated.\nBut we may simply  multiply by a truncated term $(1-\\#E(S))$.\nThe independence number can then be expressed as a global optimization on the power set of vertices:\n\\begin{equation}\\label{eq:independent-multiple}\n\\alpha(G)=\\max\\limits_{S\\subset V}\\#S(1-\\#E(S)),\n\\end{equation}\nand thus the Lov\\'asz extension can be applied.\n\\begin{proof}[Proof of Eq.~\\eqref{eq:independent-multiple}] Since $G$ is simple, $\\#S$ and $\\#E(S)$ take values in the natural numbers. Therefore,\n$$\n\\#S(1-\\#E(S))\\;\\; \\begin{cases}\\le 0,&\\text{ if } E(S)\\ne\\varnothing\\text{ or }S=\\varnothing,\\\\\n\\ge 1,&\\text{ if } E(S)=\\varnothing\\text{ and }S\\ne\\varnothing.\\end{cases}\n$\nThus, $\\max\\limits_{S\\subset V}\\#S(1-\\#E(S))=\\max\\limits_{S\\subset V \\text{ s.t. }E(S)=\\varnothing}\\#S=\\alpha(G)$.\n\\end{proof}\nHowever, Eq.~\\eqref{eq:independent-multiple} is  difficult to calculate. By the disjoint-pair Lov\\'asz extension, it equals \n$$\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{\\|\\vec x\\|_1-\\sum\\limits_{k\\in V,i\\sim j}\\min\\{|x_k|,|x_i|,|x_j|\\}}{\\|\\vec x\\|_\\infty},\n$$\nbut we don't know how to further simplify it.\n\nFortunately, there is a known  representation of the independence number as follows, and  we present a proof for convenience. \n\\begin{pro}The independence number $\\alpha(G)$ of a finite simple graph $G=(V,E)$ satisfies\n\\begin{equation}\\label{eq:independent-difference}\n\\alpha(G)=\\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right).\n\\end{equation}\n\\end{pro}\n\n\\begin{proof\nLet $A$ be an independent set of $G$, then $\\alpha(G)=\\#A=\\#A-\\#E(A)\\le \\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right)$ because there is no edge connecting points in $A$.\n\nLet $B\\subset V$ satisfy $ \\#B-\\#E(B) =\\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right)$. Assume the induced subgraph $(B,E(B))$ has $k$ connected components, $(B_i,E(B_i))$, $i=1,\\cdots,k$. Then $B=\\sqcup_{i=1}^k B_i$ and $E(B)=\\sqcup_{i=1}^k E(B_i)$. Since $(B_i,E(B_i))$ is connected, $\\# B_i\\le \\# E(B_i)+1$ and  equality holds if and only if $(B_i,E(B_i))$ is a tree.\nNow taking $B'\\subset B$ such that $\\#(B'\\cap B_i)=1$, $i=1,\\cdots,k$, then $B'$ is an independent set and thus  \\begin{align*}\n\\alpha(G)&\\ge \\#B'=k=\\sum_{i=1}^k 1\\ge \\sum_{i=1}^k (\\# B_i- \\# E(B_i))\n= \\sum_{i=1}^k\\# B_i- \\sum_{i=1}^k\\# E(B_i)\\\\&=\\#(\\cup_{i=1}^kB_i)-\\#(\\cup_{i=1}^kE(B_i)) = \\#B-\\#E(B)=\\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right).\n\\end{align*}\nAs a result, Eq.~\\eqref{eq:independent-difference} is proved.\n\\end{proof}\n\nAccording to Lov\\'asz extension, we get\n\\begin{equation}\\label{eq:independent-continuous}\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{\\|\\vec x\\|_1-\\sum\\limits_{i\\sim j}\\min\\{|x_i|,|x_j|\\}}{\\|\\vec x\\|_\\infty}.\n\\end{equation}\nBy the elementary identities: $\\sum_{i\\sim j}|x_i+x_j|+\\sum_{i\\sim j}|x_i-x_j|=2\\sum_{i\\sim j}\\max\\{|x_i|,|x_j|\\}=\\sum_{i\\sim j}\\left||x_i|-|x_j|\\right|+\\sum_i\\mathrm{deg}_i|x_i|$ and $\\sum_i\\mathrm{deg}_i|x_i|=\\sum_{i\\sim j}\\max\\{|x_i|,|x_j|\\}+\\sum_{i\\sim j}\\min\\{|x_i|,|x_j|\\}$,  Eq.~\\eqref{eq:independent-continuous} can be reduced to\n\\begin{equation}\\label{eq:independent-continuous-1}\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{2\\|\\vec x\\|_1+I^-(\\vec x)+I^+(\\vec x)- 2\\|\\vec x\\|_{1,\\mathrm{deg}}}{2\\|\\vec x\\|_\\infty},\n\\end{equation}\nwhere $I^\\pm(\\vec x)=\\sum_{i\\sim j}|x_i\\pm x_j|$ and $\\|\\vec x\\|_{1,\\mathrm{deg}}=\\sum_i\\mathrm{deg}_i|x_i|$. One would like to write Eq.~\\eqref{eq:independent-continuous-1} as\n\\begin{equation}\\label{eq:independent-continuous-2}\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{I^-(\\vec x)+I^+(\\vec x)- 2\\|\\vec x\\|_{1,\\mathrm{deg}'}}{2\\|\\vec x\\|_\\infty},\n\\end{equation}\nwhere $\\|\\vec x\\|_{1,\\mathrm{deg}'}=\\sum\\limits_{i\\in V}(\\mathrm{deg}_i-1)|x_i|$.\n\n\n\\begin{remark}\nThe maximum clique number can be reformulated in a similar way. \nIn addition, we refer to \\cite{BBcontinuous05,BBcontinuous06,SBBB20} for some other continuous formulations of the independence number. \n\\end{remark}\n\n\n\n\\paragraph{Chromatic number of a  perfect graph}\nBerge's strong perfect graph conjecture has been proved in \\cite{Annals06}.  A graph $G$ is perfect if for every induced subgraph $H$ of $G$, the chromatic number of $H$ equals the size of the largest clique of $H$. The complement of every perfect graph is perfect.\n\n\nSo for a  perfect graph, we have an easy way to calculate the  chromatic number. In a general simple graph, we refer to Section \\ref{sec:chromatic-number} for transforming the chromatic number.\n\n\\paragraph{Maximum matching }\nA matching $M$ in $G$ is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share a common vertex.\nA maximal matching  is one with the largest possible number of edges.\n\nConsider the line graph $(E,R)$ whose vertex set $E$ is the edge set of $G$, and whose edge set is $R=\\{\\{e,e'\\}:e\\cap e'\\not=\\varnothing,\\,e,e'\\in E\\}$. Then the maximum matching number of $(V,E)$ coincides with the independence number of $(E,R)$. So, we have an equivalent continuous optimization for a maximum matching problem.\n\nHall's Marriage Theorem  provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.\n\nThe Tutte-Berge formula says that the size of a maximum matching of a graph is\n$$\\frac12\\min\\limits_{U\\subset V}\\left(\\#V+\\#U-\\#\\text{ odd connected components of }G|_{V\\setminus U}\\right).$$\nCan one  transform the above discrete optimization problem into an explicit continuous optimization via some extension?\n\n\\paragraph{$k$-independence number }\nThe independence number admits several generalizations:\n the maximum size of a set of vertices in a graph whose induced subgraph has maximum degree $(k-1)$ \\cite{CaroH13}; the size of the largest $k$-colourable subgraph \\cite{Spacapan11}; the\nsize of the largest set of vertices such that any two vertices in the set are at short-path distance larger than $k$ (see \\cite{Fiol97}). For the $k$-independence number involving short-path distance, one can easily transform it into the following two continuous representations:\n$$\\alpha_k= \\max\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^V\\setminus\\{\\vec 0\\}}\\frac{\\|\\vec x\\|_1^2}{\\|\\vec x\\|_1^2-2\\sum\\limits_{\\mathrm{dist}(i,j)\\ge k+1}x_ix_j} = \\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n\\setminus\\{\\vec 0\\}}\\frac{\\sum\\limits_{\\mathrm{dist}(i,j)\\le k}(|x_i-x_j|+|x_i+x_j|)- 2\\sum\\limits_{i\\in V}(\\deg_{k}(i)-1)|x_i|}{2\\|\\vec x\\|_\\infty},$$\nwhere $\\deg_{k}(i)=\\#\\{j\\in V:\\mathrm{dist}(j,i)\\le k\\}$, $i=1,\\cdots,n$.\n\\subsection{Various and variant Cheeger problems}\n\\label{sec:variantCheeger}\n\nSeveral  Cheeger-type  constants on graphs have been proposed that are  different from  the classical one.\n\n\\paragraph{Multiplicative Cheeger constant\nFor instance\n$$h=\\min\\limits_{\\varnothing\\ne A\\subsetneqq V}\\frac{ \\#E(A,V\\setminus A) }{\\#A\\cdot\\#(V\\setminus A)}.$$\nIt is called the normalized cut problem which has many applications in image segmentation and spectral clustering \\cite{SM00,Luxburg07,HS11}.  By Proposition \\ref{pro:fraction-f\/g}, it is equal to\n$$\\min\\limits_{\\langle\\vec x,\\vec1\\rangle=0,\\vec x\\ne \\vec0}\\frac{\\sum_{i\\sim j}|x_i-x_j|}{\\sum_{i< j}|x_i-x_j|}.$$\n\n\\paragraph{Isoperimetric profile}\nThe isoperimetric profile $IP:\\mathbb{N}\\to [0,\\infty)$ is defined by\n$$IP(k)= \\inf\\limits_{A\\subset V,\\#A\\le k} \\frac{\\#E(A,V\\setminus A)}{\\#A}.$$\nThen by Lov\\'asz extension, it is equal to\n$$\\inf\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^V,\\,1\\le \\# \\ensuremath{\\mathrm{supp}}(\\vec x)\\le k}\\frac{\\sum_{\\{i,j\\}\\in E}|x_i-x_j|}{\\|\\vec x\\|_1}=\\min\\limits_{\\vec x\\in CH_k(\\ensuremath{\\mathbb{R}}^V)}\\frac{\\sum_{\\{i,j\\}\\in E}|x_i-x_j|}{\\|\\vec x\\|_1},$$\nwhere $CH_n:=\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^V,\\,\\# \\ensuremath{\\mathrm{supp}}(\\vec x)\\le k\\}$ is the union of all $k$-dimensional coordinate hyperplanes in $\\ensuremath{\\mathbb{R}}^V$.\n\n\\paragraph{Modified Cheeger constant}\n\nOn a graph $G=(V,E)$, there are three definitions of the vertex-boundary of a subset $A\\subset V$:\n\\begin{align}\n& \\partial_{\\textrm{ext}} A:=\\{j\\in V\\setminus A\\,\\left|\\,\\{j,i\\}\\in E\\text{ for some }i\\in A\\right.\\} \\label{eq:ext-vertex-boundary}\\\\\n& \\partial_{\\textrm{int}} A:=\\{i\\in A\\,\\left|\\,\\{i,j\\}\\in E\\text{ for some }j\\in V\\setminus A\\right.\\}\\label{eq:int-vertex-boundary}\\\\\n& \\partial_{\\textrm{ver}} A:=\\partial_{\\textrm{out}} A\\cup \\partial_{\\textrm{int}} A=V(E(A,V\\setminus A))=V(\\partial_{\\textrm{edge}} A)\\label{eq:vertex-boundary}\n\\end{align}\nThe  {\\sl external vertex boundary} \\eqref{eq:ext-vertex-boundary} and the  {\\sl internal vertex boundary} \\eqref{eq:int-vertex-boundary} are introduced and studied recently in \\cite{Vigolo19tams,VigoloPHD}.  Research on metric measure space \\cite{HMT19} suggests to consider the {\\sl vertex boundary} \\eqref{eq:vertex-boundary}.\n\nDenote by $N(i)=\\{i\\}\\cup\\{j\\in V:\\{i,j\\}\\in E\\}$ the 1-neighborhood of $i$. Then the Lov\\'asz extensions of $\\#\\partial_{\\textrm{ext}} A$, $\\#\\partial_{\\textrm{int}} A$ and $\\#\\partial_{\\textrm{ver}} A$ are\n$$\\sum\\limits_{i=1}^n(\\max\\limits_{j\\in N(i)}x_j-x_i),\\;\\;\\;\\sum\\limits_{i=1}^n(x_i-\\min\\limits_{j\\in N(i)}x_j)\\;\\;\\text{ and }\\;\\;\\sum\\limits_{i=1}^n(\\max\\limits_{j\\in N(i)}x_j-\\min\\limits_{j\\in N(i)}x_j),$$ respectively.\nThey can be seen as the  `total variation' of $\\vec x$ with respect to $V$ in $G$, while the usual {\\sl edge boundary} leads to $\\sum\\limits_{\\{i,j\\}\\in E}|x_i-x_j|$ which is  regarded as the total variation of $\\vec x$ with respect to $E$ in $G$.  Their disjoint-pair Lov\\'asz extensions are $$\\sum_{i=1}^n \\max\\limits_{j\\in N(i)} |x_j|-\\|\\vec x\\|_1,\\;\\;\\;\\|\\vec x\\|_1-\\sum_{i=1}^n \\min\\limits_{j\\in N(i)} |x_j|,\\;\\;\\;\\sum_{i=1}^n \\left(\\max\\limits_{j\\in N(i)} |x_j|-\\min\\limits_{j\\in N(i)} |x_j|\\right).$$\nComparing with the graph $1$-Poincare profile (see \\cite{Hume17,HMT19,Hume19arxiv}) $$P^1(G):=\\inf\\limits_{\\langle\\vec x,\\vec 1\\rangle=0,\\vec x\\ne\\vec 0}\\frac{\\sum_{i\\in V} \\max\\limits_{j\\sim i}|x_i-x_j|}{\\|\\vec x\\|_1},$$\nwe easily get the following\n\\begin{pro}\n$$ \\frac12\\max\\{h_{\\mathrm{int}}(G),h_{\\mathrm{ext}}(G)\\}\\le P^1(G)\\le h_{\\mathrm{ver}}(G):=\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\n\\frac{\\#\\partial_{\\mathrm{ver}} A}{\\min\\{\\#(A),\\#(V\\setminus A)\\}}$$\nwhere $h_{\\mathrm{int}}(G)$,  $h_{\\mathrm{ext}}(G)$ and $h_{\\mathrm{ver}}(G)$ are modified Cheeger constants w.r.t. the type of vertex-boundary.\n\\end{pro}\n\\begin{proof}\nBy Theorem \\ref{introthm:eigenvalue}, $$h_{\\mathrm{ver}}(G)=\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\n\\frac{\\#\\partial_{\\mathrm{ver}} A}{\\min\\{\\#(A),\\#(V\\setminus A)\\}}=\\inf\\limits_{\\langle\\vec x,\\vec 1\\rangle=0,\\vec x\\ne\\vec 0}\\frac{\\sum_{i\\in V} \\max\\limits_{j\\sim i}|x_i-x_j|}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}\\|\\vec x-t\\vec1\\|_1}\\ge P^1(G).$$\nOn the other hand, it is easy to check that $\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}\\|\\vec x-t\\vec1\\|_1\\ge \\frac12 \\|\\vec x\\|_1$ whenever $\\langle\\vec x,\\vec 1\\rangle=0$. Thus, $h_{\\mathrm{ver}}(G)\\le 2P^1(G)$. The proof is then completed by noting that $\\max\\{h_{\\mathrm{int}}(G),h_{\\mathrm{ext}}(G)\\}\\le h_{\\mathrm{ver}}(G)$.\n\\end{proof}\n\n\\paragraph{Cheeger-like constant}\nSome further recent results  \\cite{JM19arxiv} can be also rediscovered via  Lov\\'asz extension.\n\nA main equality in \\cite{JM19arxiv} can be absorbed into the following identities:\n\\begin{align}\n\t      \\max_{\\text{edges }(v,w)}\\biggl(\\frac{1}{\\deg v}+\\frac{1}{\\deg w}\\biggr)\n&=\n\\max_{\\gamma:E\\rightarrow\\mathbb{R}}\\frac{\\sum_{v\\in V}\\frac{1}{\\deg v}\\cdot \\biggl|\\sum_{e_{\\text{in}}: v\\text{ input}}\\gamma(e_{\\text{in}})-\\sum_{e_{\\text{out}}: v\\text{ output}}\\gamma(e_{\\text{out}})\\biggr|}{\\sum_{e\\in E}|\\gamma(e)|} \\notag\n\\\\&=\\max_{\\hat{\\Gamma}\\subset\\Gamma \\text{ bipartite}} \\frac{\\sum_{v\\in V}\\frac{\\deg_{\\hat{\\Gamma}}(v)}{\\deg_\\Gamma (v)}}{|E(\\hat{\\Gamma})|}, \\label{eq:mainJM19}\n\\end{align}\nwhere the left quantity is called a Cheeger-like constant \\cite{JM19arxiv}.\n\nIn fact, given $c_i\\ge 0$, $i\\in V$,\n$$\\max\\limits_{\\{i,j\\}\\in E}(c_i+c_j)=\\max\\limits_{E'\\subset E}\\frac{\\sum_{\\{i,j\\}\\in E'}(c_i+c_j)}{\\# E'},$$\nand then via Lov\\'asz extension, one immediately gets that the above constant equals to\n$$\n\\max\\limits_{\\vec x\\in [0,\\infty)^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum\\limits_{e=\\{i,j\\}\\in E}x_e(c_i+c_j)}{\\sum_{e\\in E}x_e}=\\max\\limits_{\\vec x\\in[0,\\infty)^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}c_i\\sum_{e\\ni i}x_e}{\\sum_{e\\in E}x_e}=\\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}c_i\\left|\\sum_{e\\ni i}x_e\\right|}{\\sum_{e\\in E}|x_e|}.\n$$\nThus, for any family $\\ensuremath{\\mathcal{E}}\\subset\\ensuremath{\\mathcal{P}}(E)$ such that $E'\\in \\ensuremath{\\mathcal{E}}$ $\\Rightarrow$ $E'\\supset \\{\\{e\\}:e\\in E\\}$, we have\n$$\\max\\limits_{\\{i,j\\}\\in E}(c_i+c_j)=\\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}c_i\\left|\\sum_{e\\ni i}x_e\\right|}{\\sum_{e\\in E}|x_e|}=\\max\\limits_{E'\\in\\ensuremath{\\mathcal{E}}}\\frac{\\sum_{\\{i,j\\}\\in E'}(c_i+c_j)}{\\# E'},$$\nwhich recovers the interesting equality \\eqref{eq:mainJM19}  by taking $c_i=\\frac{1}{\\deg i}$ and $\\ensuremath{\\mathcal{E}}$ the collections of all edge sets of bipartite subgraphs.\n\nA similar simple trick gives\n\\begin{equation*}\n    \\min_{(v,w)}\\frac{\\bigl|\\mathcal{N}(v)\\cap \\mathcal{N}(w)\\bigr|}{\\max\\{\\deg v,\\deg w\\}}=\\min\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}\\sum_{e\\ni i}\\left|x_e\\right|\\cdot\\#\\text{ triangles containing }e}{\\sum_{e=\\{i,j\\}\\in E}|x_e|\\max\\{\\deg i,\\deg j\\}}.\n\\end{equation*}\n\n\n\\subsection{Frustration in signed networks}\n\\label{sec:frustration}\nIn this section, we apply our theory to signed graphs, a concept first introduced by Harary \\cite{Harary55}.\n\\begin{defn}\n  A \\emph{signed graph} $\\Gamma$ consists of a vertex set $V$ and a set $E$ of undirected edges with a sign function\n\\begin{equation}\\label{sign1}\ns:E \\to \\{+1,-1\\}.\n\\end{equation}\nThe adjacency matrix of $(\\Gamma,s)$, is denoted by  $\\mathrm{A}^s:=(s_{ij})_{i,j\\in V}$, where $s_{ij}:=s(e)$ if $e=\\{i,j\\}\\in E$, and  $s_{ij}:=0$ otherwise.\n\\end{defn}\nWhen we replace the sign function $s$ by $-s$, we shall call the resulting graph \\emph{antisigned}.\n\\begin{defn}\n The signed cycle $C_m$ (consisting of $m$ vertices that are cyclically connected by $m$ edges) is \\emph{balanced} if \n\\begin{equation}\\label{sign3}\n\\prod_{i=1}^m s(e_i)=1.\n\\end{equation}\nA  signed graph $(\\Gamma,s)$ is \\emph{balanced}  if every cycle contained in it is balanced.\\\\\n$(\\Gamma,s)$ is \\emph{antibalanced}  if $(\\Gamma,-s)$ is balanced. \\\\\nThe frustration index of  a signed graph $\\Gamma=(V,E)$ is  \n\\begin{equation}\\label{eq:frustration}\n\\min_{x_i\\in \\{-1,1\\},\\forall i} \\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j|,\n\\end{equation}  where $s_{ij}\\in\\{-1,1\\}$  indicates the sign of the edge $(i,j)$.\n\\end{defn}\nThe frustration index then vanishes iff the graph is balanced.\\\\\n\\begin{defn}\n The  (normalized) Laplacian $\\Delta^s$ of a signed graph is defined by\n\\begin{equation}\n\\label{sign5}\n(\\Delta^s \\vec x)_i:= x_i-\\frac{1}{\\deg i}\\sum_{j \\sim i}s_{ij}x_j=\\frac{1}{\\deg i}\\sum_{j \\sim i}(x_i-s_{ij}x_j)\n\\end{equation}\nfor a vector $\\vec x\\in\\ensuremath{\\mathbb{R}}^V$.\n\\end{defn}\n\\begin{remark}\nThe Laplacian thus is of the form $\\Delta^s =\\mathrm{id}-\\mathrm{A}^s$, and \n when we change the signs of all the edges, that is, go from a signed graph to the corresponding antisigned graph, the operator becomes $\\Delta^{-s} =\\mathrm{id}+\\mathrm{A}^s$. Therefore, the eigenvalues simply  change from $\\lambda$ to $2-\\lambda$ (and therefore, also the ordering gets reversed). \n\\end{remark}\n\nBy Proposition \\ref{pro:Lovasz-eigen}, it is easy to verify that every eigenvalue of the function pair $(F,G)$ has an  eigenvector in  $\\{-1,0,1\\}^n$,  where $F(\\vec x)=\\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j|$ and $G(\\vec x)=\\|\\vec x\\|_\\infty$. One may  relax \\eqref{eq:frustration} as  \\begin{equation}\\label{eq:frustration-relax}\n\\min_{\\vec x\\in \\{-1,0,1\\}^n\\setminus\\{\\vec0\\}} \\sum_{(i,j)\\in E}|x_i-s_{ij}x_j|.    \n\\end{equation}\n\n\n\nThis suggests the  eigenvalue problem of $(F(\\vec x),\\|\\vec x\\|_\\infty)$ on a signed graph, \nwhere $F(\\vec x)=\\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j|$. Below,  we show some key properties.\n\n\n\n\n\n\n\n\\begin{itemize}\n\\item\nThe coordinate form of the eigenvalue problem $\\nabla \\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j| \\cap \\lambda\\nabla\\|\\vec x\\|_\\infty\\ne\\varnothing$ reads as\n\n $\\exists\\,z_{ij}\\in \\Sgn(x_i-s_{ij}x_j) \\mbox{ with }z_{ij}+s_{ij}z_{ji}=0$~such that\n\\begin{align}\n\\label{eq:1LinftyN1}&\\sum\\limits_{j\\sim i} z_{ij} = 0, &i\\in D_0(\\vec x),\\\\\n\\label{eq:1LinftyN2}&\\sum\\limits_{j\\sim i} z_{ij} \\in \\lambda\\ \\mathrm{sign}(x_i)\\cdot [0,1],& i\\in D_\\pm(\\vec x),\\\\\n\\label{eq:1LinftyN3}&\\sum\\limits_i^n \\big|\\sum\\limits_{j\\sim i} z_{ij} \\big|=\\lambda,&\n\\end{align}\nwhere $D_\\pm(\\vec x) =\\{i\\in V\\big| \\pm x_i = \\|\\vec x\\|\\}$,\nand $D_0(\\vec x)=\\{i\\in V\\big| |x_i|<\\|\\vec x\\|\\}$.\n\\item All eigenvalues are integers in  $\\{0,1,\\cdots,\\vol(V)\\}$. And each eigenvalue has an eigenvector in $\\{-1,0,1\\}^n$.\n\nProof: This is a direct consequence of Proposition \\ref{pro:Lovasz-eigen}. \n\n\\item The largest eigenvalue has an eigenvector in $\\{-1,1\\}^n$.\n\nProof: Let $\\vec 1_A-\\vec 1_B$ be an eigenvector  w.r.t. the largest eigenvalue. Note that $\\vec 1_A-\\vec 1_B=\\frac12(\\vec1_A-\\vec1_{V\\setminus A}+\\vec1_{V\\setminus B}-\\vec1_B)$. By the convexity of $F$, we have $F(\\vec 1_A-\\vec 1_B)\\le\\max\\{F(\\vec1_A-\\vec1_{V\\setminus A}),F(\\vec1_{V\\setminus B}-\\vec1_B)\\}$. Hence, either $\\vec1_A-\\vec1_{V\\setminus A}$ or $\\vec1_{V\\setminus B}-\\vec1_B$ is an eigenvector w.r.t. the largest eigenvalue. \n\n\\item \\textbf{The frustration index  is an eigenvalue}.\nHowever, in general, we don't know which eigenvalue the  frustration  index is.\n\n Proof: We shall check that for any $A\\subset V$, the binary vector $\\vec x:=\\vec1_A-\\vec1_{V\\setminus A}$ is an eigenvector w.r.t. the eigenvalue $\\lambda:=2(|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|)$, where  $|E_+(A,V\\setminus A)|$ indicates the number of positive edges lying between  $A$ and $V\\setminus A$, while  $|E_-(A)|$ denotes the number of negative edges lying in $A$.  Indeed, $D_+(\\vec x)=A$ and $D_-(\\vec x)=V\\setminus A$.   For $i\\in A$, taking $z_{ij}=1$ if $s_{ij}x_j<0$; and $z_{ij}=0$ if $s_{ij}x_j>0$.  Similarly, for $i\\in V\\setminus A$, letting $z_{ij}=0$ if $s_{ij}x_j<0$; and $z_{ij}=-1$ if $s_{ij}x_j>0$. It is easy to see that $z_{ij}\\in \\mathrm{Sgn}(x_i-s_{ij}x_j)$ and $z_{ij}+s_{ij}z_{ji}=0$ for any edge $ij$. Next, we verify the conditions \\eqref{eq:1LinftyN2} and \\eqref{eq:1LinftyN3}.\n\nNote that  $\\sum_{j\\sim i}z_{ij}=\\#(\\{j\\in A:ij\\text{ is negative}\\}\\cup \\{j\\in V\\setminus A:ij\\text{ is positive}\\})\\in[0,\\lambda]$ for $i\\in A$, and  $\\sum_{j\\sim i}z_{ij}=-\\#(\\{j\\in A:ij\\text{ is positive}\\}\\cup \\{j\\in V\\setminus A:ij\\text{ is negative}\\})\\in[-\\lambda,0]$ for $i\\in V\\setminus A$. \nTherefore, $\\sum_{i\\in V}|\\sum_{j\\sim i}z_{ij}|=2(|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|)=\\lambda$.\n\nIn particular, for $\\vec x\\in\\{-1,1\\}^n$ that realizes the frustration index, $\\vec x$ must be an eigenvector, and the frustration index is the corresponding eigenvalue. This fact can also be derived by  Proposition  \\ref{pro:set-pair-infty-norm}.  \n\n\\item We can use the the   Dinkelbach-type scheme in Section \\ref{sec:algo}  directly to  calculate the smallest eigenvalue. When we get an eigenvector $\\vec x$, we can take $\\vec 1_{D_+(\\vec x)}-\\vec 1_{D_-(\\vec x)}$ instead of $\\vec x$. \n\\item We construct a recursive method to approximate the frustration index:\n\\begin{itemize}\n\\item Input a signed graph $G$,  and use the   Dinkelbach-type algorithm to get a subpartition $(U_{+},U_{-})$ where $U_+=D_+(\\vec x)$ and $U_-=D_-(\\vec x)$ with $\\vec x$ being an eigenvector w.r.t. the smallest eigenvalue. \n\\item Let $G$ be the signed graph induced by $V\\setminus(U_+\\cup U_-)$, and let $(U_+',U_-')$ be the subpartition found by the  Dinkelbach-type  algorithmm; return $(U_+\\cup U_+',U_-\\cup U_-')$ or $(U_+\\cup U_-',U_-\\cup U_+')$, whichever is better.\n\\item Repeat the above process, until we get a partition $(V_+,V_-)$ of $V$, which derives an  approximate solution of the  frustration index. There are at most $n$ iterations.\n\\end{itemize}\nIn other words, the relaxation problem  \\eqref{eq:frustration-relax} can approximate the frustration index \\eqref{eq:frustration} in a recursive way.  This is inspired by the recursive spectral cut  algorithm for the   maxcut problem proposed by Trevisan  \\cite{Trevisan2012}.\n\\end{itemize}\n \n\n \nNext, we show some equivalent continuous representations of the  frustration index. Let $E_+$ (resp. $E_-$) collect all the positive (resp. negative) edges of $(V,E)$.\n  Note that up to a scale factor,  \\eqref{eq:frustration} is equivalent to solve $\\min\\limits_{A\\subset V}|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|$, where $|E_+(A,V\\setminus A)|$ denotes the number of  positive edges between $A$ and $V\\setminus A$, while $|E_-(A)|$ indicates the number of negative edges in $A$. By Lov\\'asz extension,  the frustration index  is equivalent to\n$$|E_-|+\\min\\limits_{\\vec x\\ne0}\\frac{\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|+\\sum_{i\\in V}\\deg_i|x_i|-\\sum_{\\{i,j\\}\\in E_-}(|x_i-x_j|+|x_i+x_j|)}{\\|x\\|_\\infty}.$$ Also, \\eqref{eq:frustration} is equivalent to $|E_-|+\\min\\limits_{A\\subset V}(|E_+(A,V\\setminus A)|-|E_-(A,V\\setminus A)|)$, and by Lov\\'asz extension, the frustration index equals \n$$|E_-|+\\min\\limits_{\\vec x\\ne0}\\frac{\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|-\\sum_{\\{i,j\\}\\in E_-}|x_i-x_j|}{2\\|x\\|_\\infty}.$$\nOne can then apply  the   Dinkelbach-type scheme in Section \\ref{sec:algo}   straightforwardly  to  compute the  frustration index.\n\n \n\n\n\n\\begin{remark}\nWe should point out that the notion $|E_+(A)|$ (resp. $|E_-(A)|$) indicates the number of positive (resp. negative)  edges (unordered pairs) whose vertices are in $A$. Therefore, in   our paper, the values of  $|E_+(A)|$ and  $|E_-(A)|$ are \nhalf of those of \nAtay-Liu \\cite{AtayLiu}, in which they count the  ordered pairs. \n\\end{remark}\nBesides, by Theorem \\ref{thm:tilde-fg-equal-PQ} (or Theorem \\ref{thm:tilde-H-f}), we can derive another continuous formulation of the frustration index:\n$$\\min\\limits_{A\\subset V}|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|=\\min\\limits_{x\\ne0}\\frac{\\sum\\limits_{\\{i,j\\}\\in E_+}|x_i-x_j|^\\alpha+\\sum\\limits_{\\{i,j\\}\\in E_-}(2\\|\\vec x\\|_\\infty-|x_i-x_j|)^\\alpha}{(2\\|x\\|_\\infty)^\\alpha}$$\nwhenever $0<\\alpha\\le 1$. It is interesting that by taking $\\alpha\\to 0^+$, we immediately get \n$$\\min\\limits_{A\\subset V}|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|=\\min\\limits_{x\\ne0}\\sum\\limits_{\\{i,j\\}\\in E_+}\\mathrm{sign}(|x_i-x_j|)+\\sum\\limits_{\\{i,j\\}\\in E_-}\\mathrm{sign}(2\\|\\vec x\\|_\\infty-|x_i-x_j|).$$\n\n\\subsection{Modularity measure}\\label{sec:modularity-measure}\n\nFor a weighted graph $(V,(w_{ij})_{i,j\\in V})$, the   modularity measure \\cite{TMH18} is defined as\n$$Q(A)=\\sum_{i,j\\in A}w_{ij}-\\frac{\\vol(A)^2}{\\vol(V)},\\;\\;\\text{where }A\\subset V,$$\nand it satisfies the following equalities (see Theorem 3.7 and Theorem 3.9 in \\cite{TMH18}, respectively)\n\\begin{equation}\\label{eq:Q-modularity-measure}\n\\max\\limits_{A\\subset V}Q(A)=\\max\\limits_{x\\ne 0}\\frac{\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|}{4\\|\\vec x\\|_\\infty}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:Q\/mu-modularity-measure-mu}\n\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\\frac{Q(A)}{\\mu(A)\\mu(V\\setminus A)}=\\max\\limits_{\\sum_{i\\in V} \\mu_ix_i=0}\\frac{\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|}{\\mu(V)\\sum_{i\\in V}\\mu_i|x_i|}.\n\\end{equation}\n\n\nIt is clear that  \\eqref{eq:Q-modularity-measure} can be obtained more directly by Theorem \\ref{thm:tilde-fg-equal}. We shall also state a new analog of \\eqref{eq:Q\/mu-modularity-measure-mu}: \n\\begin{equation}\\label{eq:Q\/mu-modularity-measure}\n\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\\frac{Q(A)}{\\mu(A)\\mu(V\\setminus A)}=\\max\\limits_{\\sum_{i\\in V} x_i=0}\\frac{\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|}{\\sum_{i,j\\in V}\\mu_i\\mu_j|x_i-x_j|}\n\\end{equation}\nwhich can be derived straightforwardly by Theorem \\ref{thm:tilde-fg-equal}. \n\nBy Proposition \\ref{pro:Lovasz-f-pre}, we immediately obtain  Theorem 1 in \\cite{CRT20}, i.e., \nfor any $a,b>0$, \n$$ \\max\\limits_{-a\\le x_i\\le b,\\forall i}\\frac12\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|=(a+b)\\max\\limits_{A\\subset V}Q(A).$$\n\n\n\\textbf{A relation with the frustration index}\n\n For a signed weighted graph with real weights $(w_{ij})_{i,j\\in V}$ and signs $s_{ij}=\\mathrm{sign}(w_{ij})$, we define the frustration index as \\begin{equation}\\label{eq:frustration-weight}\n\\min_{x_i\\in \\{-1,1\\},\\forall i} \\sum_{\\{i,j\\}}|w_{ij}|\\cdot|x_i-s_{ij}x_j|.\n\\end{equation} \n\nThe following result reveals an interesting relation between the modularity\nmeasure and the frustration index.\n\\begin{pro}For a weighted graph $(V,(w_{ij})_{i,j\\in V})$, let $\\tilde{w}_{ij}=w_{ij}-\\frac{\\deg(i)\\deg(j)}{\\vol(V)}$. \nIn the signed weighted graph $(V,(\\tilde{w}_{ij})_{i,j\\in V})$, \n $\\{i,j\\}$ is a positive (resp. negative) edge if $\\tilde{w}_{ij}>0$ (resp. $\\tilde{w}_{ij}<0$). Then, the frustration index of $(V,(\\tilde{w}_{ij})_{i,j\\in V})$ equals \n$2\\left(\\sum_{\\{i,j\\}:\\tilde{w}_{ij}<0}|\\tilde{w}_{ij}|-\\max\\limits_{A\\subset V}Q(A)\\right)$.\n\\end{pro}\n\n\\begin{proof}\nWe know from Section \\ref{sec:frustration} (or by Theorem \\ref{thm:tilde-fg-equal}) that the frustration index of $(V,(\\tilde{w}_{ij})_{i,j\\in V})$ equals \n$$2\\left(\\sum_{\\{i,j\\}:\\tilde{w}_{ij}<0}|\\tilde{w}_{ij}|+\\min\\limits_{\\vec x\\ne0}\\frac{\\sum_{i,j\\in V}\\tilde{w}_{ij}|x_i-x_j|}{4\\|x\\|_\\infty}\\right).$$\nThe proof is then completed by \\eqref{eq:Q-modularity-measure}.\n\\end{proof}\n\n\\subsection{Chromatic number}\\label{sec:chromatic-number}\nThe chromatic number (i.e., the smallest vertex coloring number) of a graph is the smallest number of colors needed to color the vertices\nso that no two adjacent vertices share the same color.\nGiven a simple connected graph $G=(V,E)$ with $\\#V=n$, its chromatic number $\\gamma(G)$ can be expressed as a global optimization on the $n$-power set of vertices:\n\\begin{equation}\\label{eq:coloring-number-sum}\n\\gamma(G)=\\min\\limits_{(A_1,\\cdots,A_n)\\in\\ensuremath{\\mathcal{P}}_n(V)}\\left\\{n\\sum_{i=1}^n\\#E(A_i)\n+\\sum_{i=1}^n\\sgn(\\#A_i)+n\\left(n-\\sum_{i=1}^n\\#A_i\\right)^2\\right\\}\n\\end{equation}\nand similarly, we get the following\n\\begin{pro}The chromatic number $\\gamma(G)$ of a finite simple graph $G=(V,E)$ satisfies\n\\begin{equation}\\label{eq:coloring-number}\n\\gamma(G)=\\min\\limits_{(A_1,\\cdots,A_n)\\in\\ensuremath{\\mathcal{P}}(V)^n}\\left\\{n\\sum_{i=1}^n\\#E(A_i)+\\sum_{i=1}^n\\sgn(\\#A_i)+n\\left(n-\\#\\bigcup_{i=1}^n A_i\\right)\\right\\}\n\\end{equation}\n\\end{pro}\n\n\\begin{proof\n Let $f:\\ensuremath{\\mathcal{P}}(V)^n\\to\\ensuremath{\\mathbb{R}}$ be defined by  $$f(A_1,\\cdots,A_n)=n\\sum_{i=1}^n\\#E(A_i)+\\sum_{i=1}^n\\sgn(\\#A_i)+n\\left(n-\\#\\bigcup_{i=1}^n A_i\\right).$$\n Let $\\{C_1,\\cdots,C_{\\gamma(G)}\\}$ be a proper coloring class of $G$, and set  $C_{\\gamma(G)+1}=\\cdots=C_n=\\varnothing$. Then we have $E(C_i)=\\varnothing$, $\\#\\cup_{i=1}^n C_i=n$,  $\\#C_i\\ge1$ for $1\\le i\\le \\gamma(G)$, and $\\#C_i=0$ for $i> \\gamma(G)$. In consequence, $f(C_1,\\cdots,C_n)=\\gamma(G)$. Thus, it suffices to prove $f(A_1,\\cdots,A_n)\\ge \\gamma(G)$ for any $(A_1,\\cdots,A_n)\\in\\ensuremath{\\mathcal{P}}(V)^n$.\n\nIf $\\bigcup_{i=1}^n A_i\\ne V$, then $f(A_1,\\cdots,A_n)\\ge n+1> \\gamma(G)$.\n\nIf there exist at least $\\gamma(G)+1$  nonempty sets $A_1,\\cdots,A_{\\gamma(G)+1}$, then $f(A_1,\\cdots,A_n)\\ge \\gamma(G)+1> \\gamma(G)$.\n\nSo we focus on the case that $\\bigcup_{i=1}^n A_i= V$ and $A_{\\gamma(G)+1}=\\cdots=A_n=\\varnothing$. If there further exists $i\\in\\{1,\\cdots,\\gamma(G)\\}$ such that $A_i=\\varnothing$, then by the definition of  the chromatic number, there is $j\\in \\{1,\\cdots,\\gamma(G)\\}\\setminus\\{i\\}$ with $E(A_j)\\ne \\varnothing$. So $f(A_1,\\cdots,A_n)\\ge n+1> \\gamma(G)$.\nAccordingly, each of $A_1,\\cdots,A_{\\gamma(G)}$ must be nonempty, and thus $f(A_1,\\cdots,A_n)\\ge\\gamma(G)$.\n\nAlso, when the equality $f(A_1,\\cdots,A_n)=\\gamma(G)$ holds, one may see from the above discussion that $A_1,\\cdots,A_{\\gamma(G)}$ are all independent sets of $G$ with $\\bigcup_{i=1}^n A_i\\ne V$.\n\n\\end{proof}\n\nNote that\n$$\\#\\bigcup_{i=1}^n V^t(\\vec x^i)=\\#\\{j\\in V:\\exists i \\text{ s.t. }x_{i,j}>t\\}=\\sum_{j=1}^n\\max\\limits_{i=1,\\cdots,n} 1_{x_{i,j}>t}=\\sum_{j=1}^n 1_{\\max\\limits_{i=1,\\cdots,n}x_{i,j}>t}$$\nSo the $n$-way Lov\\'asz extension of $\\#\\bigcup_{i=1}^n A_i$ is\n\\begin{align*}\n\\int_{\\min\\vec x}^{\\max\\vec x}\\#\\bigcup_{i=1}^n V^t(\\vec x^i)dt+\\min\\vec x\\#\\bigcup_{i=1}^n V(\\vec x^i)&=\\sum_{j=1}^n \\int_{\\min\\vec x}^{\\max\\vec x}1_{\\max\\limits_{i=1,\\cdots,n}x_{i,j}>t}dt+\\min\\vec x\\#V\n\\\\&=\\sum_{j=1}^n(\\max\\limits_{i=1,\\cdots,n}x_{i,j}-\\min\\vec x)+n\\min\\vec x\n\\\\&=\\sum_{j=1}^n\\max\\limits_{i=1,\\cdots,n}x_{i,j}\n\\end{align*}\nAnd the $n$-way disjoint-pair Lov\\'asz extension of $\\#\\bigcup_{i=1}^n A_i$ is $\\sum_{j=1}^n\\max\\limits_{i=1,\\cdots,n}|x_{i,j}|=\\sum_{j=1}^n\\|\\vec x^{,j}\\|_\\infty$.\n\nThe $n$-way Lov\\'asz extension of $\\sgn(\\#A_i)$ is\n\\begin{align*}\n\\int_{\\min\\vec x}^{\\max\\vec x}\\sgn(\\#V^t(\\vec x^i))dt+\\min\\vec x\\sgn(\\#V(\\vec x^i))&= \\int_{\\min\\vec x}^{\\max\\vec x^i}1dt+\\min\\vec x\\sgn(\\#V)\n\\\\&= \\max\\limits_{j=1,\\cdots,n}x_{i,j}-\\min\\vec x+\\min\\vec x= \\max\\limits_{j=1,\\cdots,n}x_{i,j}\n\\end{align*}\nand the $n$-way disjoint-pair  Lov\\'asz extension of $\\sgn(\\#A_i)$ is $\\|\\vec x^{i}\\|_\\infty$. Similarly, the $n$-way disjoint-pair Lov\\'asz extension of $\\#E(A_i)$ is $\\sum_{j\\sim j'}\\min\\{|x_{i,j}|,|x_{i,j'}|\\}$. Thus\n\\begin{align*}\nf^L(\\vec x)&=n\\sum_{i=1}^n\\sum_{j\\sim j'}\\min\\{|x_{i,j}|,|x_{i,j'}|\\} +\\sum_{i=1}^n\\|\\vec x^{i}\\|_\\infty+n\\left(n\\|\\vec x\\|_\\infty-\\sum_{j=1}^n\\|\\vec x^{,j}\\|_\\infty\\right)\n\\\\&\n=n\\sum_{i=1}^n\\left(\\|\\vec x^i\\|_{1,\\deg}-(I^+(\\vec x^i)+I^-(\\vec x^i))\/2\\right) +\\sum_{i=1}^n\\|\\vec x^{i}\\|_\\infty+n\\left(n\\|\\vec x\\|_\\infty-\\sum_{j=1}^n\\|\\vec x^{,j}\\|_\\infty\\right)\n\\\\&\n=n^2\\|\\vec x\\|_\\infty+n\\|\\vec x\\|_{1\\text{-}deg,1}+\\|\\vec x\\|_{\\infty,1}-nI_{\\pm,1}(\\vec x)-n\\|\\vec x\\|^{\\infty, 1}.\n\\end{align*}\nAccording to Proposition \\ref{pro:fraction-f\/g} on the multi-way Lov\\'asz extension, we get\n\\begin{equation}\\label{eq:chromatic-continuous}\n\\gamma(G)= n^2-\\sup\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^{n^2}\\setminus\\{\\vec 0\\}}\\frac{nI_{\\pm,1}(\\vec x)+n\\|\\vec x\\|^{\\infty, 1}-n\\|\\vec x\\|_{1\\text{-}deg,1}-\\|\\vec x\\|_{\\infty,1}}{\\|\\vec x\\|_\\infty}.\n\\end{equation}\n\n\\paragraph{Clique covering number }\n\nThe clique covering number of a graph $G$ is the minimal number of cliques in $G$ needed to cover the vertex set. It is equal to the chromatic number of the graph complement of $G$. Consequently, we can explicitly write down the continuous representation of a clique covering number by employing Theorem \\ref{thm:graph-numbers}.\n\n\n{ \\linespread{0.95} \\small ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section 1}\n\nM. J. Golay introduced Golay complementary pairs (GCPs) in his work on multislit spectrometry \\cite{Golay51}. GCPs are sequence pairs having zero aperiodic autocorrelation sums (AACS) at all non-zero time shifts \\cite{Golay61}. Due to their ideal correlation sums, GCPs have found numerous applications in modern day communication systems \\cite{Davis1999,Paterson2000,Georghiades2001,farrel2003,abdi2007,lei2014}, Radar \\cite{Spano1996,Pezeshki2008}, etc. One of the main drawbacks of the GCPs are its availability for limited lengths \\cite{Borwein2000}. To overcome this drawback and to find the sequence pairs which depicts ``closest\" autocorrelation properties to that of the GCPs Fan \\textit{et al.} proposed Z-complementary pairs (ZCPs) in 2007 \\cite{Fan2007}. ZCPs are sequence pairs having zero AACS within a certain time-shift around the in-phase position \\cite{Fan2007}. In recent years lot of research has been done on the existence \\cite{Li2011}, systematic constructions \\cite{liu20141,liu2014,Adhikary2016,Adhikary2018,Adhikary2020,chen2017,Adhikary20201} and applications of ZCPs \\cite{Adhikary20191,chen20192}.\n\n\\subsection{Sequences pairs with zero periodic crosscorrelation zone}\nSince the autocorrelation of the sequence pair sum up to zero at all non-zero time-shifts (or time-shifts within a certain region in case of ZCPs) Golay sequences have been widely used to reduce peak-to-mean envelope power ratio in orthogonal frequency division multiplexing systems \\cite{Davis1999,Paterson2000}. However, the sequences own periodic autocorrelation plays an important role in some applications like synchronization and detection of the signal. Working in this direction, Gong \\textit{et al.} \\cite{Gong2013} investigated the periodic autocorrelation behaviour of a single Golay sequence in 2013. To be more specific, Gong \\textit{et al.} presented two constructions of Golay sequences of length $2^m$ each displaying a periodic zero autocorrelation zone (ZACZ) of $2^{m-2}$, and $2^{m-3}$, respectively, around the in-phase position \\cite{Gong2013}. In \\cite{Gong2013}, the authors also discussed the application of Golay sequences with large ZACZ for ISI channel estimation. Using Golay sequences with large ZACZ as channel estimation sequences (CES), the authors analysed the performance of\nGolay-sequence-aided channel estimation in terms of the error\nvariance and the classical Cramer-Rao lower bound (CRLB). The performance was also compared with the well known sequences (Frank-Zadoff-Chu sequences and $m$-sequences) which are generally used for ISI channel estimation. It was shown in \\cite{Gong2013} that when the channel impulse response (CIR) is within the ZACZ width then the variance of the Golay sequences attains the CRLB.\n\n\n\nInspired by the work of Gong \\textit{et al.} \\cite{Gong2013}, Chen \\textit{et al.} studied the zero cross-correlation zone among the Golay sequences in 2018 and proposed Golay-ZCZ sequence sets \\cite{Chen201811}. Golay-ZCZ sequence sets are sequence sets having periodic ZACZ for each sequences, periodic zero cross-correlation zone (ZCCZ) for any two sequences and also the aperiodic autocorrelation sum is zero for all non-zero time shifts. Specifically, Chen \\textit{et al.} gave a systematic construction of Golay-ZCZ sequence set consisting $2^k$ sequences, each of length $2^m$ and $\\min\\{ZACZ,ZCCZ\\}$ is $2^{m-k-1}$ \\cite{Chen201811}. However, the lengths of the GCPs with large ZCZs discussed in the works of Gong \\textit{et al.} and Chen \\textit{et al.} are all in the powers of two \\cite{Chen201811}. To the best of the authors knowledge, the problem of investigating the individual periodic autocorrelations of the GCPs and the periodic cross-correlations of the pairs when the length of the GCPs are non-power-of-two, remains largely open. An overview of of the previous works, which considers the periodic ZACZ of the individual sequences and ZCCZ of a GCP, is given in Table \\ref{Table duibi}.\n\n\\begin{table}\n  \\centering\n  \\caption{Golay sequences with periodic ZACZ and ZCCZ.}\n  \\label{Table duibi}\n\\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n \n  Ref. & Length $N$ & Complementary sets size $M$ & ZACZ width & ZCCZ width  \\\\\\hline\n\n  \\hline\n\n  \\cite{Gong2013} & $2^m$ & $2$ & $2^{m-2}$ or $2^{m-3}$ & Not discussed  \\\\\n  \\hline\n\n  \\cite{Chen201811} & $2^m$ & $2^k$ & $2^{m-k-1}$ & $2^{m-k-1}$ \\\\\n  \\hline\n\n  Theorem 1 & $4N$ & $2$ & $N$ & $N$ \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\nBased on the discussion on using Golay sequences as CES for ISI channel estimation in \\cite{Gong2013} it can be realised that our proposed constructions will add flexibility in choosing the Golay sequences of various lengths for using it as CES. Since,\nin practical scenarios, a longer training sequence will give\nrise to a higher training overhead, therefore, selection of the\ntraining length is a trade-off between channel estimation performance\nand training overhead. For example, let us consider\nthat in a practical scenario, the CIR is a length 10 vector. If only\nthe Golay sequences in \\cite{Gong2013} or \\cite{Chen201811} are considered, then one have to use a length 64 GCP which have a ZACZ width of 16. However, by our proposed constructions, a length 40 GCP which have a ZACZ width of 10 can be used as a CES as described in \\cite{Gong2013}. This will improve the system performance.\n\n\n\\subsection{Two-dimensional complementary array pairs}\nIn 1978, Ohyama \\textit{et al.} introduced two-dimensional (2D) sequence sets with zero side lobes in their work on image processing \\cite{Ohyama1978}. In 1985, H. D. Luke presented some iterative constructions of higher-dimensional complementary codes \\cite{Luke1985}. In 1990, Bomer \\textit{et al.} proposed perfect binary arrays \\cite{bomer1990}. In search of one-dimensional sequences with low autocorrelation magnitudes at non-zero time-shifts, in 2007, Jedwab and Parker made a remarkable progress by generalising the problem to multiple dimensions \\cite{Jedwab20071}. Instead of searching sequences with low correlation properties in one dimension, the authors analysed the possibility of existence of such sequence arrays in multiple dimensions in the hope of a larger existence patterns. In \\cite{Jedwab20071} the authors presented a systematic construction of $m$ dimensional Golay complementary array pairs (GCAPs) from an $m+1$ dimensional GCAPs. 2D- GCAPs are array pairs having the 2D autocorrelations of the constituent arrays sum up to zero for all non-zero time-shifts.\n\nGeneralizing the concept of ZCZ in 2D, Fan \\textit{et. al} introduced binary array set with ZCZ in 2001 \\cite{Tang2001}. In 2002 Hayashi proposed a class of 2D- binary sequences with ZCZ \\cite{Hayashi2004}. In 2010 Cheng \\textit{et al.} proposed another new class of class of 2D- binary sequences with ZCZ \\cite{Cheng2010}. Recently in 2019 Chen \\textit{et al.} proposed a systematic construction of 2D- Z-complementary array pairs (ZCAPs) \\cite{pai2019}. However, the behavior of autocorrelation\nof a single Golay array is still unknown. To the best of the authors knowledge, the problem of investigating the periodic 2D- autocorrelations of the constituent arrays of the 2D- GCAPs and the periodic 2D- cross-correlations of the 2D- GCAPs, remains largely open.\n\n\n\\subsection{Contributions}\n\nMotivated by the works of Gong \\textit{et al.} \\cite{Gong2013} and Chen \\textit{et al.} \\cite{Chen201811} we propose a systematic construction of GCPs of length non-power-of-two, where the individual sequences have a periodic ZACZ and the periodic cross-correlation of the sequence pairs also have a ZCCZ. We also extend the ideas to construct 2D-GCAPs.  To be more specific we make  the following contributions in this paper:\n\\begin{enumerate}\n\t\\item Assuming a GCP of length $N$ exists, we  systematically construct GCPs of length $4N$. The proposed GCPs have $Z_{\\min}=N+1$, where $Z_{\\min}=\\min\\{ZACZ,ZCCZ\\}$.\n\t\\item We  also systematically constructe a 2D GCAP of size $s_1\\times 4s_2$, assuming a 2D- GCAP of size $s_1 \\times s_2$ exists. The designed 2D- GCAPs have a $2D\\text{-}Z_{\\min}=s_1\\times (s_2+1)$, where $2D\\text{-}Z_{\\min}=\\min\\{2D\\text{-}ZACZ,2D\\text{-}ZCCZ\\}$.\n\t\\item We propose a systematic construction of 2D GCAP of size $4s_1\\times 4s_2$, assuming a 2D- GCAP of size $s_1 \\times s_2$ exists. The designed 2D- GCAPs have a $2D\\text{-}Z_{\\min}=(s_1+1)\\times (s_2+1)$, where $2D\\text{-}Z_{\\min}=\\min\\{2D\\text{-}ZACZ,2D\\text{-}ZCCZ\\}$.\n\\end{enumerate}\n\n\n\n\\subsection{Organization}\nThe rest of the paper is organized as follows. In Section \\ref{section 2}, some useful notations and preliminaries are recalled.\nIn Section \\ref{section 3}, a systematic construction for GCPs of lengths non-power-of-two with large periodic ZACZ and ZCCZ is proposed.\nIn Section \\ref{section 4}, we extended the construction to 2D- GCAPs.\nFinally, we conclude the paper in Section \\ref{section 5}.\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{section 2}\n\\begin{definition}\n\tLet $\\mathcal{A}=[A_{i,j}]$ and $\\mathcal{B}=[B_{i,j}]$, for $0\\leq i <L_1$ and $0\\leq j <L_2$, be two unimodular complex-valued 2D- arrays, each of size $L_1\\times L_2$. Then, for $-L_1<\\tau_1<L_1$ and $-L_2<\\tau_2<L_2$, the 2D- periodic crosscorrelation function is defined as follows:\n\t\\begin{equation}\n      R_{\\mathcal{A},\\mathcal{B}}(\\tau_1,\\tau_2)=\\sum\\limits_{i=0}^{L_1-1}\\sum\\limits_{j=0}^{L_2-1}A_{i,j}B^*_{(i+\\tau_1)\\pmod {L_1},(j+\\tau_2)\\pmod {L_2}},\n\t\\end{equation}\n\twhere $(^*)$ denotes the complex conjugate of $B_{(i+\\tau_1)\\pmod {L_1},(j+\\tau_2)\\pmod {L_2}}$. When $L_1=1$, then the above definition reduces to a 1D- periodic cross-correlation function. When $\\mathcal{A}=\\mathcal{B}$, it is called periodic autocorrelation function, and is denoted by $R_{\\mathcal{A}}$ for short.\n\\end{definition}\n\n\\begin{definition}\n\tFor $0\\leq i <L_1$ and $0\\leq j <L_2$, let $\\mathcal{A}=[A_{i,j}]$ and $\\mathcal{B}=[B_{i,j}]$ be two unimodular complex-valued 2D- arrays, each of size $L_1\\times L_2$. Then the 2D- aperiodic crosscorrelation function is defined as follows:\n\t\\begin{equation}\n\tC_{\\mathcal{A},\\mathcal{B}}(\\tau_1,\\tau_2)=\\begin{cases}\n\t\\sum\\limits_{i=0}^{L_1-1-\\tau_1}\\sum\\limits_{j=0}^{L_2-1-\\tau_2}A_{i,j}B^*_{(i+\\tau_1),(j+\\tau_2)}, & 0\\leq \\tau_1<L_1,0\\leq \\tau_2<L_2;\\\\\n\t\\sum\\limits_{i=0}^{L_1-1+\\tau_1}\\sum\\limits_{j=0}^{L_2-1-\\tau_2}A_{i-\\tau_1,j}B^*_{i,(j+\\tau_2)}, & -L_1< \\tau_1<0,0\\leq \\tau_2<L_2;\\\\\n\t\\sum\\limits_{i=0}^{L_1-1-\\tau_1}\\sum\\limits_{j=0}^{L_2-1+\\tau_2}A_{i,j-\\tau_2}B^*_{(i+\\tau_1),j}, & 0\\leq \\tau_1<L_1,-L_2< \\tau_2<0;\\\\\n\t\\sum\\limits_{i=0}^{L_1-1+\\tau_1}\\sum\\limits_{j=0}^{L_2-1+\\tau_2}A_{i-\\tau_1,j-\\tau_2}B^*_{i,j}, & -L_1< \\tau_1<0,-L_2< \\tau_2<0;\n\t\\end{cases}\n\t\\end{equation}\n\twhere $(^*)$ denotes the complex conjugate of the element. When $L_1=1$, then the above definition reduces to a 1D- aperiodic cross-correlation function. When $\\mathcal{A}=\\mathcal{B}$, it is called aperiodic autocorrelation function, and is denoted by $C_{\\mathcal{A}}$.\n\\end{definition}\n\n\n\n\n\n\n\n\nUsing the above definitions, we have the following property.\n\\begin{property}\nLet $\\mathcal{A}$ be a 2D- array with size $L_1\\times L_2$, then for $0\\leq \\tau_1<L_1$ and $0\\leq \\tau_2<L_2$, we have\n  \\begin{equation}\n  \\begin{split}\n  C_{\\mathcal{A}}(\\tau_{1},\\tau_2) &=C_{\\mathcal{A}}^*(-\\tau_{1},-\\tau_2);\\\\\n  R_{\\mathcal{A}}(\\tau_{1},\\tau_2) &=R_{\\mathcal{A}}^*(-\\tau_{1},-\\tau_2).\n  \\end{split}\n  \\end{equation}\n\\end{property}\n\n\n\\begin{definition}\\label{Definition-GCAP}\nA pair of 2D- arrays $\\mathcal{A}$ and $\\mathcal{B}$, each of size $L_1\\times L_2$ is called a Golay complementary array pair (GCAP) if\n\\begin{equation}\n  C_\\mathcal{A}(\\tau_1,\\tau_2)+C_\\mathcal{B}(\\tau_1,\\tau_2)=0, \\hbox{ for all } (\\tau_1,\\tau_2)\\neq(0,0).\n\\end{equation}\nIn particular, when $L_1=1$, GCAP is reduced to a Golay complementary pair (GCP), denoted by $(\\mathbf{a,b})$.\n\\end{definition}\n\n\\begin{definition}\\label{Definition-GAMate}\nLet $(\\mathcal{A,B})$ and $(\\mathcal{C,D})$ be two GCAPs, each of size $L_1\\times L_2$. Then\n$(\\mathcal{C,D})$ is called a Golay array mate of $(\\mathcal{A,B})$ if \\cite{Matsufuji2010}\n$$C_\\mathcal{A,C}(\\tau_1,\\tau_2)+C_\\mathcal{B,D}(\\tau_1,\\tau_2)=0,$$\nfor any $-L_1<\\tau_1<L_1$ and $-L_2<\\tau_2<L_2$.\nIn particular, when $L_1=1$, $(\\mathcal{A,B})$ reduces to a GCP and $(\\mathcal{C,D})$ reduces to mate of the GCP $(\\mathcal{A,B})$.\n\\end{definition}\n\n\\begin{lemma}\nLet $(\\mathcal{A,B})$ be GCAP of size $L_1\\times L_2$. Then one of the mate of $(\\mathcal{A,B})$ is as follows:\n\\begin{equation}\n\t(\\mathcal{C,D}) =(\\overleftarrow{\\mathcal{B^*}},-\\overleftarrow{\\mathcal{A^*}}),\n\\end{equation}\nwhere $\\overleftarrow{\\mathcal{A}}$ denote reverse of array $\\mathcal{A}$ at every dimension and $(^*)$ represents complex conjugation.\n\\end{lemma}\n\n\n\\begin{definition}\\label{Definition-GolayZCZ}\nLet ($\\mathcal{A}$,$\\mathcal{B}$) be a GCAP of size $L_1\\times L_2$. Let the 2D- periodic autocorrelation function of $\\mathcal{A}$ and $\\mathcal{B}$ have a 2D- ZACZ of $Z_{k_0}$ and $Z_{k_1}$, respectively. Also let the 2D- periodic cross-correlation function have a 2D- ZCCZ of $Z_{k_2}$ when the cross-correlation between $\\mathcal{A}$ and $\\mathcal{B}$ is considered. Then ($\\mathcal{A}$,$\\mathcal{B}$) is a GCAP having a 2D- periodic ZCZ of $Z_{\\min}=\\min\\{Z_{k_0},Z_{k_1},Z_{k_2}\\}$ and we denote it as\n$(L_1\\times L_2,Z_{\\min})$- GCAP. In other words, the 2D- array pair ($\\mathcal{A}$,$\\mathcal{B}$) is a $(L_1\\times L_2,Z_{\\min})$- GCAP, having 2D- periodic ZCZ $Z_{\\min}$ if\n\\begin{equation}\n\\begin{split}\n&\\text{C1: }C_\\mathcal{A}(\\tau_1,\\tau_2)+C_\\mathcal{B}(\\tau_1,\\tau_2)=0, \\hbox{ for all } (\\tau_1,\\tau_2)\\neq(0,0);\\\\\n&\\text{C2: }R_\\mathcal{A}(\\tau_1,\\tau_2)=R_\\mathcal{B}(\\tau_1,\\tau_2)=0, \\hbox{ for } |\\tau_1|<Z_{\\min},|\\tau_2|<Z_{\\min} \\text{ and }(\\tau_1,\\tau_2)\\neq(0,0);\\\\\n&\\text{C3: }R_{\\mathcal{A},\\mathcal{B}}(\\tau_1,\\tau_2)=0, \\hbox{ for } |\\tau_1|<Z_{\\min},|\\tau_2|<Z_{\\min},\n\\end{split}\n\\end{equation}\nwhere, $Z_{\\min}$ is defined as above. When $L_1=1$, we get the conditions for $(L_2,Z_{\\min})$- GCPs having 1D- periodic ZCZ width of $Z_{\\min}$.\n\\end{definition}\n\n\n\n\\section{New Construction of Golay Complementary Pair with ZCZ}\\label{section 3}\nIn this section we propose a new class of GCPs, where each of the sequences have a periodic ZACZ and also have a periodic ZCCZ when the cross-correlation between the pairs are considered. Please note that GCPs are 1D- GCAPs where $L_1=1$ and $L_2=N$ (in line with Section II). In this section, $\\mathbf{a}\\|\\mathbf{b}$ denotes the concatenation of sequences $\\mathbf{a}$ and $\\mathbf{b}$ and $Z_{\\min}$ is usually 1D- $Z_{\\min}$. Before introduce the construction, we need the following lemma.\n\n\\begin{lemma}\\label{lem2}\n\tLet $(\\mathcal{A,B})$ a be GCAP of size $s_1\\times s_2$ and $(\\mathcal{C,D})$ be one of the Golay array mate of $(\\mathcal{A,B})$. Let\n\t\\begin{equation}\n\t\\mathcal{P}=\n\t\\left[\n\t\\begin{array}{rrrr}\n\tx_1\\mathcal{A} & x_2\\mathcal{B} & x_3\\mathcal{A} & x_4\\mathcal{B} \\\\\n\t\\end{array}\n\t\\right],\n\t\\mathcal{Q}=\n\t\\left[\n\t\\begin{array}{rrrr}\n\tx_1\\mathcal{C} & x_2\\mathcal{D} & x_3\\mathcal{C} & x_4\\mathcal{D} \\\\\n\t\\end{array}\n\t\\right],\n\t\\end{equation}\n\twhere $x_1,x_2,x_3,x_4\\in\\{+1,-1\\}$. Then $(\\mathcal{P,Q})$ is a GCAP of size $s_1\\times 4s_2$, if the following condition holds:\n\t\\begin{equation}\n\t\tx_1x_2+x_3x_4=0.\n\t\\end{equation}\n\\end{lemma}\n\\begin{IEEEproof} Let us consider $0\\leq\\tau_1\\leq s_1-1$ and $0\\leq\\tau_2\\leq s_2-1$. Then we have\n\t\\begin{equation}\n\t\\begin{split}\n\tC_\\mathcal{P}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{A}(\\tau_1,\\tau_2) +C_\\mathcal{B}(\\tau_1,\\tau_2)\\big)+(x_1x_2+x_3x_4)C_\\mathcal{B,A}^*(\\tau_1,s_2-\\tau_2) +x_2x_3\\rho_\\mathcal{A,B}^*(\\tau_1,s_2-\\tau_2), \\\\\n\tC_\\mathcal{Q}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{C}(\\tau_1,\\tau_2) +C_\\mathcal{D}(\\tau_1,\\tau_2)\\big)+(x_1x_2+x_3x_4)C_\\mathcal{D,C}^*(\\tau_1,s_2-\\tau_2) +x_2x_3\\rho_\\mathcal{C,D}^*(\\tau_1,s_2-\\tau_2).\n\t\\end{split}\n\t\\end{equation}\n\tHence, for $0\\leq\\tau_1\\leq s_1-1$ and $0\\leq\\tau_2\\leq s_2-1$, we have\n\t\\begin{equation}\n\t\tC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=4\\big(C_\\mathcal{A}(\\tau_1,\\tau_2) +C_\\mathcal{B}(\\tau_1,\\tau_2)\\big).\n\t\\end{equation}\n\nConsider $0\\leq\\tau_1\\leq s_1-1$ and $s_2\\leq\\tau_2\\leq 2s_2-1$. Then one has\n\\begin{equation}\n\\begin{split}\nC_\\mathcal{P}(\\tau_1,\\tau_2)=& (x_1x_2+x_3x_4)C_\\mathcal{A,B}(\\tau_1,\\tau_2-s_2) +x_2x_3C_\\mathcal{B,A}(\\tau_1,\\tau_2-s_2) +x_1x_3C_\\mathcal{A}^*(\\tau_1,2s_2-\\tau_2)\\\\&\\hspace{1in} +x_2x_4C_\\mathcal{B}^*(\\tau_1,2s_2-\\tau_2), \\\\\nC_\\mathcal{Q}(\\tau_1,\\tau_2)=& (x_1x_2+x_3x_4)C_\\mathcal{C,D}(\\tau_1,\\tau_2-s_2) +x_2x_3C_\\mathcal{D,C}(\\tau_1,\\tau_2-s_2) +x_1x_3C_\\mathcal{C}^*(\\tau_1,2s_2-\\tau_2)\\\\&\\hspace{1in} +x_2x_4C_\\mathcal{D}^*(\\tau_1,2s_2-\\tau_2).\n\\end{split}\n\\end{equation}\n\nHence, for $0\\leq\\tau_1\\leq s_1-1$ and $s_2\\leq\\tau_2\\leq 2s_2-1$, we have\n\\begin{equation}\nC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=0.\n\\end{equation}\n\n\nConsider $0\\leq\\tau_1\\leq s_1-1$ and $2s_2\\leq\\tau_2\\leq 3s_2-1$. Then we have\n\\begin{equation}\n\\begin{split}\nC_\\mathcal{P}(\\tau_1,\\tau_2)=& x_1x_3C_\\mathcal{A}(\\tau_1,\\tau_2-2s_2) +x_2x_4C_\\mathcal{B}(\\tau_1,\\tau_2-2s_2) +x_1x_4C_\\mathcal{B,A}^*(\\tau_1,3s_2-\\tau_2), \\\\\nC_\\mathcal{Q}(\\tau_1,\\tau_2)=& x_1x_3C_\\mathcal{C}(\\tau_1,\\tau_2-2s_2) +x_2x_4C_\\mathcal{D}(\\tau_1,\\tau_2-2s_2) +x_1x_4C_\\mathcal{D,C}^*(\\tau_1,3s_2-\\tau_2).\n\\end{split}\n\\end{equation}\n\nHence, for $0\\leq\\tau_1\\leq s_1-1$ and $2s_2\\leq\\tau_2\\leq 3s_2-1$, we have\n\\begin{equation}\nC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=0.\n\\end{equation}\n\nConsider $0\\leq\\tau_1\\leq s_1-1$ and $3s_2\\leq\\tau_2\\leq 4s_2-1$. Then we have\n\\begin{equation}\n\\begin{split}\nC_\\mathcal{P}(\\tau_1,\\tau_2)=& x_1x_4C_\\mathcal{A,B}(\\tau_1,\\tau_2-3s_2), \\\\\nC_\\mathcal{Q}(\\tau_1,\\tau_2)=& x_1x_4C_\\mathcal{C,D}(\\tau_1,\\tau_2-3s_2),.\n\\end{split}\n\\end{equation}\n\nHence, for $0\\leq\\tau_1\\leq s_1-1$ and $3s_2\\leq\\tau_2\\leq 4s_2-1$, we have\n\\begin{equation}\nC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=0.\n\\end{equation}\n\nThe other cases can also be proved similarly.\n\\end{IEEEproof}\n\n\n\n\\begin{construction}\\label{Construction-seq-ZCZ}\nLet $(\\mathbf{a,b})$ be a GCP of length $N$ and $(\\mathbf{c,d})$ be the Golay mate of $(\\mathbf{a,b})$, then define\n\\begin{equation}\n  \\begin{split}\n    \\mathbf{p}=~&x_1\\cdot\\mathbf{a}~\\|~x_2\\cdot\\mathbf{b}~\\|~x_3\\cdot\\mathbf{a}~\\|x_4\\cdot\\mathbf{b}, \\\\\n    \\mathbf{q}=~&x_1\\cdot\\mathbf{c}~\\|~x_2\\cdot\\mathbf{d}~\\|~x_3\\cdot\\mathbf{c}~\\|x_4\\cdot\\mathbf{d},\n  \\end{split}\n\\end{equation}\nwhere $x_1,x_2,x_3,x_4\\in\\{+1,-1\\}$.\n\\end{construction}\n\n\\begin{theorem}\n  Let $(\\mathbf{p,q})$ be a sequence pair generated via Construction \\ref{Construction-seq-ZCZ}. Then $(\\mathbf{p,q})$ is a GCP of length $4N$ with $Z_{\\min}=N+1$, if $x_1,x_2,x_3,x_4$ if the following condition holds:\n  \\begin{equation}\n  \tx_1x_2+x_3x_4=0.\n  \\end{equation}\n\\end{theorem}\n\\begin{IEEEproof}\nUsing Lemma \\ref{lem2} for 1D cases, i.e., considering $s_1=1$ and $s_2=N$, we can prove that $(\\mathbf{p,q})$ is a GCP of length $4N$.\t\n\nNext, we have to prove the other two conditions of Definition \\ref{Definition-GolayZCZ}. Consider $1\\leq\\tau\\leq N$, then we have\n\\begin{equation}\n\\begin{split}\nR_\\mathbf{p}(\\tau)=& \\sum_{k=0}^{4N-1} p_kp_{k+\\tau}^*\\\\\n=& \\sum_{k=0}^{N-1-\\tau} a_ka_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_1a_kx_2^*b_{k-(N-\\tau)}^* +\\sum_{k=0}^{N-1-\\tau} b_kb_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_2b_kx_3^*a_{k-(N-\\tau)}^* +\\\\\n& \\sum_{k=0}^{N-1-\\tau} a_ka_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_3a_kx_4^*b_{k-(N-\\tau)}^* +\\sum_{k=0}^{N-1-\\tau} b_kb_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_4b_kx_1^*a_{k-(N-\\tau)}^* \\\\\n=&2\\big(\\rho_\\mathbf{a}(\\tau) +\\rho_\\mathbf{b}(\\tau)\\big). \\\\\n\\end{split}\n\\end{equation}\nSimilarly for $1\\leq\\tau\\leq N$, we have\n\\begin{equation}\n\\begin{split}\nR_\\mathbf{q}(\\tau)=&2\\big(C_\\mathbf{c}(\\tau) +C_\\mathbf{d}(\\tau)\\big), \\\\\nR_\\mathbf{p,q}(\\tau)=&2\\big(C_\\mathbf{a,c}(\\tau) +C_\\mathbf{b,d}(\\tau)\\big).\n\\end{split}\n\\end{equation}\nSince $(\\mathbf{a},\\mathbf{b})$ is a GCP and $(\\mathbf{c},\\mathbf{d})$ is one of the complementary mates of $(\\mathbf{a},\\mathbf{b})$, $R_\\mathbf{p}(\\tau)=R_\\mathbf{q}(\\tau)=0,\\hbox{ for all } 1\\leq\\tau\\leq N,$ and $R_\\mathbf{p,q}(\\tau)=0,\\hbox{ for all } 0\\leq\\tau\\leq N$.\n\nTherefore, $(\\mathbf{p,q})$ is a GCP of length $4N$ with $Z_{\\min}=N+1$.\n\\end{IEEEproof}\n\n\n\n\\begin{example}\\label{ex1 binary}\nLet $(\\mathbf{a,b})$ be a binary GCP of length 10, given by $\\mathbf{a}=(1,1,-1,1,1,1,1,1,-1,-1),\\mathbf{b}=(1,1,-1,1,-1,1,-1, -1,1,1)$.\nThen $(\\mathbf{c,d})=(\\overleftarrow{\\mathbf{b}^*},-\\overleftarrow{\\mathbf{a}^*})$ is a Golay mate of $(\\mathbf{a,b})$. Define\n\\begin{equation}\n  \\begin{split}\n    \\mathbf{p}=~&\\mathbf{a}~\\|~\\mathbf{b}~\\|~\\mathbf{a}~\\|-\\mathbf{b}, \\\\\n    \\mathbf{q}=~&\\mathbf{c}~\\|~\\mathbf{d}~\\|~\\mathbf{c}~\\|-\\mathbf{d}.\n  \\end{split}\n\\end{equation}\nThen $(\\mathbf{p,q})$ is a GCP of length $40$ with $Z_{\\min}=11$, because\n\\begin{equation}\n  \\begin{split}\n    \\big(R_\\mathbf{p}(\\tau)\\big)_{\\tau=0}^{39}=&(40,\\mathbf{0}_{10},-4,-8,4,8, -4,0,4,0,12,0,12,0,4,0,-4,8,4,-8,-4,\\mathbf{0}_{10}), \\\\\n    \\big(R_\\mathbf{q}(\\tau)\\big)_{\\tau=0}^{39}=&(40,\\mathbf{0}_{10},4,8,-4,-8, 4,0,-4,0,-12,0,-12,0,-4,0,4,-8,-4,8,4,\\mathbf{0}_{10}), \\\\\n    \\big(R_\\mathbf{p,q}(\\tau)\\big)_{\\tau=0}^{39}=&(\\mathbf{0}_{11},-4,-8,4, 16,4,0,4,-8,-4,0,4,-8,12,0,12,0,-4,8,4,\\mathbf{0}_{10}).\n  \\end{split}\n\\end{equation}\n\\end{example}\n\n\n\n\\begin{example}\\label{ex1}\nLet $\\mathbf{a}=(1,1,-1),\\mathbf{b}=(1,i,1)$, then $(\\mathbf{a,b})$ be a quadriphase GCP of length 3.\nThen $(\\mathbf{c,d})=(\\overleftarrow{\\mathbf{b}^*},-\\overleftarrow{\\mathbf{a}^*})$ is a Golay mate of $(\\mathbf{a,b})$. Define\n\\begin{equation}\n  \\begin{split}\n    \\mathbf{p}=~&\\mathbf{a}~\\|~\\mathbf{b}~\\|~\\mathbf{a}~\\|-\\mathbf{b} =(1,1,-1,1,i,1,1,1,-1,-1,-i,-1), \\\\\n    \\mathbf{q}=~&\\mathbf{c}~\\|~\\mathbf{d}~\\|~\\mathbf{c}~\\|-\\mathbf{d} =(1,-i,1,1,-1,-1,1,-i,1,-1,1,1).\n  \\end{split}\n\\end{equation}\nThen $(\\mathbf{p,q})$ is a GCP of length $12$ with $Z_{\\min}=4$, because\n\\begin{equation}\n  \\begin{split}\n    \\big(R_\\mathbf{p}(\\tau)\\big)_{\\tau=0}^{11}=&(12,0,0,0,-4,0,0,0,-4,0,0,0), \\\\\n    \\big(R_\\mathbf{q}(\\tau)\\big)_{\\tau=0}^{11}=&(12,0,0,0,4,0,0,0,4,0,0,0), \\\\\n    \\big(R_\\mathbf{p,q}(\\tau)\\big)_{\\tau=0}^{11}=&(0,0,0,0,-4,4-4i,4i,4+4i,4,0,0,0).\n  \\end{split}\n\\end{equation}\nThe periodic correlation magnitudes of $(12,4)$- GCP ($\\mathbf{p,q}$) are shown in Fig. \\ref{fig1}.\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=.7\\textwidth]{Example1.pdf}\n  \\caption{A glimpse of the 1D- periodic correlations of the proposed GCP given in Example \\ref{ex1}.}\\label{fig1}\n\\end{figure}\n\\end{example}\n\n\\begin{remark}\n\tPlease note that binary GCP of length 40 and quadriphase GCP of length 12 with ZCZ has not been previously reported in the literature. By taking GCP $(\\mathbf{a,b})$ of length $2^m$, then length of $(\\mathbf{p,q})$ is $2^{m+2}$ and the generated GCP $(\\mathbf{p,q})$ is reduced to the GCP in \\cite{Gong2013}.\n\\end{remark}\n\n\n\n\\section{New Construction of Golay Complementary Array with ZCZ}\\label{section 4}\nIn this section we have proposed new sets of GCAPs, where each of the arrays have a 2D- periodic ZACZ and also have a 2D- periodic ZCCZ when the cross-correlation between the array pairs are considered. In this section $Z_{\\min}$ are usually 2D- $Z_{\\min}$.\n\\begin{construction}\\label{Construction-array-ZCZ1}\nLet $(\\mathcal{A,B})$ be a GCAP of size $s_1\\times s_2$ and $(\\mathcal{C,D})$ be the Golay array mate of $(\\mathcal{A,B})$, then define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      x_1\\mathcal{A} & x_2\\mathcal{B} & x_3\\mathcal{A} & x_4\\mathcal{B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      x_1\\mathcal{C} & x_2\\mathcal{D} & x_3\\mathcal{C} & x_4\\mathcal{D} \\\\\n    \\end{array}\n  \\right],\n\\end{equation}\nwhere $x_1,x_2,x_3,x_4\\in\\{+1,-1\\}$.\n\\end{construction}\n\n\\begin{theorem}\\label{Theorem-array-ZCZ1}\n  Let $(\\mathcal{P},\\mathcal{Q})$ be an array pair constructed via Construction \\ref{Construction-array-ZCZ1}. Then $(\\mathcal{P},\\mathcal{Q})$ is a GCAP of size $s_1\\times 4s_2$ with $Z_{\\min}=s_1\\times (s_2+1)$, if $x_1,x_2,x_3,x_4$ if the following condition holds:\n  \\begin{equation}\n  \tx_1x_2+x_3x_4=0.\n  \\end{equation}\n\\end{theorem}\n\\begin{IEEEproof}\n\tUsing Lemma \\ref{lem2}, we can show that $(\\mathcal{P,Q})$ is a GCAP of size $s_1\\times4s_2$.\n\t\n\t\nNext, we have to prove the other two conditions of Definition \\ref{Definition-GolayZCZ}. Consider\t$-(s_1-1)\\leq\\tau_1\\leq s_1-1,-(s_2-1)\\leq\\tau_2\\leq (s_2-1)$. Then we have\n\\begin{equation}\n\\begin{split}\nR_\\mathcal{P}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{A}(\\tau_1,\\tau_2) +C_\\mathcal{B}(\\tau_1,\\tau_2) +C_\\mathcal{A}(\\tau_1-s_1,\\tau_2) +C_\\mathcal{B}(\\tau_1-s_1,\\tau_2)\\big), \\\\\nR_\\mathcal{Q}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{C}(\\tau_1,\\tau_2) +C_\\mathcal{D}(\\tau_1,\\tau_2) +C_\\mathcal{C}(\\tau_1-s_1,\\tau_2) +C_\\mathcal{D}(\\tau_1-s_1,\\tau_2)\\big), \\\\\nR_\\mathcal{P,Q}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{A,C}(\\tau_1,\\tau_2) +C_\\mathcal{B,D}(\\tau_1,\\tau_2) +C_\\mathcal{A,C}(\\tau_1-s_1,\\tau_2) +C_\\mathcal{B,D}(\\tau_1-s_1,\\tau_2)\\big),\n\\end{split}\n\\end{equation}\nSince $(\\mathcal{A},\\mathcal{B})$ is a GCAP and $(\\mathcal{C},\\mathcal{D})$ is one of the complementary mates of $(\\mathcal{A},\\mathcal{B})$, $R_\\mathcal{P}(\\tau_1,\\tau_2)=R_\\mathcal{Q}(\\tau_1,\\tau_2)=0,\\hbox{ for all } -(s_1-1)\\leq\\tau_1\\leq (s_1-1),-(s_2-1)\\leq\\tau_2\\leq (s_2-1) \\text{ except }(\\tau_1,\\tau_2)\\neq(0,0),$ and $R_\\mathcal{P,Q}(\\tau_1,\\tau_2)=0,\\hbox{ for all } -(s_1-1)\\leq\\tau_1\\leq s_1-1,-(s_2-1)\\leq\\tau_2\\leq (s_2-1)$.\n\nTherefore, $(\\mathcal{P,Q})$ is a GCAP with periodic $Z_{\\min}=s_1\\times (s_2+1)$.\n\\end{IEEEproof}\n\n\\begin{example}\\label{ex2}\n  Let $\\mathcal{A}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ -1 & -i & -1 \\\\ \\end{array}\\right],\\mathcal{B}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ 1 & i & 1 \\\\ \\end{array}\\right]$, then $(\\mathcal{A,B})$ is a quaternary GCAP of size $2\\times3$.\n  Then $(\\mathcal{C,D})= (\\overleftarrow{\\mathcal{B^*}},-\\overleftarrow{\\mathcal{A^*}})$ is a Golay array mate of $(\\mathcal{A,B})$. Define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n    \\end{array}\n  \\right].\n\\end{equation}\nThen $(\\mathcal{P,Q})$ is a GCAP of size $2\\times12$ with $Z_{\\min}=2\\times4$.\nThe 2D- periodic correlation magnitudes of $(2\\times12,2\\times4)$- GCAP $(\\mathcal{P,Q})$ are shown in Fig. \\ref{fig2}.\n\\begin{figure}[ht]\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex2P.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{P}$}\n\t\t\\label{fig:sub-first}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex2Q.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{Q}$}\n\t\t\\label{fig:sub-second}\n\t\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n\t\\centering\n\n\t\\includegraphics[width=\\textwidth]{Ex2PQ.pdf}\n\t\\caption{2D- periodic cross-correlation of $\\mathcal{P}$ and $\\mathcal{Q}$}\n\t\\label{fig:sub-second}\n\\end{subfigure}\n\t\\caption{A glimpse of the 2D- periodic correlations of the GCAP given in Example \\ref{ex2}.}\n\t\\label{fig2}\n\\end{figure}\n\\end{example}\n\n\\begin{construction}\\label{Construction-array-ZCZ2}\nLet $(\\mathcal{A,B})$ be a GCAP of size $s_1\\times s_2$ and $(\\mathcal{C,D})$ be the Golay array mate of $(\\mathcal{A,B})$, then define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{-A} & \\mathcal{B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{-A} & \\mathcal{-B} & \\mathcal{A} & \\mathcal{-B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{-C} & \\mathcal{D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{-C} & \\mathcal{-D} & \\mathcal{C} & \\mathcal{-D} \\\\\n    \\end{array}\n  \\right].\n\\end{equation}\n\\end{construction}\n\n\\begin{theorem}\n  For the array pair $(\\mathcal{P},\\mathcal{Q})$ generated by Construction \\ref{Construction-array-ZCZ2}, it's a GCAP of size $4s_1\\times 4s_2$ with periodic zero correlation zone $(s_1+1)\\times (s_2+1)$.\n\\end{theorem}\n\\begin{IEEEproof}\nThe proof is similar to the one of Theorem \\ref{Theorem-array-ZCZ1}.\n\\end{IEEEproof}\n\n\\begin{example}\\label{ex3}\n  Let $\\mathcal{A}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ -1 & -i & -1 \\\\ \\end{array}\\right],\\mathcal{B}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ 1 & i & 1 \\\\ \\end{array}\\right]$, then $(\\mathcal{A,B})$ is a quaternary GCAP of size $2\\times3$.\n  Construct $(\\mathcal{C,D})= (\\overleftarrow{\\mathcal{B^*}},-\\overleftarrow{\\mathcal{A^*}})$ as a Golay array mate of $(\\mathcal{A,B})$. Define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{-A} & \\mathcal{B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{-A} & \\mathcal{-B} & \\mathcal{A} & \\mathcal{-B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{-C} & \\mathcal{D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{-C} & \\mathcal{-D} & \\mathcal{C} & \\mathcal{-D} \\\\\n    \\end{array}\n  \\right].\n\\end{equation}\nThen $(\\mathcal{P,Q})$ is a GCAP of size $8\\times12$ with $Z_{\\min}=3\\times4$.\nThe 2D- periodic correlation magnitudes of $(8\\times12,3\\times4)$- GCAP $(\\mathcal{P,Q})$ are shown in Fig. \\ref{fig3}.\n\\begin{figure}[ht]\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex3P1.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{P}$}\n\t\t\\label{fig:sub-first}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex3Q1.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{Q}$}\n\t\t\\label{fig:sub-second}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex3PQ.pdf}\n\t\t\\caption{2D- periodic cross-correlation of $\\mathcal{P}$ and $\\mathcal{Q}$}\n\t\t\\label{fig:sub-second}\n\t\\end{subfigure}\n\t\\caption{A glimpse of the 2D- periodic correlations of the GCAP given in Example \\ref{ex3}.}\n\t\\label{fig3}\n\\end{figure}\n\\end{example}\n\\section{Conclusion}\\label{section 5}\n\nIn this paper, we have made two  contributions. Firstly, we have systematically constructed GCPs of lengths non-power-of-two where each of the constituent sequences have a periodic ZACZ and the pair have a periodic ZCCZ. This partially fills the gap left by the previous remarkable works of Gong \\textit{et al.} and Chen \\textit{et. al}, in terms of the lengths of the GCP. Secondly, by extending our construction to higher dimensions, we construct 2D- GCAPs. For the first time, the periodic 2D-autocorrelations of the constituent arrays and the periodic 2D-cross-correlation of the 2D-GCAP has been analysed. All the proposed 2D- GCAPs have large periodic 2D autocorrelation and cross-correlation zones. An interesting future\nwork will be to design Golay complementary sequence sets, consisting sequences of length non-power of two, which will have periodic ZACZ and ZCCZ around the in-phase position.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\vskip 10 pt\n\nThis paper is based on three talks given at the PASI conference in Olinda, Brazil in the summer of\n2009. One point of the paper is  to introduce students to  one aspect of characteristic $p$\nmethods in commutative algebra. Such methods have been among the most powerful in the\nfield. The basic method of reduction to\ncharacteristic $p$ is used to prove results for arbitrary Noetherian rings containing a field; the field\ncan be characteristic $0$. Thus, even though one is often working in positive characteristic, ones main\ninterest might well be in rings containing the rationals. In the late 1980's, ``tight closure theory\" was\ndiscovered by M. Hochster and myself. This theory synthesized many existing reduction to characteristic $p$\nproofs into one theory, which has now grown to encompass a large number of different directions.  In this paper,\nwe'll concentrate on a result which Karen Smith has refered to as a ``crown jewel\" of\ntight closure theory, namely the fact that the absolute integral closure of a complete local domain\nof positive characteristic is Cohen-Macaulay. We first introduce basic concepts, and discuss some rather\namazing properties of absolute integral closures. Then we prove some classical theorems in dimension one.\nThe main part of the paper will give a recent proof of the Cohen-Macaulayness of the absolute integral closure\nin positive characteristic. Applications will be given in Section 6. Some further thoughts\nabout directions to go will be presented in the last section. \nFor unexplained results or definitions, we refer the reader to Eisenbud's book \\cite{Ei}.\n\\section{Basic Concepts}\n\n\nWe begin with a definition:\n\n\\begin{definition} {\\rm Let $R$ be a domain, and let $R\\subset S$, a ring. The \\it integral closure \\rm\nof $R$ in $S$ is the set of all elements $s\\in S$ which satisfy an equation of the form\n$$s^n + r_1s^{n-1} + ... + r_n = 0,$$ where $r_i\\in R$ for all $i$.}\n\\end{definition}\n\nThe set of elements in $S$ which are integral over $R$  form a ring, called the \\it integral closure of\n$R$ in $S$\\rm.  When $S$ is the fraction field of $R$, the resulting ring is called the\nintegral closure of $R$. The ring $R$ is said to be \\it integrally closed \\rm if \n$R$ is equal to its integral closure. For example, a well-known result in commutative algebra says that\nevery UFD is integrally closed. So, e.g., the integers are integrally closed. Another\nbasic result is states that if $R$ is integrally closed, so is\nthe ring of polynomials $R[X]$ as well as the ring of formal power series $R[[X]]$.  An obvious induction shows that also\n$R[X_1,...,X_n]$ and $R[[X_1,...,X_n]]$ are integrally closed for all $n\\geq 1$. \n\nThe main object we will study is given in the following definition.\n\n\\begin{definition} {\\rm Let $R$ be an integral domain with fraction field $K$. Let $\\overline{K}$ be a\nfixed algebraic closure of $K$. The integral closure of $R$ in $\\overline{K}$, denoted $R^+$, is\ncalled the \\it absolute integral closure \\rm of $R$.}\n\\end{definition}\n\nWe simply say ``R-plus\" to mean the absolute integral closure. This ring has been well-studied in\nseveral different contexts. \n\nFor other interesting results concerning $R^+$ which we will not be writing about, see\n\\cite{A},\\cite{AH}, and \\cite{Sm}.\n\n\\medskip\n\n\n\\medskip\n\n\\begin{disc} {\\rm Whenever we study $R^+$, we might as well assume that $R$ is integrally closed itself; if $S$ is\nthe integral closure of $R$, then $R\\subset S\\subset R^+$, and $S^+ = R^+$. More generally,\nif $S$ is \\it any \\rm finite integral extension of $R$ which is a domain, then the fraction field\nof $S$ is algebraic over the fraction field of $R$, and hence there is an isomorphic copy of $S$ which\ncontains $R$ and sits inside $R^+$. By an abuse of language, we'll just assume that $S\\subset R^+$. \nIn this case, $S^+ = R^+$.  Studying $R^+$ is basically studying all integral extensions of $R$\nat the same time.}\n\\end{disc}\n\n\\medskip\n\n\\begin{remark}  {\\rm Continuing the discussion from above, if $R$ is local and complete in the ${\\gothic{m}}$-adic topology and contains\na field, then $R$ is actually a finite integral extension of a formal power series ring over a\nfield $k$. Namely, the Cohen structure theorem gives that there exists a field $k$ inside\n$R$ such that the composition of maps $k\\rightarrow R\\rightarrow R\/{\\gothic{m}}$ is an isomorphism.\nMoreover, if $x_1,...,x_d$ is a system of parameters for $R$, then the complete local subring of $R$\ngenerated by these elements, $k[[x_1,...,x_d]] = A$ is isomorphic with a formal power series ring\nover $k$, and $R$ is module-finite over $A$.\nHence $A^+ = R^+$. \nThus, if we fix a copy of the residue field, say $k$,  sitting inside $R$,  there is really only one $R^+$ for each dimension; it is the absolute integral closure of\nthe formal power series ring over  the field $k$. In other words, for each dimension, we are really just studying\none ring, the absolute integral closure of the power series ring over $k$. There is a classical theorem\nwhich gives the structure of such a ring in dimension one over a field of characteristic $0$. See Theorem~\\ref{newton}.  } \n\\end{remark}\n\n\\medskip\n\n \\bigskip\n\n\\begin{disc} {\\rm It is not difficult to prove that if $R$ is an integral domain, then a domain $S$ integral\nover $R$ is isomorphic with $R^+$ if and only if monic polynomials over $S$ factor into monic linear polynomials over $S$.\nUsing this criterion, one can easily prove two very basic properites of $R^+$. \n\n\\medskip\n$\\bullet$ Firstly, if $q\\in \\text{Spec}(R)$, then $(R_q)^+\\cong (R^+)_q$, where the latter localization is at the multiplicatively closed set\n$R-q$.\n\n\\medskip\n\n$\\bullet$ Secondly, if $Q\\in \\text{Spec}(R^+)$, then $R^+\/Q\\cong (R\/(Q\\cap R))^+$.  Thus every quotient of $R^+$ by a prime ideal is itself\nthe absolute integral closure of an appropriate quotient of $R$.}\n\\end{disc}\n\nWe leave these statements as exercises below.\n\nThese properties not only allow us to mod out by primes and localize by changing the absolute integral closure\nin a similar way, they will give very strong properites. For example, the main result of Section 4 states\nthat the absolute integral closure of an excellent local domain in positive characteristic is Cohen-Macaulay\nin a suitable sense. The second property discussed above then says that the same is true modulo every prime\nideal of the absolute integral closure! This is remarkable.\n\nM. Artin \\cite{Ar} proved another amazing property of $R^+$:\n\n\\begin{thm} Let $R$ be a domain. If $P,Q$ are prime ideals in $R^+$, then either $P+Q = R^+$, or\n$P+Q$ is prime.\n\\end{thm}\n\nThis is very far from what happens in rings we typically study. For example in the polynomial ring $k[x,y]$,\nthe ideals $P = (x+y^2)$ and $Q = (x)$ are prime, but the sum is not. The proof we give is taken from\n\\cite{HH}.\n\n\\begin{proof}\nTo prove Artin's result, let $ab\\in P + Q$. Set $z= b-a$. Notice that $a^2 + za = ab = u+v$ for some $u\\in P, v\\in Q$.\nThe equation $X^2+zX= u$ has a solution $x\\in R$. Since $u\\in P$, it follows that either $x+z\\in P$ or $x\\in P$. But\n$(x^2+zx)-(a^2+za) = u-(u+v)\\in Q$, so either $x-a\\in Q$ or $x+a+z\\in Q$. In each of the four different cases, we\nobtain that either $a\\in P+Q$ or $b\\in P+Q$. For example, if $x+z\\in P$ and $x-a\\in Q$, then $b= (x+z)-(x-a)\\in Q$.\n\\end{proof}\n\\medskip\n\n\nNotice that if $R$ is complete and local, then every finite extension domain of $R$ is also local. In particular,\n$R^+$ is a local ring, i.e., has a unique maximal ideal. Therefore the possibility that $P+Q = R$ cannot happen\nin this case, and the sum of every set of prime ideals is prime.\n\n \\bigskip\n\n\n\\centerline{\\bf Exercises}\n\\bigskip\n\n1. Let $R$ be a domain with fraction field $K$, and let $L$ be an algebraic extension of $K$.\nProve that $L$ is algebraically closed if and only if every monic polynomial with coefficients in $R$\nhas a root in $L$.\n\n\\medskip\n\n2. Let $R$ be an integral domain. Prove that a domain $S$ integral over $R$ is isomorphic with $R^+$\nif and only if every monic polynomial over $S$ factors into monic linear polynomials over $S$.\n\n\\medskip\n\n3. Prove that Artin's theorem can be extended to primary ideals: the sum of any two primary ideals in $R^+$\nis again primary or the whole ring.\n\\medskip\n\n4.  Let $R$ be a domain of positive dimension. Prove that $R^+$ is not Noetherian.\n\n\\medskip\n\n5. Suppose that $R$ is an integral domain, and $R\\subseteq S\\subseteq R^+$. Prove that $R^+ = S^+$.\n\n\\medskip\n\n6. If $q\\in \\text{Spec}(R)$, then $(R_q)^+\\cong (R^+)_q$, where the latter localization is at the multiplicatively closed set\n$R-q$.\n\n\\medskip\n\n7. If $Q\\in \\text{Spec}(R^+)$, then $R^+\/Q\\cong (R\/(Q\\cap R))^+$.  Thus every quotient of $R^+$ by a prime ideal is itself\nthe absolute integral closure of an appropriate quotient of $R$.\n\n\\vskip.5truein\n\n\\section{Dimension One}\n\\bigskip\n\n Let $k$ be a field of characteristic $0$, and consider the formal power series ring $k[[t]]$, whose fraction field is\ndenoted $k((t))$. A famous theorem due to Puiseux, but apparently known to Newton, is that the algebraic closure of\nthe field $k((t))$ is the union of the fields $k((t^{1\/n}))$ for $n\\geq 1$.  A recent paper by Kedlaya (\\cite{K}) gives\na characterization of this algebraic closure in the case $k$ has positive characteristic; it is quite hard even to describe.\nChevalley  \\cite{C} pointed out that the Artin-Schreirer polynomial $x^p-x-t^{-1}$ has no root in the Newton-Puiseux field\n$\\bigcup_n k((t^{1\/n}))$. \nIn fact, as we shall see, there are remarkable differences in $R^+$ depending on the characteristic of the ground field.\n\nWe give a proof of the  Newton-Puiseux theorem, following a treatment in \\cite{No}.\n\n\\begin{thm}\\label{newton} Let $k$ be an algebraically closed field of characteristic $0$, and let $k((t))$ be the fraction field\nof the formal power series ring $k[[t]]$. Then an algebraic closure of $k((t))$ is the Newton-Puiseux field $\\cup_n k((t^{1\/n}))$.\n\\end{thm}\n\n\\begin{proof} We need to prove that every monic polynomial $P(z)\\in B[z]$ is reducible, where $B = \\cup_nk[[t^{1\/n}]]$.\nWrite $P = z^n+ a_1z^{n-1} + ... + a_n$, where $a_i\\in B$.  By using a transformation $z' = z+\\frac{1}{n}a_1$, we can\nassume without loss of generality that $a_1 = 0$. This is called the ``Tschirhausen transformation\". Furthermore, by replacing\n$k[[t]]$ by $k[[t^{1\/n}]]$ for some large $n$, we can change variables and assume that all $a_i\\in k[[t]]$.   Let $r_k = \\text{ord}(a_k)$,\nso that $a_k = t^{r_k}u_k(t)$, where $u_k(t)$ is a unit. Here ``ord\" denotes the $t$-adic order of an element. \n\nWe want to factor $P(z)$, and to do so we will use Hensel's lemma\\footnote{ The version of Hensel's lemma\nwe use is the following theorem:\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a local ring which is complete in the ${\\gothic{m}}$-adic topology. If $f(T)$ is a monic polynomial\nwith coefficients in $R$ such that $f(T)$ factors modulo ${\\gothic{m}}$ into the product of two relatively prime monic\npolynomials of positive degree, then this factorization lifts to a factorization of $f(T)$ into two monic polynomials.\n\\end{thm}}; it suffices to factor $P(z)$ into two relatively prime polynomials\nafter going modulo $t$. This is always possible unless after reduction mod $t$, $P(z)$ becomes of the form $(t-\\alpha)^n$ for\nsome $\\alpha\\in k$. However, since $a_1 = 0$, the only possible such $\\alpha$ is $0$. Thus we are done unless for every $k$, $2\\leq k\\leq n$,\n$a_i(0) = 0$.  We want to make a change of variables where this does not occur. \n\nSet $z = t^ry$, for some rational $r$ to be chosen later. Substituting, we see that\n$P(z) = t^{rn}y^n + a_2t^{r(n-2)}y^{n-2} + ... + a_n$. We wish to factor out $t^{rn}$ from every term in such a way that\nat least one term becomes a unit. The power of $t$ dividing\nthe $i$th term is $t^{r_i}t^{(n-i)r}$, so what we need to do is to choose $r$ in such a way that \n$r_i + (n-i)r \\geq rn$, i.e., so that $r_i\\geq ri$. We set $r = \\text{min}\\{\\frac {r_i}{i}\\}$. Then we can rewrite\n$P(z) = t^{rn}Q(y,t)$, where $Q = y^n + b_2y^{n-1} + ... + b_n$. By again replacing $t$ by $t^{1\/m}$ for suitably large $m$,\nwe can assume that each $b_j\\in k[[t]]$. But now for at least one $b_i$, $b_i(0)\\ne 0$; this occurs for every term where\n$r_i\/i = r$. It follows that we can factor $Q$ over the residue field into relatively prime polynomials, and by Hensel's Lemma\nthis lifts to a factorization of $Q$ and hence also of $P$. \n\\end{proof} \n\n\nThis theorem has the following almost immediate corollary:\n\n\\begin{cor} Let $k$ be an algebraically closed field of characteristic $0$. Then $k[[t]]^+ = \\cup_nk[[t^{1\/n}]]$.\n\\end{cor}\n\n\\begin{proof} The ring on the right side of the equation is clearly integral over $k[[t]]$, and the fraction field\nof it is clearly the Newton-Puiseux field, which is an algebraic closure of $k((t))$.\nTo finish the proof of the corollary, we need to prove that $\\cup_nk[[t^{1\/n}]]$ is integrally closed. But since\nit is a union of discrete valuation rings, each integrally closed, it is also. \\end{proof}\n\n\\begin{example} {\\rm As an example, consider the\nequation $X^2 - X = t^{-1}$. Using the quadratic formula and\nTaylor series shows that the roots of this polynomial\nare a power series in $t^{-1\/2}$. One can solve recursively for the\ncoefficients. }\n\\end{example}\n\\medskip\n\nThe situation in positive characteristic is drastically different. We use the discussion in the  paper \\cite{K} in what follows.\n A \\it generalized power series \\rm is an\nexpression of the form, $\\sum_{i\\in \\mathbb Q} c_it^i$, with $c_i\\in k$, such that the set of $i$ with $x_i\\ne 0$\nis a well-ordered subset of $\\mathbb Q$, i.e., every non-empty subset has a least element. Such generalized power\nseries form a ring in a natural way; it makes sense to multiply and add them in the obvious way.\nAbyhankar pointed out that with this generalization of the idea of a power series,\nthe Chevalley polynomial has the root $t^{-1\/p} + t^{-1\/p^2}+...$.\n\nAre there enough generalized power series to obtain an algebraic closure of $k((t))$?  The following result was proved independently\nby Huang \\cite{Hua}, Rayner \\cite{Ray}, and \\c Stef\\u anescu \\cite{St}:\n\n\\begin{thm} Let $k$ be an algebraically closed field of positive characteristic, and let $K$ be the set of generalized power\n series of the form $f = \\sum_{i\\in S} c_it^i$,\nwith $c_i\\in k$, where the set $S$ has the following properties:\n\\begin{enumerate}[\\quad\\rm (1)]\n\\item  $S$ is a subset of $\\mathbb Q$ which depends on $f$.\n\\item  Every nonempty subset of $S$ has a least element.\n\\item  There exists a natural number $m$ such that every element of $mS$ has denominator a power of $p$.\n\\end{enumerate}\nThen $K$ is an algebraically closed field containing $k((t))$.\n\\end{thm}\n\n\nKedlaya \\cite{K} gives a construction  of the algebraic closure of $k((t))$ when $k$ is algebraically closed of positive characteristic.\nBy the above theorem, one needs to identify generalized power series which are algebraic over $k((t))$. \nHis result is quite complicated to even describe, but a glimpse of the issues which arise can be seen\nin the following result of Huang \\cite{Hua} and \\c Stef\\u anescu \\cite{St}:\n\n\\begin{thm} The series $\\sum_{i = 0}^{\\infty} c_it^{-1\/p^i} $ with $c_i\\in \\overline{F_p}$, the\nalgebraic closure of the field with $p$ elements, is algebraic over $\\overline{F_p}((t))$ if and only if \nthe sequence $\\{c_i\\}$ is eventually periodic.\n\\end{thm}\n\nA consequence of the main result of \\cite{K} is the following nice theorem, which perhaps can be\nproved directly.\n\n\\begin{thm} Let $k$ be an algebraically closed field of positive characteristic, and let $\\sum_i c_it^i$ be a generalized power series which is algebraic over $k((t))$. Then for\nevery real number $\\alpha$,  $\\sum_{i < \\alpha} c_it^i$ is also algebraic over $k((t))$.\n\\end{thm}\n\n\nWe now move away from the local case. \nAnother classical result in dimension one concerns the ring of all algebraic integers, $\\mathbb Z^+$.\nTo prove this theorem, we first recall some results about\nclass groups. \n\nLet $D$ be a ring of algebraic integers. This means that the fraction field $K$ of $D$ is a finite\nextension of $\\mathbb Q$, and $D$ is the integral closure of $\\mathbb Z$ in $K$. Necessarily\n$D$ is a one-dimensional integrally closed domain, i.e., a Dedekind domain. Moreover, a classical\nresult is that $D$ has a torsion class group.  In particular this means that every ideal $I$ in\n$D$ has a power which is a principal ideal. This fact leads to the following theorem\nabout $\\mathbb Z^+$.\n\n\n\\begin{thm}\\label{bezout} Every finitely generated ideal of\n$\\mathbb Z^+$ is principal (a domain with this property is said to be a B\\'ezout domain).\n\\end{thm}\n\n\\begin{proof} Let $I\\subset \\mathbb Z^+$ be generated by $t_1,...,t_m$. There is a finite field extension $L$\nof $\\mathbb Q$   containing all of these elements. The integral closure of $\\mathbb Z$\nin $L$, say $D$, is a Dedekind domain which contains $t_1,...,t_m$. Let $J$ be the ideal they\ngenerate in $D$. By the discussion above, some power of $J$ is principal, say $J^n = (d)$.\nWe claim then that $d^{\\frac{1}{n}}\\mathbb{Z}^+ = I$. Let $i\\in I$. We can write $i = \\sum_j e_jt_j$, where the\n$e_j\\in \\mathbb{Z}^+$. Again, there is a finite extension $T$ of $D$, with $T$ a Dedekind domain, such that \n$i\\in I\\cap T = (t_1,...,t_m)T = JT$. In a Dedekind domain there is unique factorization into prime ideals.\nWrite $JT = P_1^{a_1}\\cdots P_k^{a_k}$. Then $J^nT = dT = P_1^{na_1}\\cdots P_k^{na_k}$. By unique factorization,\nit follows that $d^{\\frac{1}{n}}T = P_1^{a_1}\\cdots P_k^{a_k} = JT$, so that $i\\in d^{\\frac{1}{n}}\\mathbb{Z}^+$.\n\\end{proof}\n\n\\bigskip\n\\bigskip\n\\section{Regular Sequences}\n\n\\medskip\n\nWe summarize some of the basic notions we will use to analyze $R^+$. \n\n\\begin{definition}{\\rm A sequence of elements $x_1,...,x_d$ in a ring $R$ is said to be\na \\it regular sequence \\rm if $rx_i\\in (x_0,...,x_{i-1})$ implies that\n$r\\in (x_0,...,x_{i-1})$ for all $1\\leq i\\leq d-1$ (here we set $x_0 = 0$), and\n$(x_1,...,x_d)R \\ne R$.}\n\\end{definition}\n\nAnother definition we will need is the following.\n\n\\begin{definition} {\\rm Let $(R,{\\gothic{m}})$ be a Noetherian local ring of dimension $d$. Elements\n$x_1,...,x_d$ are said to be a \\it system of parameters \\rm if the nilradical\nof the ideal they generate is ${\\gothic{m}}$.\nA Noetherian local ring is said to be \\it Cohen-Macaulay \\rm if some (equivalently) every\nsystem of parameters forms a regular sequence.}\n\\end{definition}\n\n\nWe are aiming at a theorem which shows that in characteristic $p$, $R^+$ has regular sequences having\nmaximal length. It turns out that this is equivalent to being flat in an appropriate sense. The next proposition\nproves this.\n\n\\medskip\n\n\\begin{prop} Let $A = k[[x_1,...,x_d]]$, where $k$ is a field of characteristic $p > 0$. Then the following two conditions\nare equivalent:\n\n(1) $x_1,...,x_d$ form a regular sequence on $A^+$ (by an abuse of language we say that $A^+$ is Cohen-Macaulay).\n\\newline\n(2) $A^+$ is flat over $A$.\n\\end{prop}\n\n\\begin{proof}\nWe prove the equivalence. First assume (1).\nIf $A^+$ is not flat over $A$, choose $i\\geq 1$ as large as possible so that\n$\\operatorname{Tor{}}_i^A(A\/P, A^+)\\ne 0$ for some prime $P$ in $A$. Such a choice is possible because $A$ is regular\nand large Tors vanish. If $y_1,...,y_s$ is a maximal regular sequence\nin $P$, then one can embed $A\/P$ in $A\/(y_1,...,y_s)$ with cokernel $C$. But since our assumption\nforces $\\operatorname{Tor{}}_{i+1}^R(C,A^+) =  0$ (as $C$ has a prime filtration), and $y_1,...,y_s$ form a\nregular sequence on $A^+$, we obtain that $\\operatorname{Tor{}}_i^A(A\/P, A^+) = 0$, a contradiction.\n\nTo see that $y_1,...,y_s$ form a regular sequence, extend them to a system of parameters, and let\n$B = k[[y_1,...,y_d]]$. Then $B^+ = A^+$, and our hypothesis says that the $y's$ form a regular\nsequence.\n\nAssume (2). Flat maps preserve regular sequence in general.\n\\end{proof}\n\n\nOur method of studying regular sequences relies on local cohomology. We only need the description below.\n\n For $x\\in R$, let $K^{\\bullet}(x;R)$ denote the\ncomplex $0\\rightarrow R\\rightarrow R_x\\rightarrow 0$, graded so that the degree $0$ piece of the complex is\n$R$, and the degree $1$ is $R_x$. If $x_1,...,x_n\\in R$, let $K^{\\bullet}(x_1,x_2,...,x_n;R)$ denote\nthe complex $K^{\\bullet}(x_1;R)\\otimes_R ...\\otimes_R K^{\\bullet}(x_n;R)$, where in general recall that\nif $(C^{\\bullet},d_C)$ and $(D^{\\bullet}, d_D)$ are complexes, then the tensor product\nof these complexes, $(C\\otimes_RD, \\Delta)$ is by definition the complex whose ith\ngraded piece is $\\sum_{j+k = i} C_j\\otimes D_k$ and whose differential is determined\nby the map from $ C_j\\otimes D_k\\rightarrow (C_{j+1}\\otimes D_k) \\oplus (C_j\\otimes D_{k+1})$\ngiven by $\\Delta(x\\otimes y) = d_C(x)\\otimes y + (-1)^k x\\otimes d_D(y)$.\n\nThe modules in this complex, called the Koszul cohomology complex, are\n$$0\\rightarrow R\\rightarrow \\oplus\\sum_i R_{x_i}\\rightarrow \\oplus\\sum_{i< j} R_{x_ix_j}\\rightarrow ...\\rightarrow R_{x_1x_2\\cdots x_n}\\rightarrow 0$$\nwhere the differentials are the natural maps induced from  localization, but with signs attached.\nIf $M$ is an $R$-module, we set $K^{\\bullet}(x_1,x_2,...,x_n; M) = K^{\\bullet}(x_1,x_2,...,x_n;R)\\otimes_RM$.\nWe denote the cohomology of $K^{\\bullet}(x_1,x_2,...,x_n; M)$ by $H^i_{I}(M)$, called\nthe $i$th local cohomology of $M$ with respect to $I = (x_1,...,x_d)$.\nIt is a fact that this module only depends on the ideal generated by the $x_i$ up to radical.\nWe summarize some useful information concerning these modules.\n\n\\begin{prop}\\label{propbasechange} Let $R$ be a Noetherian ring, $I$ and ideal and $M$ and $R$-module.\nLet $\\phi: R\\rightarrow S$ be a homomorphism and let $N$ be an $S$-module.\n\\begin{enumerate}[\\quad\\rm (1)]\n\\item If $\\phi$ is flat, then $H^j_I(M)\\otimes_RS\\cong H^j_{IS}(M\\otimes_RS)$. In particular,\nlocal cohomology commutes with localization and completion.\n\\item (Independence of Base) $H^j_I(N)\\cong H^j_{IS}(N)$, where the first local cohomology\nis computed over the base ring $R$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof} Choose generators $x_1,...,x_n$ of $I$. The first claim follows at once from the\nfact that $K^{\\bullet}(x_1,...,x_n;M)\\otimes_RS = K^{\\bullet}(\\phi(x_1),...,\\phi(x_n);M)\\otimes_RS)$, and that\n$S$ is flat over $R$, so that the cohomology of $K^{\\bullet}(x_1,...,x_n;M)\\otimes_RS$ is the cohomology\nof $K^{\\bullet}(x_1,...,x_n;M)$ tensored over $R$ with $S$.\n\nThe second claim follows from the fact that $$K^{\\bullet}(x_1,...,x_n;N) = K^{\\bullet}(x_1,...,x_n;R)\\otimes_RN =\n(K^{\\bullet}(x_1,...,x_n;R)\\otimes_RS)\\otimes_SN$$\n$$ = K^{\\bullet}(\\phi(x_1),...,\\phi(x_n);S)\\otimes_RN =\nK^{\\bullet}(\\phi(x_1),...,\\phi(x_n);N).$$\n\\end{proof}\n\\bigskip\n\n\\section{$R^+$ is Cohen-Macaulay in Positive Characteristic}\n\n\\medskip\n\nLet $R$ be a commutative ring containing a field of characteristic $p>0$, let $I\\subset R$\nbe an ideal, and let $R'$ be an $R$-algebra. The Frobenius ring homomorphism\n$f:R'\\stackrel{r\\mapsto r^p}{\\to}R'$ induces a map $f_*:H^i_I(R')\\to H^i_I(R')$ on all\nlocal cohomology modules of $R'$ called the action of the Frobenius on $H^i_I(R')$.\nFor an element $\\alpha\\in H^i_I(R')$ we denote $f_*(\\alpha)$ by $\\alpha^p$. This follows since\nthe Frobenius extends to localization of $R$ in the obvious way, and commutes with the\nmaps in the Koszul cohomology complex, which are simply signed natural maps.\n\nThe main result is that if $R$ is a local Noetherian domain which is a homomorphic image of a\nGorenstein local ring and has positive characteristic, then $R^+$ is Cohen-Macaulay in the\nsense that every system of parameters of $R$ form a regular sequence in $R^+$. To prove this\nresult we use the proof given in \\cite{HL}. The original proof, with slightly different assumptions, was given\nin 1992 in \\cite{HH}, as a result of developments from tight closure theory. Although tight closure has\nnow disappeared from the proof, it remains an integral part of the theory. A critical point is that we must find some\nway of annihilating nonzero local cohomology classes. \nThe next lemma is essentially the only way known to do this.\n\n\\begin{lemma} \\label{element} Let $R$ be a commutative Noetherian domain containing a field of\ncharacteristic $p>0$, let $K$ be the fraction field of $R$ and let $\\overline K$ be the algebraic closure\nof $K$. Let $I$ be an ideal of $R$ and let $\\alpha\\in H^i_I(R)$ be an element such that the elements\n$\\alpha, \\alpha^p,\\alpha^{p^2},\\dots,\\alpha^{p^t},\\dots$ belong to a finitely generated $R$-submodule of $H^i_I(R)$.\nThere exists an $R$-subalgebra $R'$ of $\\overline K$ (i.e. $R\\subset R'\\subset \\overline K$) that is finite\nas an $R$-module and such that the natural map $H^i_I(R)\\to H^i_I(R')$ induced by the natural\ninclusion $R\\to R'$ sends $\\alpha$ to 0.\n\\end{lemma}\n\\emph{Proof.} Let $A_t=\\sum_{i=1}^{i=t}R\\alpha^{p^i}$ be the $R$-submodule of $H^i_I(R)$ generated\nby $\\alpha,\\alpha^p,\\dots,\\alpha^{p^t}$. The ascending chain $A_1\\subset A_2\\subset A_3\\subset\\dots$\nstabilizes because $R$ is Noetherian and all $A_t$ sit inside a single finitely generated $R$-submodule\nof $H^i_I(R)$. Hence $A_s=A_{s-1}$ for some $s$, i.e. $\\alpha^{p^s}\\in A_{s-1}$. Thus there exists an\nequation $\\alpha^{p^s}=r_1\\alpha^{p^{s-1}}+r_2\\alpha^{p^{s-2}}+\\dots+r_{s-1}\\alpha$ with $r_i\\in R$\nfor all $i$. Let $T$ be a variable and let $g(T)=T^{p^s}-r_1T^{p^{s-1}}-r_2^{p^{s-2}}-\\dots-r_{s-1}T$.\nClearly, $g(T)$ is a monic polynomial in $T$ with coefficients in $R$ and $g(\\alpha)=0$.\n\nLet $x_1,\\dots, x_d\\in R$ generate the ideal $I$. Recall that we can calculate the local cohomology\nfrom the Koszul cohomology complex\n$C^{\\bullet}(R)$, \n$$0\\to C^0(R)\\to\\dots \\to C^{i-1}(R)\\stackrel{d_{i-1}}{\\to} C^i(R)\\stackrel{d_i}{\\to}\nC^{i+1}(R)\\to\\dots\\to C^d(R)\\to 0$$\nwhere $C^0(R)=R$ and $C^i(R)=\\oplus_{1\\leq j_1<\\dots<j_{i}\\leq d}R_{x_{j_1}\\cdots x_{j_{i}}}$,\nand $H^i_I(M)$ is the $i$th cohomology module of $C^{\\bullet}(R)$.\n\nLet $\\tilde \\alpha\\in C^i(R)$ be a cycle (i.e. $d_i(\\tilde \\alpha)=0$) that represents $\\alpha$.\nThe equality $g(\\alpha)=0$ means that $g(\\tilde \\alpha)=d_{i-1}(\\beta)$ for some $\\beta\\in C^{i-1}(R)$.\nSince $C^{i-1}(R)=\\oplus_{1\\leq j_1<\\dots<j_{i-1}\\leq d}R_{x_{j_1}\\cdots x_{j_{i-1}}}$, we may write\n$\\beta=(\\frac{r_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}})$\nwhere $r_{j_1,\\dots,j_{i-1}}\\in R$, the integers $e_1,\\dots, e_{i-1}$ are non-negative, and\n$\\frac{r_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}}\\in R_{x_{j_1}\\cdots x_{j_{i-1}}}$.\n\nConsider the equation $g(\\frac{Z_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}})\n-\\frac{r_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}}=0$ where\n$Z_{j_1,\\dots,j_{i-1}}$ is a variable. Multiplying this equation by\n$(x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}})^{p^s}$ produces a monic polynomial equation in\n$Z_{j_1,\\dots,j_{i-1}}$ with coefficients in $R$. Let $z_{j_1,\\dots,j_{i-1}}\\in \\overline K$ be a root\nof this equation and let $R''$ be the $R$-subalgebra of $\\overline K$ generated by all the $z_{j_1,\\dots,j_{i-1}}$s,\ni.e. by the set $\\{z_{j_1,\\dots,j_{i-1}}|1\\leq j_1<\\dots<j_{i-1}\\leq d\\}$. Since each $z_{j_1,\\dots,j_{i-1}}$\nis integral over $R$ and there are finitely many $z_{j_1,\\dots,j_{i-1}}$s, the $R$-algebra $R''$ is finite\nas an $R$-module.\n\nLet $\\tilde{\\tilde\\alpha} =\n(\\frac{z_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\n\\cdots x_{j_{i-1}}^{e_{i-1}}})\\in C^{i-1}(R'')$. The natural inclusion $R\\to R''$ makes $C^{\\bullet}(R)$\ninto a subcomplex of $C^{\\bullet}(R'')$ in a natural way, and we identify $\\tilde \\alpha\\in C^i(R)$\nand $\\beta\\in C^{i-1}(R)$ with their natural images in $C^i(R'')$ and $C^{i-1}(R'')$ respectively.\nWith this identification, $\\tilde \\alpha\\in C^i(R'')$ is a cycle representing the image of $\\alpha$\nunder the natural map $H^i_I(R)\\to H^i_I(R'')$, and so is $\\overline \\alpha=\\tilde\\alpha-d_{i-1}(\\tilde{\\tilde\\alpha})\n\\in C^i(R'')$. Since $g(\\tilde{\\tilde\\alpha})=\\beta$ and $g(\\tilde \\alpha)=d_{i-1}(\\beta)$, we conclude that\n$g(\\overline \\alpha)=0$. Let $\\overline\\alpha=(\\rho_{j_1,\\dots,j_i})$ where $\\rho_{j_1,\\dots,j_i}\n\\in R''_{x_{j_1}\\cdots x_{j_i}}$. Each individual $\\rho_{j_1,\\dots,j_i}$ satisfies the equation\n$g(\\rho_{j_1,\\dots,j_i})=0$. Since $g(T)$ is a monic polynomial in $T$ with coefficients in $R$,\neach $\\rho_{j_1,\\dots,j_i}$ is an element of the fraction field of $R''$ that is integral over $R$.\nLet $R'$ be obtained from $R''$ by adjoining all the $\\rho_{j_1,\\dots,j_i}$.\ndirect sum of all such copies of $R'$. This subcomplex is exact because its cohomology groups are the\ncohomology groups of $R'$ with respect to the unit ideal. Since $\\overline \\alpha$ is a cycle and belongs to\nthis exact subcomplex, it is a boundary, hence it represents the zero element in $H^i_I(R')$. \\qed\n\n\n\\medskip\n\nWe recall that for a Gorenstein local ring $A$ of dimension $n$, local duality says that\nthere is an isomorphism of functors $D({\\rm Ext}^{n-i}_A(-, A))\\cong H^i_{\\mathfrak m}(-)$\non the category of finite $A$-modules, where $D={\\rm Hom}_A(-,E)$ is the Matlis duality\nfunctor (here $E$ is the injective hull of the residue field of $A$ in the category of\n$A$-modules). \n\n\\begin{thm}\\label{module}\\cite{HL}\nLet $R$ be a commutative Noetherian local domain containing a field of characteristic $p>0$, let\n$K$ be the fraction field of $R$ and let $\\overline K$ be the algebraic closure of $K$. Assume $R$ is a surjective\nimage of a Gorenstein local ring $A$. Let $\\mathfrak m$ be the maximal ideal of $R$. \nLet $i< \\dim R$\nbe a non-negative integer. There is an $R$-subalgebra $R'$ of $\\overline K$ (i.e. $R\\subset R'\\subset \\overline K$)\nthat is finite as an $R$-module and such that the natural map $H^i_{\\mathfrak m}(R)\\to H^i_{\\mathfrak m}(R')$\nis the zero map.\n\\end{thm}\n\n\\emph{Proof.} The proof comes from \\cite{HL}. Let $n=\\dim A$ and let $N={\\rm Ext}^{n-i}_A(R,A)$. \nClearly $N$ is a finite $A$-module.\n\nLet $d=\\dim R$. We use induction on $d$. For $d=0$ there is nothing to prove, so we assume that\n$d>0$ and the theorem proven for all smaller dimensions. Let $P\\subset R$ be a non-maximal prime ideal.\nWe claim there exists an $R$-subalgebra $R^P$ of $\\overline K$  such that\n$R^P$ is a finite $R$-module and for every $R^P$-subalgebra $R^*$ of $\\overline K$ (i.e. $R^P\\subset R^*\\subset \\overline K$)\nsuch that $R^*$ is a finite $R$-module, the image $\\mathcal I\\subset N$ of the natural map\n${\\rm Ext}^{n-i}_A(R^*,A)\\to N$ induced by the natural inclusion $R\\to R^*$ vanishes after localization at $P$,\ni.e. $\\mathcal I_P=0$. Indeed, let $d_P=\\dim R\/P$. Since $P$ is different from the maximal ideal,\n$d_P>0$. As $R$ is a surjective image of a Gorenstein local ring, it is catenary, hence the dimension\nof $R_{P}$ equals $d-d_P$, and $i<d$ implies $i-d_P<d-d_P=\\dim R_{P}$. By the induction hypothesis\napplied to the local ring $R_{P}$, \nthere is an $R_{P}$-subalgebra $\\tilde R$ of $\\overline K$, which is finite as an $R_{P}$-module, such that\nthe natural map $H^{i-d_P}_{P}(R_P)\\to H^{i-d_P}_{P}(\\tilde R)$ is the zero map.\nLet $\\tilde R=R_{P}[z_{1},z_{2},\\dots,z_{t}]$, where $z_{1},z_{2},\\dots,z_{t}\\in \\overline K$ are\nintegral over $R_{P}$. Multiplying, if necessary, each $z_{j}$ by some element of $R\\setminus P$,\nwe can assume that each $z_{j}$ is integral over $R$. We set $ R^P=R[z_{1},z_{2},\\dots,z_{t}]$.\nClearly, $R^P$ is an $R'$-subalgebra of $\\overline K$ that is finite as $R$-module.\n\nNow let $R^*$ be both\nan $R^P$-subalgebra of $\\overline K$ (i.e. $R^P\\subset R^*\\subset \\overline K$) and a finite $R$-module.\nThe natural inclusions $R\\to R^P\\to R^*$ induce natural maps\n$${\\rm Ext}^{n-i}_A(R^*,A)\\to {\\rm Ext}^{n-i}_A(R^P,A)\\to N.$$ This implies that\n$\\mathcal I\\subset \\mathcal J$, where $\\mathcal J$ is the image of the natural map\n$\\phi:{\\rm Ext}^{n-i}_A(R^P,A)\\to N$. Hence it is enough to prove that $\\mathcal J_{P}=0$.\nLocalizing this map at $P$ we conclude that $J_P$ is the image of the natural map\n$\\phi_P:{\\rm Ext}^{n-i}_{A_P}(\\tilde R,A_P)\\to {\\rm Ext}^{n-i}_{A_P}(R_P,A_P)$ induced by the\nnatural inclusion $R_P\\to \\tilde R$ (by a slight abuse of language we identify the prime ideal\n$P$ of $R$ with its full preimage in $A$). Let $D_P(-)={\\rm Hom}_{A_P}(-, E_P)$ be the Matlis duality\nfunctor in the category of $R_P$-modules, where $E_P$ is the injective hull of the residue field of\n$R_P$ in the category of $R_P$-modules. Local duality implies that $D_P(\\phi_P)$ is the natural map\n$H^{i-d_P}_{P}(R_P)\\to H^{i-d_P}_{P}(\\tilde R)$ which is the zero map by construction\n(note that $i-d_P=\\dim A_P-(n-i)$). Since $\\phi_P$ is a map between finite $R_P$-modules\nand $D_P(\\phi_P)=0$, it follows that $\\phi_P=0$. This proves the claim.\n\nSince $N$ is a finite $R$-module, the set of the associated primes of $N$ is finite.\nLet $P_1,\\dots,P_s$ be the associated primes of $N$ different from $\\mathfrak m$. For each $j$\nlet $R^{P_j}$ be an $R$-subalgebra of $\\overline K$ corresponding to $P_j$, whose existence is guaranteed\nby the above claim. Let $\\overline R=R[R^{P_1},\\dots, R^{P_s}]$ be the compositum of all the $R^{P_j}$, $1\\leq j\\leq s$.\nClearly, $\\overline R$ is an $R$-subalgebra of $\\overline K$. Since each $R^{P_j}$\nis a finite $R$-module, so is $\\overline R$. Clearly, $\\overline R$  contains every $R^{P_j}$. Hence the above\nclaim implies that $\\mathcal I_{P_j}=0$ for every $j$, where $\\mathcal I\\subset N$ is the image of the\nnatural map ${\\rm Ext}^{n-i}_A(\\overline R,A)\\to N$ induced by the natural inclusion $R\\to\\overline R$. It follows\nthat  not a single $P_j$ is an associated prime of $\\mathcal I$. But $\\mathcal I$ is a submodule of $N$,\nand therefore every associated prime of $\\mathcal I$ is an associated prime of $N$.\nSince $P_1,\\dots, P_s$ are all the associated primes of $N$ different from $\\mathfrak m$, we conclude that if\n$\\mathcal I\\ne 0$, then $\\mathfrak m$ is the only associated prime of $\\mathcal I$. Since $\\mathcal I$,\nbeing a submodule of a finite $R$-module $N$, is finite, and since $\\mathfrak m$ is the only associated\nprime of $\\mathcal I$, we conclude that $\\mathcal I$ is an $R$-module of finite length.\n\nWriting the natural map ${\\rm Ext}^{n-i}_A(\\overline R,A)\\to N$ as the composition of two maps\n$${\\rm Ext}^{n-i}_A(\\overline R,A)\\to \\mathcal I\\to N,$$ the first of which is surjective and the second injective,\nand applying the Matlis duality functor $D$, we get that the natural map\n$\\varphi:H^i_{\\mathfrak m}(R)\\to H^i_{\\mathfrak m}(\\overline R)$ induced by the inclusion $R\\to \\overline R$ is the\ncomposition of two maps $H^i_{\\mathfrak m}(R)\\to D(\\mathcal I)\\to H^i_{\\mathfrak m}(\\overline R)$, the first\nof which is surjective and the second injective. This shows that the image of $\\varphi$ is isomorphic to\n$D(\\mathcal I)$ which is an $R$-module of finite length since so is $\\mathcal I$. In particular, the image\nof $\\varphi$ is a finitely generated $R$-module. Let $\\alpha_1,\\dots, \\alpha_s\\in\nH^i_{\\mathfrak m}(\\overline R)$ generate Im$\\varphi$.\n\nThe natural inclusion $R\\to \\overline R$ is compatible with the Frobenius homomorphism, i.e. with\nthe raising to the $p$th power on $R$ and $\\overline R$. This implies that $\\varphi$ is compatible\nwith the action of the Frobenius $f_*$ on $H^i_{\\mathfrak m}(R)$ and $H^i_{\\mathfrak m}(\\overline R)$,\ni.e. $\\varphi(f_*(\\alpha))=f_*(\\varphi(\\alpha))$ for every $\\alpha\\in H^i_{\\mathfrak m}(R)$,\nwhich, in turn, implies that Im$\\varphi$ is an $f_*$-stable $R$-submodule of $H^i_{\\mathfrak m}(\\overline R)$,\ni.e. $f_*(\\alpha)\\in {\\rm Im}\\varphi$ for every $\\alpha\\in {\\rm Im}\\varphi$. We finish the proof\nby applying Lemma~\\ref{element} to each element of  a finite generating set $\\alpha_1,...,\n\\alpha_s$ of Im$\\varphi$.\nApplying Lemma~\\ref{element} we obtain a\n$\\overline R$-subalgebra $R_j$ of $\\overline K$ (i.e. $\\overline R\\subset R_j\\subset \\overline K$) such that\n$R'_j$ is a finite $R$-module and the natural map $H^i_{\\mathfrak m}(\\overline R)\\to H^i_{\\mathfrak m}(R_j)$\nsends $\\alpha_j$ to zero. Let $R'=R[R_1,\\dots, R_s]$ be the compositum of all the $R_j$.\nThen $R'$ is an $R$-subalgebra of $\\overline K$ and is a finite $R$-module since so is each $R_j$.\nThe natural map $H^i_{\\mathfrak m}(\\overline R)\\to H^i_{\\mathfrak m}(R')$ sends every $\\alpha_j$ to zero,\nhence it sends the entire Im$\\varphi$ to zero. Thus the natural map\n$H^i_{\\mathfrak m}(R)\\to H^i_{\\mathfrak m}(R')$ is zero. \\qed\n\n\\medskip\n\n\n\n\n\\begin{cor}\\label{corcm}\nLet $R$ be a commutative Noetherian local domain containing a field of characteristic $p>0$.\nAssume that $R$ is a surjective image of a Gorenstein local ring. Then the following hold:\n\n(a) $H^i_{\\mathfrak m}(R^+)=0$ for all $i<\\dim R$, where $\\mathfrak m$ is the maximal ideal of $R$.\n\n(b) Every system of parameters of $R$ is a regular sequence on $R^+$.\n\\end{cor}\n\n\\emph{Proof.} (a) $R^+$ is the direct limit of the finitely generated $R$-subalgebras $R'$, hence $H^i_{\\mathfrak m}(R^+)=\\varinjlim H^i_{\\mathfrak m}(R')$.\nBut Theorem~\\ref{module} implies that for each $R'$ there is $R''$ such that the map \n $H^i_{\\mathfrak m}(R')\\to H^i_{\\mathfrak m}(R'')$ in the inductive system is zero. Hence the limit is zero.\n\n(b) Let $x_1,..., x_d$ be a system of parameters of $R$. We prove that $x_1,...,x_j$ is a regular\nsequence on $R^+$ by induction on $j$. The case $j=1$ is clear, since $R^+$ is a domain.\nAssume that $j>1$ and $x_1,\\dots, x_{j-1}$ is a regular sequence on $R^+$. Set $I_t = (x_1,...,x_t)$.\nThe fact that $H^i_{\\mathfrak m}(R^+)=0$ for all $i<d$ and the short exact sequences\n$$0\\to R^+\/I_{t-1}R^+\\stackrel{\\rm x_t}{\\longrightarrow} R^+\/I_{t-1}R^+\\to R^+\/I_{t}R^+\\to 0$$ for\n$t\\leq j-1$ imply by induction on $t$ that $H^q_{\\mathfrak m}(R^+\/(x_1,...,x_{t})R^+)=0$ for $q<d-t$.\nIn particular, $H^0_{\\mathfrak m}(R^+\/(x_1,...,x_{j-1})R^+)=0$ since $0<d-(j-1)$.\nHence $\\mathfrak m$ is not an associated prime of $R^+\/(x_1,...,x_{j-1})R^+$. This implies that the\nonly associated primes of $R^+\/(x_1,...,x_{j-1})R^+$ are the minimal primes of $R\/(x_1,...,x_{j-1})R$.\nIndeed, if there is an embedded associated prime, say $P$, then $P$ is the maximal ideal of the ring\n$R_P$ whose dimension is bigger than $j-1$ and $P$ is an associated prime of\n$(R^+\/(x_1,...,x_{j-1})R^+)_P=(R_P)^+\/(x_1,...,x_{j-1})(R_P)^+$ which is impossible by the above.\nHence every element of $\\mathfrak m$ not in any minimal prime of $R\/(x_1,...,x_{j-1})R$, for example,\n$x_j$, is a regular element on $R^+\/(x_1,...,x_{j-1})R^+$.\\qed\n\n\\medskip\n\n\\begin{disc}\n{\\rm It is important to understand the huge differences between characteristic $p$, characteristic $0$, and\nmixed characteristic.\nSuppose that $k$ has characteristic $0$. Let $A$ be a complete Noetherian local integrally closed domain with residue field $k$,\nand fraction field $K$. \nIf $L$ is any finite field extension of $K$ and $B$ is the integral closure of $A$\nin $L$, then the reduced trace map\\footnote{The \\it reduced trace \\rm is defined as $\\frac{1}{n} Tr_{L\/K}$ where\n$L$ is a finite field extension of a field $K$, and $[L:K] = n$. This map fixes the ground field $K$. If\n$R$ is an integrally closed domain with fraction field $K$ and $S$ is the integral closure of $R$ in $L$,\nthen the reduced trace sends $S$ to $R$ and fixes $R$.} gives a splitting of $A$ from $B$, i.e., $B\\cong A\\oplus N$ as\nan $A$-module for some module $A$-module $N$. Then $H^i_{{\\gothic{m}}}(A)$ splits out of $H^i_{{\\gothic{m}}}(B)$, so\nthat the map $H^i_{{\\gothic{m}}}(A)\\to H^i_{{\\gothic{m}}}(B)$ is \\it never \\rm zero unless $H^i_{{\\gothic{m}}}(A) = 0$. Thus the\nexact opposite holds in characteristic $0$.}\n\\end{disc}\n\nWhat happens in mixed characteristic is a great mystery. One of the great results in recent years was that\nof Heitmann. He proved the following theorem:\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a complete three-dimensional Noetherian local integrally closed domain of\nmixed characteristic $p\\in \\mathbb N$. (This means that $p\\in {\\gothic{m}}$.) Then for all $n\\geq 1$,\n$p^{\\frac {1}{n}}$ annihilates the local cohomology $H^2_{{\\gothic{m}}}(R^+)$.\n\\end{thm}\n\nHeitmann's theorem is slightly stronger than this, but this is the essential result of \\cite{He}. In characteristic\n$p$ we know this local cohomology module is zero, but this is not known in mixed characteristic.\n\nOne can hope that if $R$ has mixed characteristic $p$, then $R^+\/pR^+$ is Cohen-Macaulay. Of course, this\nring has positive characteristic. However, perhaps this is too much to hope for. The next best result would\nbe to conjecture that $R^+\/\\sqrt{pR^+}$ is Cohen-Macaulay, a question raised by Lyubeznik. He has a partial\nresult in this direction \\cite{Ly}:\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a Noetherian local excellent domain of mixed characteristic $p$. Assume the\ndimension of $R$ is at least $3$.  Set $\\overline{R} = R\/\\sqrt{pR}$ and $\\overline{R^+} = R^+\/\\sqrt{pR^+}$.\nThen $H^1_{{\\gothic{m}}}(\\overline{R^+}) = 0$, and every part of a system of parameters $a,b$ of $\\overline{R}$\nform a regular sequence on $\\overline{R^+}$.\n\\end{thm}\n\n\n\n\\bigskip\n\n\nAnother interesting question is whether or not one must use inseparable extensions to trivialize local cohomology.\nIn fact, Anurag Singh \\cite{Si} found the following nice trick to change inseparable elements to separable.\n\n\\begin{prop} Let $R$ be an excellent domain of characteristic $p > 0$, and let $I$ be an ideal of $R$. Suppose that\n$z\\in R$ is such that $z^q\\in I^{[q]}$, where $ q = p^e$ is a power of $p$. Then there exists an integral domain\n$S$, which is a module-finite separable extension of $R$, such that $z\\in IS$.\n\\end{prop}\n\\bigskip\n\nThe point here is that there clearly a finite inseparable extension of $R$, say $T$, such that $z\\in IT$. Simply\ntake $qth$ roots of the elements $a_j$ such that $z = \\sum a_jx_j^q$ where $x_j\\in I$. \n\n\\begin{proof}  Write $z = \\sum_{1\\leq j\\leq n} a_jx_j^q$ where $x_j\\in I$ as above. Consider the equations for $2\\leq i\\leq n$,\n$$ U_i^q + U_ix_1^q-a_i = 0.$$\nThese are monic separable equations and therefore have roots $u_i$ in a separable field extension of the fraction field\nof $R$. Let $S$ be the integral closure of the ring $R[u_2,...,u_n]$. \nSince $R$ is excellent, $S$ is finite as an $R$-module. We claim that $z\\in IS$. Set\n$$u_1 = (z- \\sum_{2\\leq i\\leq n} x_iu_i)\/x_1.$$\nNote that $u_1$ is an element of the fraction field of $S$. Taking $qth$ powers we see that\n$$u_1^q = a_1 + \\sum_{2\\leq i\\leq n} u_ix_i^q.$$\nTherefore $u_1$ is integral over $S$. As $S$ is integrally closed, $u_1\\in S$. This implies that $$z = \\sum_{1\\leq i\\leq n} u_ix_i$$\nand so $z\\in IS$. \\end{proof}\n\n\\begin{disc}{\\rm  There is an interesting property pertaining to our main theorem. Suppose that\n$(R,{\\gothic{m}})$ is a complete local Noetherian domain of positive characteristic, and let $x_1,...,x_d$ form\na regular sequence. If $x_1,...,x_d$ is a system of parameters, and if $R$ is not Cohen-Macaulay, then\nthere is a non-trivial relation $r_1x_1+...+r_dx_d = 0$. Non-trivial means that it does not\ncome from the Koszul relations. Since $R^+$ is Cohen-Macaulay, we can trivialize this relation in $R^+$,\nand therefore in some finite extension ring $S$ of $R$, $R\\subseteq S\\subseteq R^+$. But Theorem~\\ref{module} does not say\nwhether or not there is a fixed finite extension ring $T$, $R\\subseteq T\\subseteq R^+$ in which all relations on all\nparameters of $R$ become simultaneously trivial. Even if such a ring $T$ exists, this does not mean $T$ is\nitself Cohen-Macaulay; new relations coming from elements of $T$ may be introduced. However, there is a finite\nextension which simultaneously trivializes all relations on systems of parameters. This fact has been proved by\nMelvin Hochster and Yongwei Yao \\cite{HY}.}\n\\end{disc}\n\n\\section{Applications}\n\\bigskip\n\nThe existence of a big Cohen-Macaulay algebra has a great many applications. In some sense\nit repairs the failure of a ring to be Cohen-Macaulay. Hochster proved and used the existence of big Cohen-Macaulay\nmodules (the word ``big\" refers to the fact the modules may not be finitely generated) to prove many of the homological\nconjectures. For a modern update, see \\cite{Ho1}. In general, if you can prove a theorem in the\nCohen-Macaulay case, you should immediately try to use $R^+$ to try to prove it in general.\nWe give several examples of this phenomena in this section.  As examples, we will prove some\nof the old homological conjectures using this approach; this is not new, but there are currently\na growing number of new homological conjectures, and it could be that characteristic $p$ methods\napply.\n\nOf course, some of the homological conjectures deal directly with systems of parameters. These are\neasy to prove once one has a Cohen-Macaulay module. For example, the next theorem gives the\nmonomial conjecture.\n\n\\begin{thm} Let $R$ be a local Noetherian ring of dimension $d$ and positive characteristic $p$.\nLet $x_1,...,x_d$ be a system of parameters. Then for all $t\\geq 1$, $(x_1\\cdots x_d)^t$ is not\nin the ideal generated by $x_1^{t+1},...,x_d^{t+1}$.\n\\end{thm}\n\n\\begin{proof} We use induction on the dimension $d$ of $R$. The case $d = 1$ is trivial.\nSuppose by way of contradiction that $d > 1$ and  $(x_1\\cdots x_d)^t\\in (x_1^{t+1},...,x_d^{t+1})$.\nThis is preserved after completion, and is further preserved after moding out a minimal prime $P$\nsuch that the dimension of the completion modulo $P$ is still $d$. After these operations,\nthe images of the elements $x_i$ still form a system of parameters as well.  Thus we may assume that\n$R$ is a complete local domain. We apply Theorem~\\ref{module} to conclude that $x_1,...,x_d$ is\na regular sequence in $R^+$. Write\n$$(x_1\\cdots x_d)^t = \\sum_i s_ix_i^{t+1},$$\nwhere $s_i\\in R$. Then $x_d^t((x_1\\cdots x_{d-1})^t - s_dx_d)\\in (x_1^{t+1},...,x_{d-1}^{t+1})$.\nSince the powers of the $x_i$ also form a regular sequence in $R^+$, we conclude that\n$(x_1\\cdots x_{d-1})^t - s_dx_d\\in (x_1^{t+1},...,x_{d-1}^{t+1})R^+$. It follows that\nthere is a Noetherian complete local domain $S$ containing $R$ and module-finite over $R$\nsuch that $(x_1\\cdots x_{d-1})^t\\in (x_1^{t+1},...,x_{d-1}^{t+1},x_d)S$. But now\n$(x_1\\cdots x_{d-1})^t$ is in the ideal $(x_1^{t+1},...,x_{d-1}^{t+1})$ in the ring $S\/x_dS$,\nwhich has dimension $d-1$. Our induction shows that this is impossible. \\end{proof}\n\nNext, we apply Theorem~\\ref{module} it to various intersection theorems. One of the first such intersection conjectures\nwas: \n\n\\begin{conj} Let $(R,{\\gothic{m}})$ be a local Noetherian ring, and let $M,N$ be two finitely\ngenerated nonzero $R$-modules such that $M\\otimes_RN$ has finite length. Then\n$$\\dim N \\leq \\text{pd}_R(M).$$\n\\end{conj}\n\nOf course there is nothing to prove if the projective dimension of $M$ is infinite.\nWe prove (see \\cite{Ho}):\n\n\\begin{thm}\\label{newintersection} Let $(R,{\\gothic{m}})$ be a local Noetherian ring of positive prime characteristic $p$, and let $M,N$ be two finitely\ngenerated nonzero $R$-modules such that $M\\otimes_RN$ has finite length. Then\n$$\\dim N \\leq \\text{pd}_R(M).$$\n\\end{thm}\n\n\\begin{proof}\nOne can begin by making some easy reductions. These types of reduction are very good practice\nin commutative algebra. First, note that the assumption that the tensor product has finite\nlength is equivalent to saying that $I+J$ is ${\\gothic{m}}$-primary, where $I = \\text{Ann}(N)$ and\n$J = \\text{Ann}(M)$. Then we can choose a prime $P$ containing $I$ such that $\\dim(R\/P) =\n\\dim (N)$, and observe that we can replace $N$ by $R\/P$ without loss of generality.\nIt is more difficult to change $M$, since the property of being finite projective dimension\ndoes not allow many changes.  \n\nLet's just suppose for a moment that $R\/P$ is Cohen-Macaulay. Since $P+J$ is ${\\gothic{m}}$-primary,\nwe can always choose $x_1,...,x_d\\in J$ whose images in $B = R\/P$ form a system of parameters\n(and thus are a regular sequence in $R\/P$). If the projective dimension of $M$\nis smaller than $d = \\dim(R\/P)$, then  $Tor_d^{R}(B\/(x_1,...,x_d)B, M) = 0$.  \nNotice that $Tor_0^{R}(B, M) \\ne 0$. We claim by induction that for $0\\leq i\\leq d$,\n$Tor_i^{R}(B\/(x_1,...,x_i)B, M) \\ne 0$. When $i=d$ we arrive at a contradiction.\nSuppose we know this for $i < d$.  Set $B _i = B\/(x_1,...,x_i)$. The short exact sequence\n$0\\to B_i\\to B_i\\to B_{i+1}\\to 0$ obtained by multiplication\nby $x_{i+1}$ on $B_i$ induces a map of Tors when tensored with $M$. Since all $x_i$ kill $M$,\nwe obtain a surjection of $Tor_{i+1}^{R}(B_{i+1}, M)$ onto $Tor_i^{R}(B_i, M)$.\nThis finishes the induction.\n\nOf course, we don't know that $R\/P$ is Cohen-Macaulay, and in general it won't be. But now\nsuppose that we are in positive characteristic. We can first complete $R$ before beginning\nthe proof. Now $R\/P$ is a complete local domain, and $S = (R\/P)^+$ is Cohen-Macaulay in\nthe sense that $x_1,..,x_d$ form a regular sequence in this ring. The same proof works\nverbatium, provided we know that $S\\otimes_RM\\ne 0$. But this is easy; it is even nonzero\nafter passing to the residue field of $S$.  \\end{proof}\n\n\nAs a corollary, we get a favorite of the old Chicago school of commutative algebra, \nthe zero-divisor conjecture (now a theorem):\n\n\n\\bigskip\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a Noetherian local ring of characteristic $p$, and let\n$M$ be a nonzero finitely generated $R$-module having finite projective dimension. If\n$x$ is a non-zerodivisor on $M$, then $x$ is a non-zerodivisor on $R$.\n\\end{thm}\n\n\\begin{proof} This proof is taken from \\cite{PS}.  First observe that the statement of the theorem is equivalent to saying\nthat every associated prime of $R$ is contained in an associated prime of $M$. We induct on\nthe dimension of $M$ to prove this statement. If $\\dim M = 0$, then the only associated\nprime of $M$ is ${\\gothic{m}}$, which clearly contains every prime of $R$. Hence we may assume that\n$\\dim M > 0$. Let $P\\in \\text{Ass}(R)$. First suppose that there is a prime $Q\\in \\text{Supp}(M), Q\\ne {\\gothic{m}},$\nsuch that $P\\subseteq Q$. Then we can change the ring to $R_Q$ and the module to $M_Q$. By induction, $P_Q$ is\ncontained in an associated prime of $M_Q$, so lifting back gives us that $P$ is in an associated\nprime of $M$. We have reduced to the case in which $R\/P\\otimes_RM$ has finite length. By \nTheorem~\\ref{newintersection}, $\\dim (R\/P)\\leq \\text{pd}_R(M) =  \\text{depth}(R) - \\text{depth}(M)$. Since\n$P$ is associated to $R$,  $\\dim (R\/P)\\geq \\text{depth}(R)$ (exercise). It follows that\nthe depth of $M$ is $0$, and hence the maximal ideal is associated to $M$ (and contains $P$).\n\\end{proof} \n\n\\medskip\n\nFor a completely different type of application, we consider an old result of Grothendieck's concerning\nwhen the punctured spectrum of a local ring is connected.\nThere is a beautiful proof of Grothendieck's result in all characteristics due to Brodmann and Rung \\cite{BR}. The main\npoint here is that if $R$ is Cohen-Macaulay and the $x_i$ are parameters, then there is a very easy\nproof. It turns out that one can always assume that the $x_i$ are parameters, and then the proof\nof the Cohen-Macaulay case directly generalizes to one in characteristic $p$ using $R^+$. \nThe exact statement is:\n\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a complete local Noetherian domain of dimension $d$, and let\n$x_1,...,x_k\\in {\\gothic{m}}$, where $k\\leq d-2$. Then the punctured spectrum of $R\/(x_1,...,x_k)$ is\nconnected.\n\\end{thm}\n\n\\begin{proof} We take the proof from \\cite{HH}.\nFirst assume that the $x_i$ are parameters.\nLet $I$ and $J$ give a disconnection of the punctured spectrum\nof $R\/(x_1,...,x_k)$. Choose elements $u+v$, $y+z$ which together with the $x_i$ form parameters\nsuch that $u,y\\in I$ and $v,z\\in J$. Modulo $(x_1,...,x_k)$ one has the relation\n$y(u+v) - u(y+z) = 0$. Since the parameters form a regular sequence in $R^+$, we obtain that\n$y\\in (y+z,x_1,...,x_k)R^+$. Similarly, $z\\in (y+z,x_1,...,x_k)R^+$. Write $y = c(y+z)$ modulo\n$(x_1,...,x_k)$ and $z = d(y+z)$ modulo $(x_1,...,x_k)$. Then $(1-c-d)(y+z)\\in (x_1,...,x_k)R^+$\nso that $1-c-d\\in (x_1,...,x_k)R^+$. At least one of $c$ or $d$ is a unit in $R^+$, say $c$. But then\n$y$ is not a zerodivisor modulo $(x_1,...,x_k)R^+$ and this implies that $J\\subseteq (x_1,...,x_k)R^+$.\nThen $I$ must be primary to $mR^+$ which is a contradiction since the height of $I$ is too small.\n\n\nIt remain to reduce to the case in which the $x_i$ are parameters.\n\nWe claim that any $k$-elements are up to radical in an ideal generated by $k$-elements which are\nparameters.\n The key point is to prove this for $k = 1$. Suppose that $x = x_1$ is given. If $x$\nalready has height one we are done. If $x$ is nilpotent, choose $y$ to be any parameter in $I$. So assume\nthat $x$ is not in every minimal prime. For $n\\gg 0$, $0:x^n = 0:x^{n+1}$, and changing $x$ to $x^n$,\nwe obtain that $x$ is not a zero divisor on $R\/(0:x)$. Then there is an element $s\\in 0:x$ such that\n$y = x +s$ has height one, and we may multiply $s$ by a general element of $I$ to obtain that\n$y\\in I$. But $xy = x^2$ so that $x$ is nilpotent on $(y)$. Inductively choose\n$y_1,...,y_{k-1}$ which are parameters such that the ideal they generate contains $x_1,...,x_{k-1}$ up\nto radical. Replace $R$ by $R\/(y_1,...,y_{k-1})$, and repeat the $k = 1$ step.\n\nNow if\n$I$ and $J$ disconnect the punctured spectrum of $R\/(x_1,...,x_k)$ choose any ideals $I'$ and $J'$\nof height at least $k$, not primary to the maximal ideal such that $I'$ contains $I$ up to radical\nand $J'$ contains $J$ up to radical. Choose parameters in $I'\\cap J'$ such that the $x_i$ are in the\nradical $K$ of these parameters. Then $K + I$ and $K + J$ disconnect the punctured spectrum of $R\/K$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{General remarks}\nA large open problem of classical general relativity is the\ncharacterization of the structure of a spacetime by initial data. The\nflat case, Minkowski spacetime, is geodesically complete. To the\nother extreme the singularity theorems by R. Penrose and S. Hawking show\nthat the spacetime cannot be geodesically complete if the data are\nlarge \\cite{HaE73TL}.\n\\\\\nIn the last years there has been remarkable progress in describing\nwhat happens if one goes from data for flat space to large data: The\nfuture of small data evolving in accordance with\nthe Einstein equation with various matter models as sources, vacuum\nand Einstein-Maxwell-Yang-Mills, looks like the future of data for\nflat space \\cite{ChK93TG,Fr93as}. Nevertheless many problems are\nstill unsolved.\n\\\\\nThose results were significantly improved by D.~Christodoulou for\nspherically symmetric models with a massless Klein-Gordon scalar field as\nsource. He was able to relate properties of the initial data to\nproperties of singularities. But even in this case of high symmetry\nthe questions left are still numerous as numerical\nsimulations by M.~Choptuik show \\cite{Ch93ua}. He found very\ninteresting properties, the so called echoing effect, for models which\nare in the parameter space of initial data near to the boundary which\nseparates regular from singular spacetimes.\n\\\\\nIn this paper conformal techniques are used to analyze the\nhyperboloidal initial value problem with scalar fields as\nmatter models --- for data near Minkowskian data the future of the\ninitial value\nsurface possesses a smooth future null infinity and a regular timelike\ninfinity, for large data a smooth future null infinity exists for at\nleast some time. In the second part of the introduction more about\nconformal techniques and their application for a mathematical\ndescription of asymptotically flat spacetimes will be said.\n\\\\\nAlthough the primarily treated matter model is that of the conformally\ninvariant scalar field, whose equations can be written as\n\\begin{subequations}\n\\label{model}\n\\begin{eqnarray}\n\\label{Wllngl}\n  \\tilde{\\vphantom{\\phi}\\Box} \\mbox{$\\tilde{\\phi}$} - \\frac{\\mbox{$\\tilde{R}$}}{6} \\, \\mbox{$\\tilde{\\phi}$} & = & 0\n\\\\\n\\label{EinstPhys}\n  ( 1 - \\frac{1}{4} \\mbox{$\\tilde{\\phi}$}^2 ) \\, \\mbox{$\\tilde{R}$}_{ab} & = &\n       \\left(\n          (\\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}) - \\frac{1}{2} \\, \\mbox{$\\tilde{\\phi}$} \\, \\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}\n          - \\frac{1}{4} \\, \\mbox{$\\tilde{g}$}_{ab} (\\mbox{$\\tilde{\\nabla}$}^c \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_c \\mbox{$\\tilde{\\phi}$})\n       \\right),\n\\end{eqnarray}\nthe results obtained apply to a larger class of scalar field models,\ngiven by the class of actions (\\ref{ScalarAction}), including the massless\nKlein-Gordon field, as shown in section \\ref{SkalarAequiv}. Note that\nan arbitrary factor can be absorbed into $\\mbox{$\\tilde{\\phi}$}{}$ which changes the\ncoefficients in (\\ref{EinstPhys}). My notational conventions are\ndescribed in the appendix, the $\\tilde{\\vphantom{H}}$ marks quantities\nin the physical spacetime (see definition \\ref{asymFlat}). The\nenergy-momentum tensor for the conformally invariant scalar field can\nbe written as\n\\begin{equation}\n\\label{TkonfS}\n  \\mbox{$\\tilde{T}$}_{ab} = (\\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}) - \\frac{1}{2} \\, \\mbox{$\\tilde{\\phi}$} \\, \\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}\n        + \\frac{1}{4} \\, \\mbox{$\\tilde{\\phi}$}^2 \\mbox{$\\tilde{R}$}_{ab} -\n        \\frac{1}{4} \\, \\mbox{$\\tilde{g}$}_{ab}\n          \\left( (\\mbox{$\\tilde{\\nabla}$}^c \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_c \\mbox{$\\tilde{\\phi}$}) + \\frac{1}{6} \\, \\mbox{$\\tilde{\\phi}$}^2 \\mbox{$\\tilde{R}$}\n          \\right).\n\\end{equation}\n\\end{subequations}\nThe analytic investigation presented in this paper show the well\nposedness of the initial value problem in unphysical spacetime, which\nis a technical construct to ``compactified'' asymptotically flat\nspacetimes in analogy to the compactification of the plane of complex\nnumbers ($\\Bbb{R}^2$) into the Riemann sphere ($\\Bbb{S}^2$).\n\\\\\nOne goal\nof this work was making myself familiar with the system in unphysical\nspacetime as a preparation for numerical work showing\nthat the conformal techniques are well suited for a numerical\ninvestigation of global spacetime structure and gravitational\nradiation \\cite{HuXXIP,Hu93nu}. To lower the computational\nresources required these calculations have been done for spherical\nsymmetry. It is well known that spherically symmetric, uncharged vacuum\nmodels are Schwarzschild. The inclusion of matter removes\nthat obstacle, the spacetime may evolve dynamically. Furthermore there\nis no gravitational radiation in spherically symmetric models. Therefore\nthe matter model should also be a model for radiation. Scalar fields\nwith wave equations are choices for matter which model also radiation.\nThe conformally invariant scalar field has been\nchosen since the matter equations are form invariant under rescalings\nof the metric and an appropriate transformation of the scalar field\nas the name already suggests.\n\\\\\nThe scalar fields are interesting from the analytic viewpoint since\nfor the first time conformal techniques could be used for matter\nmodels whose energy-momentum tensor has non-vanishing trace.\n\\subsection{Asymptotically flat spacetimes}\nIn this paper a geometrical, coordinate independent definition of\nasymptotical flatness along the lines suggested by R.~Penrose\nwill be used. A more thorough discussion\nof the ideas and the interpretation can be found at various places in\nthe literature, e.g.\\ \\cite{Ge76as,Pe64ct}. The\ndefinitions of asymptotical flatness given in the literature differ\nslightly. The following will be used here:\n\\begin{Def}\n\\label{asymFlat}\n  A spacetime $(\\tilde M, \\mbox{$\\tilde{g}$}_{ab})$ is called {\\bf asymptotically\n    flat} if there is another ``unphysical'' spacetime $(M,g_{ab})$\n  with boundary \\mbox{$\\cal J$}{} and a smooth embedding by which $\\tilde M$\n  can be identified with $M-\\mbox{$\\cal J$}{}$ such that:\n  \\begin{enumerate}\n  \\item There is a smooth function $\\Omega$ on $M$ with\n    \\begin{equation*}\n      \\Omega \\mid_{\\tilde M} > 0 \\qquad \\mbox{and} \\qquad\n      g_{ab} \\mid_{\\tilde M} = \\Omega^2 \\mbox{$\\tilde{g}$}_{ab}.\n    \\end{equation*}\n  \\item On \\mbox{$\\cal J$}{}\n    \\begin{equation*}\n      \\Omega = 0 \\qquad \\mbox{and} \\qquad \\nabla_a \\Omega \\ne 0.\n    \\end{equation*}\n  \\item \\label{nullCompleteness} Each null geodesic in $(\\tilde\n    M,\\tilde g_{ab})$ acquires a past and a future endpoint on \\mbox{$\\cal J$}{}.\n  \\end{enumerate}\n\\end{Def}\nBecause of item~\\ref{nullCompleteness} null geodesically incomplete\nspacetimes like Schwarzschild are not asymptotically flat. The next\ndefinition includes those spacetimes which have only an\nasymptotically flat part:\n\\begin{Def}\n\\label{weakasymFlat}\n  A spacetime is called {\\bf weakly asymptotically flat} if\n  definition~\\ref{asymFlat} with the exception\n  of item~\\ref{nullCompleteness} is fulfilled.\n\\end{Def}\nDefinition~\\ref{asymFlat} and~\\ref{weakasymFlat} classify spacetimes,\nthey do not require that Einstein's equation is fulfilled. One would\nlike to know:\n\\\\\nAre they compatible with the Einstein equation with sources?\nNeither definition~\\ref{asymFlat} nor \\ref{weakasymFlat} is in an\ninitial value problem form: A given spacetime is or is not classified\nas asymptotically flat. But for a physical problem one would like to\ngive ``asymptotically flat data'' and have guaranteed that they evolve\ninto an at least weakly asymptotically flat spacetime.\n\\\\\nNevertheless the geometrically description was extremely helpful in\nanalyzing asymptotically flat spacetimes and it can be successfully\nused as guideline to construct a formalism which is better suited for\nanalyzing initial value problems. This method has been developed and\napplied to various matter sources by H.~Friedrich~\n\\cite{Fr81on,Fr83cp,Fr85ot,Fr86op,Fr86ot,Fr88os,Fr91ot}. In this paper\nit will be applied to general relativistic scalar field models.\n\\\\\nThe idea is to choose a spacelike initial value surface in the\nunphysical spacetime $(M,g_{ab})$ and to evolve it. The problems to\nbe faced are:\n\\\\\nFor Minkowski space the unphysical spacetime $(M,g_{ab})$ can be\nsmoothly extended with three points, future $(i^+)$ and past  $(i^-)$\ntimelike infinity, the end respectively the starting point of all\ntimelike geodesics of $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$, and spacelike infinity\n$(i^0)$,\nthe\nend point of all spacelike geodesics of $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$. The point $i^0$\ndivides \\mbox{$\\cal J$}{} into two disjunct parts, future $(\\mbox{$\\cal J$}^+)$ and past\n$(\\mbox{$\\cal J$}^-)$ null infinity. It is well known and has been discussed\nelsewhere that there are unsolved problems in smoothly\nextending a ``normal'' Cauchy hypersurface of $\\tilde M$ to $i^0$ if the\nspacetime has non-vanishing ADM mass. Certain curvature quantities\nblow up at $i^0$, reflecting the non-invariance of the mass under\nrescalings.\n\\\\\nBy choosing a spacelike (with respect to $g_{ab}$) hypersurface $S$\nnot intersecting $i^0$ but $\\mbox{$\\cal J$}^+$ ($\\mbox{$\\cal J$}^-$) we avoid the problems\nwith $i^0$. $S$ is called a hyperboloidal hypersurface --- the\ncorresponding initial value problem is called a hyperboloidal initial\nvalue (a detailed definition for the scalar field models is given in\nsection~\\ref{HypInitValProblSec}, definition~\\ref{HypInitValProbl}).\nThe domain of dependence $D(S)$ of\n$S$ will not contain the whole spacetime. The interior of $S$\ncorresponds to an everywhere spacelike hypersurface in the physical\nspacetime which approaches a null hypersurface $N$ asymptotically. If $N$\nis a light cone $L$ then the domain of dependence of $S$ is $L$. Therefore\nthe hyperboloidal initial value problem is well suited to describe the\nfuture (past) of data on the spacelike hypersurface $S$, e.~g.~a\nstellar object and the gravitational radiation caused by its\ntime evolution. It is not well suited to investigate the structure\nnear $i^0$.\n\\\\\nBut even for the hyperboloidal initial value problem there are\n``regularity'' problems at \\mbox{$\\cal J$}{}: Transforming the Einstein equation\nfrom physical to unphysical spacetime an equation ``singular'' for\n$\\Omega=0$ results. That problem is solved in this paper in analogy to\nH.~Friedrich's work. A new set of equations\nfor the unphysical spacetime will be derived, its equivalence to the\nEinstein equation on $\\tilde M$ proven. This new set of equations is\nused to prove the consistency of the hyperboloidal initial value\nproblem for scalar fields with (weakly) asymptotical flatness and the\nexistence of a regular future (past) timelike infinity\nfor data sufficiently close to data for Minkowski spacetime.\n\\section{Regularizing the unphysical field equations}\nA first attempt for equations determining $(M,g_{ab})$ is the rescaled\nform of the field equation in physical spacetime. A closer look at\nthe transformation of the Einstein tensor under rescalings\n$g_{ab}=\\Omega^2\\,\\mbox{$\\tilde{g}$}_{ab}$,\n\\begin{equation}\n\\label{GabTransf}\n  \\mbox{$\\tilde{G}$}_{ab} =\n  G_{ab}\n  + 2 \\, \\Omega^{-1} \\left(\n    \\nabla_a \\nabla_b \\Omega - (\\nabla^c \\nabla_c \\Omega ) \\, g_{ab}\n    \\right)\n  + 3 \\, \\Omega^{-2} ( \\nabla^c \\Omega ) ( \\nabla_c \\Omega ) \\, g_{ab},\n\\end{equation}\nshows that this first\nattempt fails. Either there are terms proportional to $\\Omega^{-2}$\nand $\\Omega^{-1}$, which need special care on the set ${\\cal I}$ of\npoints where $\\Omega=0$, including\n\\mbox{$\\cal J$}{}, which is part of $M$. Or alternatively, the highest (second\norder) derivatives of the metric, hidden in the Einstein tensor, are\nmultiplied by a factor of $\\Omega^2$ and then the principal part of\nthe second order equation for the metric components vanishes on\n${\\cal I}$. This behaviour of an equation will be called singular on\n${\\cal I}$.\n\\\\\nIn this section a system of equations without the singularity on\n${\\cal I}$ will be derived from the rescaled Einstein\nequation by introducing new variables and equations.\n\\\\\nThe set of equations together with the equations for the matter\nvariables may be a system with a very complicated principal part ---\nas it is the case for a conformally invariant scalar field as\nmatter model. A procedure is carried out to simplify the principal\npart to a form in which no equation contains both derivatives of\nmatter variables as well as derivatives of geometry variables and the\nprincipal part of the subsystem for the geometry variables is the same\nas for the vacuum case (``standard form''). All variables already\npresent in the vacuum case are called geometry variables.\n\\\\\nIt is shown that the procedure works for the conformally invariant\nscalar field. The procedure described does not use very restrictive\nassumptions --- it is very general --- and may work for most matter\nmodels, for which the unphysical matter equations can be regularized on\n${\\cal I}$.\n\\subsection{The geometry part of the system}\nAccording to the definition of asymptotical flatness\n(definition \\ref{asymFlat}) the unphysical spacetime is connected with the\nphysical spacetime through the rescaling\n\\begin{equation}\n  g_{ab} \\mid_{\\tilde M} = \\Omega^2 \\, \\mbox{$\\tilde{g}$}_{ab}.\n\\end{equation}\nThis rule also determines the transformation of the connection\nand the curvature.\n\\\\\nAdditionally the transformation of the matter variables $\\tilde\n\\Phi{}$ under rescaling must be specified,\n\\begin{equation}\n  \\Phi \\mid_{\\tilde M} = \\Phi[\\mbox{$\\tilde{g}$}_{ab},\\Omega,\\tilde \\Phi].\n\\end{equation}\nIt is assumed that $\\Phi{}$ has a smooth limit on \\mbox{$\\cal J$}{}, the\nrescaled equations for the matter variables are regular on ${\\cal\n  I} $\\footnote{In the general case it is not known how to achieve\n  that.},\nand there exists a tensor $T_{ab}$, which is independent of derivatives\nof $\\Omega{}$ and derivatives of curvature terms, fulfills\n\\begin{equation}\n  T_{ab} \\mid_{\\tilde M} = \\Omega^{-2}\\, \\mbox{$\\tilde{T}$}_{ab},\n\\end{equation}\nand has a limit on \\mbox{$\\cal J$}{} \\footnote{From the definition of\n  asymptotical flatness and the Einstein equation it follows that\n  $\\Omega^{-1}\\, \\mbox{$\\tilde{T}$}_{ab}$ has a limit on \\mbox{$\\cal J$}{}~\\cite{AsS80ni}. The\n  faster fall off and the requirements on the form of $T_{ab}$ have\n  technical reasons.}.\nThe conditions required may seem very restrictive but they\ncan be fulfilled for Yang-Mills fields~\\cite{Fr91ot} and for the\nconformally invariant scalar field.\n\\\\[0.1cm]\nThe Riemann tensor will be split into its irreducible parts, the\nconformal Weyl tensor\n\\begin{equation}\n\\label{ddef}\nC_{abc}{}^d=:\\Omega\\,d_{abc}{}^d,\n\\end{equation}\nthe trace free part $\\mbox{$\\hat{R}$}_{ab}$ of the Ricci tensor $R_{ab}$ and the\nRicci scalar $R$:\n\\begin{equation}\n\\label{Ralg}\n  R_{abcd} =\n  \\Omega \\, d_{abcd}\n  + g_{c[a} \\mbox{$\\hat{R}$}_{b]d} - g_{d[a} \\mbox{$\\hat{R}$}_{b]c}\n  + \\frac{1}{6} g_{c[a} g_{b]d} R,\n\\end{equation}\nA $\\hat{\\phantom{H}}$ is used as an indication for trace free parts of\n\ntensors.\n\\\\\nThe irreducible decomposition of the energy-momentum tensor is\n\\begin{equation}\n\\label{IrredT}\n  T_{ab} = \\mbox{$\\hat{T}$}_{ab} + \\frac{1}{4} \\, g_{ab} \\, T.\n\\end{equation}\nThe irreducible parts transform under rescalings according to\n$$\n  \\mbox{$\\hat{T}$}_{ab} = \\Omega^{-2} \\mbox{$\\tilde{\\hT}$}_{ab}\n$$\nand\n$$\n  T = \\Omega^{-4} \\mbox{$\\tilde{T}$}.\n$$\nThe vanishing of the divergence of $\\mbox{$\\tilde{T}$}_{ab}$ becomes\n\\begin{equation}\n\\label{transdivT}\n  0 = \\tilde\\nabla^a \\tilde T_{ab} =\n  \\Omega^4 \\, \\nabla^a \\mbox{$\\hat{T}$}_{ab} +\n  \\frac{1}{4} \\, \\Omega^4 \\, \\nabla_b T +\n  \\Omega^3 \\, T \\, \\nabla_b \\Omega.\n\\end{equation}\nFor energy-momentum tensors with non-vanishing trace equation\n(\\ref{transdivT}) as an equation for the components of the\nirreducible parts of the energy-momentum tensor $T_{ab}$\nis singular on ${\\cal I}$. Since (\\ref{transdivT}) should be in same way\npart of the matter equations problems in regularizing the matter\nequations are to be expected.\n\\subsubsection{A regular system}\nThe part of (\\ref{GabTransf}) proportional to $\\mbox{$\\Omega$}^{-2}$ is a pure\ntrace, thus the $\\mbox{$\\Omega$}^{-2}$ singularity is absent in the trace free equation.\nA decomposition into the trace and the trace free part moves the\nworst term into one equation.\n\\\\\n{}From the rescaling rule for the Ricci scalar and tensor,\n\\begin{equation}\n\\label{Rtrans}\n  \\mbox{$\\tilde{R}$} = \\Omega^2 R + 6 \\, \\Omega \\, \\nabla^a \\nabla_a \\Omega -\n        12 \\, (\\nabla^a \\Omega) \\, (\\nabla_a \\Omega),\n\\end{equation}\nand\n\\begin{eqnarray}\n  \\tilde{\\mbox{$\\hat{R}$}}_{ab} & := & \\mbox{$\\tilde{R}$}_{ab} - \\frac{1}{4} \\mbox{$\\tilde{g}$}_{ab} \\mbox{$\\tilde{R}$}\n\\nonumber \\\\\n   & = & \\mbox{$\\hat{R}$}_{ab} + 2 \\, \\Omega^{-1} \\nabla_a \\nabla_b \\Omega\n         - \\frac{1}{2} \\Omega^{-1} (\\nabla^c \\nabla_c \\Omega) \\,\n           g_{ab},\n\\end{eqnarray}\n$\\mbox{$\\tilde{G}$}_{ab} = \\mbox{$\\tilde{T}$}_{ab}$, ${\\mbox{$\\tilde{G}$}=-\\mbox{$\\tilde{R}$}}$, and ${\\mbox{$\\tilde{T}$} = \\mbox{$\\tilde{G}$}}$\nit follows\n\\begin{equation}\n\\label{SpGl}\n  \\Omega \\, R + 6 \\, \\nabla^a \\nabla_a \\Omega\n    - 12 \\, \\Omega^{-1} \\, (\\nabla^a \\Omega) \\, (\\nabla_a \\Omega)\n  = - \\, \\Omega^3 \\, T\n\\end{equation}\nand\n\\begin{equation}\n\\label{sfGl}\n  \\Omega \\, \\mbox{$\\hat{R}$}_{ab} + 2 \\, \\nabla_a \\nabla_b \\Omega -\n  \\frac{1}{2} (\\nabla^c \\nabla_c \\Omega) \\, g_{ab} =\n    \\Omega^3 \\, T_{ab}.\n\\end{equation}\nEquation (\\ref{SpGl}) can be dealt with by the following lemma:\n\\begin{Lemma}\n  From $\\mbox{$\\tilde{R}$}+\\mbox{$\\tilde{T}$}=0$ ($\\hat{=}$ (\\ref{SpGl})) at one point,\n  $\\hat{\\mbox{$\\tilde{G}$}}_{ab}=\\hat{\\mbox{$\\tilde{T}$}}_{ab}$ ($\\hat{=}$ (\\ref{sfGl})), and $\\mbox{$\\tilde{\\nabla}$}^b\n  {\\mbox{$\\tilde{T}$}_{ab}}=0$ $\\mbox{$\\tilde{R}$}+\\mbox{$\\tilde{T}$}=0$ follows everywhere.\n\\end{Lemma}\nProof:\n\\begin{equation*}\n  \\mbox{$\\tilde{\\nabla}$}^a \\mbox{$\\tilde{T}$}_{ab} = \\mbox{$\\tilde{\\nabla}$}^a \\mbox{$\\tilde{\\hT}$}_{ab} + \\frac{1}{4} \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{T}$} = 0.\n\\end{equation*}\nCombined with\n\\begin{eqnarray*}\n  0 & = & \\mbox{$\\tilde{\\nabla}$}^a \\mbox{$\\tilde{G}$}_{ab} \\\\\n    & = & \\mbox{$\\tilde{\\nabla}$}^a \\tilde{\\mbox{$\\hat{G}$}}_{ab} + \\frac{1}{4} \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{G}$}\n\\end{eqnarray*}\ngives\n\\begin{equation*}\n  \\mbox{$\\tilde{\\nabla}$}_b (\\mbox{$\\tilde{T}$} + \\mbox{$\\tilde{R}$}) = 0,\n\\end{equation*}\ni.e. $\\mbox{$\\tilde{T}$} + \\mbox{$\\tilde{R}$} $ is constant.\n\\\\\nEquation (\\ref{SpGl}) will not be used any longer since $\\mbox{$\\tilde{\\nabla}$}^b\n{\\mbox{$\\tilde{T}$}_{ab}}=0$ can be derived from the remaining equations, contract\n(\\ref{quaSysd}) or see the discussion following (\\ref{IntegrBed}).\n\\\\\nIn the following the Ricci scalar $R$ will be regarded as an\narbitrary, given function.\nIt fixes part of the gauge freedom on the transition from the\nphysical to the unphysical spacetime as follows:\nThe equations (\\ref{SpGl}) and (\\ref{sfGl}) are invariant under\nrescalings ${(g_{ab},\\Omega)}\\mapsto{(\\bar g_{ab},\\bar\n  \\Omega)}:={(\\Theta^2 g_{ab},\\Theta\\,\\Omega)}$ with $\\Theta>0$. All\nthe unphysical spacetimes $(M,\\Theta^2 g_{ab}, \\Theta\\,\\Omega)$ belong\nto the same physical spacetime $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$.\n\\\\\nUnder the rescaling $\\bar g_{ab} = \\Theta^2\\,g_{ab}$, $R$ and $\\bar R$\nare connected by\n\\begin{equation}\n\\label{conformalGauge}\n  6\\, \\nabla^a \\nabla_a \\Theta = \\Theta R - \\Theta^3 \\bar R,\n\\end{equation}\nwhich is equation (\\ref{Rtrans}) where the covariant derivatives\n$\\nabla_a$ now corresponds to the unscaled metric. Solving\n(\\ref{conformalGauge}) for a spacetime ($M$, $g_{ab}$) and data\nfor $\\Theta{}$ and $\\dot \\Theta{}$ on\na spacelike surface $S$ we get at least locally a unphysical\nspace time with arbitrary Ricci scalar $\\bar{R}$.\n\\\\\nThere is still conformal gauge freedom left as every rescaling with\n$\\Theta>0$ and\n\\begin{equation}\n\\label{ConformalGaugeFix}\n  \\nabla^a \\nabla_a \\Theta = \\frac{1}{6} \\Theta\\,R\\,\\left(1-\\Theta^2\\right)\n\\end{equation}\nleaves the Ricci scalar unchanged.\n\\\\[2\\parskip]\nEquation (\\ref{sfGl}) serves as regular equation for\n$\\Omega{}$. Substituting ${\\omega = \\frac{1}{4} \\, \\nabla^c \\nabla_c\n  \\Omega}$ yields\n\\begin{equation}\n\\label{OmGl}\n  \\nabla_a \\nabla_b \\Omega = - \\, \\frac{1}{2} \\, \\Omega \\, \\mbox{$\\hat{R}$}_{ab}\n                      + \\omega \\, g_{ab}\n                      + \\frac{1}{2} \\, \\Omega^3 \\mbox{$\\hat{T}$}_{ab},\n\\end{equation}\nwhich is a second order equation for $\\Omega{}$.\n\\\\\nThe next step is to find equations for the metric and the quantities\nderived therefrom. Expressing the once contracted, second Bianchi\nidentity (${\\nabla_{[a} R_{bc]d}{}^a =0}$)  in terms of $\\mbox{$\\hat{R}$}_{ab}$ and\n$d_{abc}{}^d$\nresults in\n\\begin{equation}\n\\label{hRGl}\n  \\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c} = - \\, \\frac{1}{12} \\, (\\nabla_{[a} R) g_{b]c}\n                          - (\\nabla_d \\Omega) \\, d_{abc}{}^d\n                          - \\Omega \\, \\nabla_d d_{abc}{}^d.\n\\end{equation}\nThe once contracted second Bianchi identity in the physical spacetime,\n$$\n  \\mbox{$\\tilde{\\nabla}$}_d \\mbox{$\\tilde{C}$}_{abc}{}^d =\n  - \\mbox{$\\tilde{\\nabla}$}_{[a} (\\mbox{$\\tilde{R}$}_{b]c} - \\frac{1}{6} \\mbox{$\\tilde{g}$}_{b]c} \\mbox{$\\tilde{R}$}),\n$$\ntogether with\n$$\n  \\Omega^{-1} \\mbox{$\\tilde{\\nabla}$}_d \\mbox{$\\tilde{C}$}_{abc}{}^d = \\nabla_d(\\Omega^{-1}\n                                    C_{abc}{}^d),\n$$\nand the Einstein equation in physical spacetime provide us with\nan equation for $d_{abc}{}^d$:\n\\begin{eqnarray}\n\\label{dGl}\n  \\nabla_d d_{abc}{}^d\n                       & = & - \\, \\Omega \\, \\nabla_{[a} \\mbox{$\\hat{T}$}_{b]c}\n                             - 3 \\, (\\nabla_{[a} \\Omega) \\, \\mbox{$\\hat{T}$}_{b]c}\n                             + g_{c[a} \\mbox{$\\hat{T}$}_{b]d} \\, (\\nabla^d \\Omega)\n                                                        \\nonumber  \\\\\n                       &   & + \\frac{1}{3} \\, (\\nabla_{[a} \\Omega) \\,\n                       T \\, g_{b]c}\n                             + \\frac{1}{12} \\Omega \\, (\\nabla_{[a}T)\n                               \\, g_{b]c}  \\; =: \\;  t_{abc}.\n\\end{eqnarray}\nWe can now derive the missing equation for $\\omega{}$ from the\nintegrability condition for (\\ref{OmGl}) and by substituting (\\ref{hRGl}):\n\\begin{eqnarray}\n\\label{omGl}\n  \\nabla_a \\omega & = & - \\frac{1}{2} \\mbox{$\\hat{R}$}_{ab} \\, \\nabla^b \\Omega\n                    - \\frac{1}{12} \\, R \\, \\nabla_a \\Omega\n                    - \\frac{1}{24} \\Omega \\, \\nabla_a R\n                    + \\frac{1}{2} \\Omega^2 \\, \\mbox{$\\hat{T}$}_{ab} \\, \\nabla^b \\Omega\n                                                       \\nonumber   \\\\\n                  &  &  - \\frac{1}{6} \\Omega^2 \\, (\\nabla_a \\Omega) \\, T\n                        - \\frac{1}{24} \\Omega^3 \\, \\nabla_a T.\n\\end{eqnarray}\nIn the following in addition to the abstract indices (small Latin\nletters) frame (underlined indices) and coordinate indices (Greek\nletters) are used. The used conventions are explained in the appendix\nin more detail.\n\\\\\nUsing $\\Omega_a := \\nabla_a \\Omega{}$, the frame  $e_{\\f{i}}{}^a$, and\nthe Ricci rotation coefficients $\\gamma^a{}_{\\f{i}\\f{j}}$ as further\nvariables, we get the following first order system of tensor equations\nfor  $\\Omega{}$, $\\Omega_a$, $\\omega$, $e_{\\f{i}}{}^a$,\n$\\gamma^a{}_{\\f{i}\\f{j}}$, $\\mbox{$\\hat{R}$}_{ab}$, and $d_{abc}{}^d$: \\footnote{\nThe symbol $E$ stands for equation, the first index reminds to the\nquantity for which a temporary equation will be formed by setting the\ntensor $E$ equal to $0$. The tensors providing the eventually used\nforms of\nthe\nequations are named with $\\N{}$ standing for null quantities.}\n\\begin{subequations}\n\\label{quaSys}\n\\begin{eqnarray}\n  \\label{quaSysOm}\n  \\label{NO}\n  \\N{\\Omega}_a = E\\I{\\Omega}_a & = & \\nabla_a \\Omega - \\Omega_a = 0   \\\\\n  \\label{quaSysDOm}\n  \\label{NDO}\n  \\N{D\\Omega}_{ab} = E\\I{D\\Omega}_{ab} & = &\n    \\nabla_a \\Omega_b + \\frac{1}{2} \\Omega \\mbox{$\\hat{R}$}_{ab}\n    - \\omega g_{ab} - \\frac{1}{2} \\Omega^3 \\mbox{$\\hat{T}$}_{ab} = 0           \\\\\n  \\label{No}\n  \\N{\\omega}_{ab} = E\\I{\\omega}_a & = &\n    \\nabla_a \\omega + \\frac{1}{2} \\mbox{$\\hat{R}$}_{ab} \\Omega^b\n    + \\frac{1}{12} R \\Omega_a + \\frac{1}{24} \\Omega \\nabla_a R\n    - \\frac{1}{2} \\Omega^2 \\mbox{$\\hat{T}$}_{ab} \\Omega^b           \\nonumber  \\\\\n    && \\qquad + \\frac{1}{6} \\Omega^2 \\Omega_a T\n    + \\frac{1}{24} \\Omega^3 \\nabla_a T     =  0                   \\\\\n  \\label{quaSysDe}\n  \\label{Ne}\n  \\N{e}^a{}_{bc} = E\\I{e}^a{}_{bc} & = & T^a{}_{bc}   = 0 \\\\\n  \\label{quaSysDgamma}\n  \\label{Ng}\n  \\N{\\gamma}_{abc}{}^d = E\\I{\\gamma}_{abc}{}^d & = &\n    R\\I{diff}_{abc}{}^d - R\\I{alg}_{abc}{}^d  = 0 \\\\\n  \\label{quaSysR}\n  E\\I{R}_{abc} & = & \\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}\n    + \\frac{1}{12} (\\nabla_{[a} R) g_{b]c} + \\Omega_d d_{abc}{}^d\n    + \\Omega \\, t_{abc} = 0  \\\\                         \n  \\label{quaSysd}\n  E\\I{d}_{abc} & = &  \\nabla_d d_{abc}{}^d  - t_{abc} = 0\n\\end{eqnarray}\n\\end{subequations}\nwhere (\\ref{quaSysDe}) means vanishing torsion $T^a{}_{bc}$, expressed\nin frame index form,\n\\begin{equation*}\n    T^{\\f{i}}{}_{\\f{j}\\f{k}} =\n    \\left( e_{\\f{j}}(e_{\\f{k}}{}^\\mu) -\n           e_{\\f{k}}(e_{\\f{j}}{}^\\mu) \\right)\n      e^{\\f{i}}{}_\\mu\n  + \\gamma^{\\f{i}}{}_{\\f{j}\\f{k}} - \\gamma^{\\f{i}}{}_{\\f{k}\\f{j}},\n\\end{equation*}\nand (\\ref{quaSysDgamma}) means that the curvature tensor in terms of\nthe Ricci rotation coefficients, in frame index form\n\\begin{eqnarray*}\n    R\\I{diff}_{\\f{i}\\f{j}\\f{k}}{}^{\\f{l}} & = &\n  e_{\\f{j}}(\\gamma^{\\f{l}}{}_{\\f{i}\\f{k}})\n    - e_{\\f{i}}(\\gamma^{\\f{l}}{}_{\\f{j}\\f{k}})\n  - \\gamma^{\\f{l}}{}_{\\f{i}\\f{m}} \\gamma^{\\f{m}}{}_{\\f{j}\\f{k}}\n  + \\gamma^{\\f{l}}{}_{\\f{j}\\f{m}} \\gamma^{\\f{m}}{}_{\\f{i}\\f{k}}\n  \\nonumber  \\\\\n  && \\qquad\n  + \\gamma^{\\f{m}}{}_{\\f{i}\\f{j}} \\gamma^{\\f{l}}{}_{\\f{m}\\f{k}}\n  + \\gamma^{\\f{m}}{}_{\\f{j}\\f{i}} \\gamma^{\\f{l}}{}_{\\f{m}\\f{k}}\n  - \\gamma^{\\f{l}}{}_{\\f{m}\\f{k}} T^{\\f{m}}{}_{\\f{j}\\f{i}},\n\\end{eqnarray*}\nshould equal the combination\n\\begin{equation*}\n  \\Omega \\, d_{abcd}\n  + g_{c[a} \\mbox{$\\hat{R}$}_{b]d} - g_{d[a} \\mbox{$\\hat{R}$}_{b]c}\n  + \\frac{1}{6} g_{c[a} g_{b]d} R =: R\\I{alg}_{abc}{}^d,\n\\end{equation*}\nwhich is the irreducible decomposition of a tensor with the symmetry\nof the Riemann tensor (\\ref{Ralg}). Hence (\\ref{quaSysDe}) and\n(\\ref{quaSysDgamma})\nensure that $R\\I{alg}_{abc}{}^d$ is the curvature tensor corresponding\nto the connection given by the Ricci rotation coefficients which again\nis the torsion free connection coming from the metric (frame).\n\\subsubsection{Complications by the matter terms}\nThe final goal is to use the terms $\\nabla_a \\Omega{}$,  $\\nabla_a\n\\Omega_b$, $\\nabla_a \\omega{}$, $\\left( e_{\\f{j}}(e_{\\f{k}}{}^\\mu) -\ne_{\\f{k}}(e_{\\f{j}}{}^\\mu) \\right)  e^{\\f{i}}{}_\\mu{}$,\\hfill\n${e_{\\f{j}}(\\gamma^{\\f{l}}{}_{\\f{i}\\f{k}})  -\ne_{\\f{i}}(\\gamma^{\\f{l}}{}_{\\f{j}\\f{k}})}$, $\\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}$, and\n$\\nabla_d d_{abc}{}^d$ in (\\ref{quaSys}) as principal\npart for the geometry variables of the system. I will call these terms\nleft side of the equations, the remaining terms right side. The left side\ndoes not contain the complete principle part of the system yet as the\nenergy momentum\ntensor $T_{ab}$ and its derivatives $\\nabla_{[a} T_{b]c}$ may contain\nderivatives of the matter and geometry variables.\n\\\\\nIn the case of the conformally invariant scalar field the field equation\n(\\ref{model}) remains invariant under the rescaling\n\\begin{equation*}\n  \\phi = \\Omega^{-1} \\, \\mbox{$\\tilde{\\phi}$},\n\\end{equation*}\ni.e.\n\\begin{equation*}\n  {\\vphantom{\\phi}\\Box} \\phi - \\frac{R}{6} \\, \\phi = 0.\n\\end{equation*}\nThe physical energy-momentum tensor $\\mbox{$\\tilde{T}$}_{ab}$ fulfills the assumed properties,\n\\begin{eqnarray*}\n  \\mbox{$\\tilde{T}$}_{ab} & = &  \\mbox{$\\Omega$}^2\n               \\left[\n                  (\\nabla_a \\phi) (\\nabla_b \\phi)\n                  - \\frac{1}{2} \\phi \\nabla_a \\nabla_b \\phi\n                  + \\frac{1}{4} \\phi^2 R_{ab}\n                  - \\frac{1}{4} g_{ab}\n                  \\left( (\\nabla^c \\phi) (\\nabla_c \\phi)\n                         + \\frac{1}{6} \\phi^2 R \\right)\n               \\right]  \\nonumber \\\\\n                   & =: &  \\mbox{$\\Omega$}^2 \\, T_{ab}.\n\\end{eqnarray*}\n\\\\\nThe mentioned complications in (\\ref{quaSys}) by the right sides are\nnow obvious. Firstly $\\nabla_{[a} T_{b]c}$ contains $\\nabla_{[a}\n\\nabla_{b]} \\nabla_c \\phi $ terms which are eliminated with the\nidentity $\\nabla_{[a} \\nabla_{b]} \\nabla_c \\phi{} = \\frac{1}{2}\nR_{abc}{}^d \\nabla_d \\phi $. To get rid of the second and first\norder derivatives of $\\phi $ we use the first order system\n\\begin{subequations}\n\\label{NM}\n\\begin{eqnarray}\n\\label{Np}\n  \\N{\\phi}_a & = & \\nabla_a \\phi - \\phi_a = 0\n\\\\\n\\label{NDp}\n  \\N{D\\phi}_{ab} & = &\n  \\nabla_a \\phi_b - \\hat{\\phi}_{ab} - \\frac{1}{4} \\phi_c{}^c \\, g_{ab} = 0\n\\\\\n\\label{NBp}\n  \\N{\\Box\\phi} & = & \\phi_a{}^a - \\frac{R}{6} \\, \\phi = 0\n\\\\\n\\label{NDDp}\n  \\N{DD\\phi}_{abc} & = &\n  \\nabla _{[a} \\hat{\\phi}_{b]c}\n  + \\frac{1}{6} \\, ( \\phi\\,\\nabla_{[a} R + R \\, \\phi_{[a} ) g_{b]c}\n  - \\frac{1}{2} \\, R\\I{alg}_{abc}{}^d \\phi_d = 0\n\\\\\n\\label{NDBp}\n  \\N{D\\Box\\phi}_a & = &\n  \\nabla _{a} \\phi_b{}^b\n  - \\frac{1}{6} \\, ( \\phi\\, \\nabla_a R + R \\, \\phi_{a} ) = 0.\n\\end{eqnarray}\n\\end{subequations}\nfor the variables $\\phi$, $\\phi_a$, the trace free symmetric tensor\n$\\hat{\\phi}_{ab}$ and the trace $\\phi_a{}^a$. The system is\nderived from $\\nabla_a\n\\left( {\\vphantom{\\phi}\\Box} \\phi - \\frac{R}{6} \\, \\phi \\right) =0$.\nSystem (\\ref{NM}) also serves as matter part of the system for the\nunphysical spacetime.\n$t_{abc}$ is now written in a form which does not contain any\nderivatives of matter variables explicitly.\n\\\\\n$\\nabla_{[a} T_{b]c}$ and thus $t_{abc}$ still contain\nderivatives $\\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}$ of the trace free Ricci tensor.\nBy combining (\\ref{quaSysR}) and (\\ref{quaSysd}) the\nderivatives of $\\mbox{$\\hat{R}$}_{ab}$ and $d_{abc}{}^d$ can be decoupled.\n(\\ref{quaSysR}) and (\\ref{quaSysd}) become\n\\begin{equation}\n\\label{quaSysRvar}\n  E'\\I{R}_{abc} = \\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}\n    + \\frac{1}{12} (\\nabla_{[a} R) g_{b]c} - \\Omega_d d_{abc}{}^d\n    + \\Omega m_{abc} = 0\n\\end{equation}\nand\n\\begin{equation}\n\\label{quaSysdvar}\n  E'\\I{d}_{abc} = \\nabla_d d_{abc}{}^d  - m_{abc} = 0,\n\\end{equation}\nwith\n\\begin{eqnarray*}\n  \\lefteqn{ m_{abc} = \\frac{1}{1-\\frac{1}{4} \\, \\Omega^2\\phi^2} \\qquad * }  \\\\\n  \\lefteqn{ \\Big( \\, \\Omega } && \\qquad\n  \\big[ \\frac{3}{2}  \\, \\phi_{[a} \\phi_{b]c}\n    - \\frac{1}{2}  \\, g_{c[a} \\phi_{b]d} \\phi^d\n    + \\frac{1}{4}  \\, \\phi  \\, \\Omega d_{abc}{}^d \\phi_d\n    + \\frac{1}{4}  \\, \\phi  \\, g_{c[a} \\mbox{$\\hat{R}$}_{b]}{}^d \\phi_d\n    - \\frac{3}{4}  \\, \\phi  \\, \\phi_{[a} \\mbox{$\\hat{R}$}_{b]c}        \\\\ && \\qquad \\quad\n    - \\frac{1}{12}  \\, \\phi  \\, \\phi_{[a} g_{b]c} R\n    + \\frac{1}{4}  \\, \\Omega  \\, \\phi^2 d_{abc}{}^d \\Omega_d  \\big]\n\\\\ && \\quad\n  - 3  \\, \\Omega_{[a} \\big[ \\phi_{b]} \\phi_c\n    - \\frac{1}{2}  \\, \\phi  \\, \\phi_{b]c}\n    + \\frac{1}{4}  \\, \\phi^2 \\mbox{$\\hat{R}$}_{b]c}\n    + \\frac{1}{36}  \\, \\phi^2 g_{b]c} R\n    - \\frac{1}{3}  \\, g_{b]c}  \\, \\phi^d \\phi_d \\big]\n\\\\ && \\quad\n  + \\Omega^d g_{c[a} \\big[ \\phi_{b]} \\phi_d\n    - \\frac{1}{2}  \\, \\phi  \\, \\phi_{bd}\n    + \\frac{1}{4}  \\, \\phi^2 \\mbox{$\\hat{R}$}_{b]d}    \\big]     \\quad     \\Big).\n\\end{eqnarray*}\nNote that $m_{abc}$ may become singular for\n$1-\\frac{1}{4}\\,\\Omega^2\\phi^2= 1-\\frac{1}{4}\\,\\mbox{$\\tilde{\\phi}$}^2=0$. In the\nEinstein equations for the physical spacetime (\\ref{EinstPhys})\n$\\mbox{$\\tilde{R}$}_{ab}$ carries a factor $1-\\frac{1}{4} \\mbox{$\\tilde{\\phi}$}^2$ too.\nWe will need later that\n\\begin{equation*}\n  \\N{m}_{abc} := t_{abc} - m_{abc} =\n  - \\frac{1}{4}  \\, \\Omega  \\, \\phi^2\n    \\left( \\N{R}_{abc} + \\frac{2}{3} \\, \\Omega \\, m_{[a|d|}{}^d \\,\n    g_{b]c}\n\\right),\n\\end{equation*}\nwhere $\\N{R}_{abc}$ is the null quantity representing the final form\nof the equation for $\\mbox{$\\hat{R}$}_{ab}$ (\\ref{NR}).\nThe final form of  the equation for $d_{abc}{}^d$ is\nobtained from (\\ref{quaSysdvar}) by replacing $E'\\I{d}_{abc}=0$ with\n\\begin{equation}\n\\label{Nd}\n  \\N{d}_{abc} := E'\\I{d}_{abc} + \\frac{2}{3} m_{[a|d|}{}^d g_{b]c} = 0.\n\\end{equation}\nThis gives $\\N{d}_{abc}$ the same index symmetry properties as the Weyl tensor.\nThat replacement does not change the equation\nsince $m_{ab}{}^b=0$ as will be seen later.\n\\\\\nAnalogously we replace (\\ref{quaSysRvar}) with\n\\begin{equation}\n\\label{NR}\n  \\N{R}_{abc} :=\n    E'\\I{R}_{abc} - \\frac{2}{3} \\, \\Omega \\, m_{[a|d|}{}^d g_{b]c} = 0,\n\\end{equation}\nthe contraction $\\N{R}_{ab}{}^b=0$ is then the contracted second\nBianchi identity.\n\\section{Evolution equations and constraints}\nIn the following I will assume a system $\\N{}=0$ of the form (\\ref{quaSysOm})\n-- (\\ref{quaSysDgamma}),  (\\ref{Nd}), (\\ref{NR}),  the geometry part,\nand a matter part, in the case of the scalar field model system\n(\\ref{NM}). $m_{abc}$ and $t_{abc}$ are assumed to differ only by terms\nexpressible as null quantities. The energy-momentum tensor\n$T_{ab}$, its derivatives $\\nabla_a T_{bc}$, $m_{abc}$, and $t_{abc}$\nare assumed to be expressed in variables and thus do not contain any\nexplicit derivative of variables. By these assumptions the\nprincipal part of the system has block form, the geometry block and\nthe matter block. The two blocks are coupled through the right sides.\n\\\\\nIn this chapter the system $\\N{}=0$ will be reduced to a system of\nsymmetric hyperbolic time evolution equations, the subsidiary system.\nSufficient conditions for the equivalence of the subsidiary system and\n$\\N{}=0$ are given as conditions on $m_{abc}$. If the system can be put\ninto the described block form there do not arise any more conditions\nfrom the geometry part of the system for any matter.\n\\\\\nThe explicit carry out is technical and lengthy, the idea can be\nsummarized as follows: All the equations of the system are\nregarded as null quantities. By requiring the vanishing of some of these null\nquantities and by choosing an appropriate gauge condition for the\ncoordinates and the frame we get a symmetric hyperbolic\nsubsidiary system of evolution equations.\nWhich null quantities to choose can best be seen by a decomposition into\nthe irreducible parts in the spinor calculus as performed\nin~\\cite{Fr91ot}.\n\\\\\nThe solution of this symmetric hyperbolic subsystem exists and is\nunique. To complete the proof we must show\nthat the solution obtained in this way is consistent with the rest of\nthe equations, i.e.\\ that all null quantities remain zero if they are\ninitially zero (``propagation of the constraints''). For that purpose a\nsymmetric hyperbolic system of time evolution equations for\nthe remaining null quantities is derived. Sufficient conditions for the\npropagation of the constraints are firstly the homogeneity of the evolution\nequations for the remaining null quantities in the null quantities\nsince then the unique solution of these evolution equations is the vanishing\nof all null quantities for all times if they vanish on the initial\nsurface and secondly that the domain of dependence of $S$ with respect\nto the equations for the propagation of the constraints is a\nsuperset of the domain of dependence of $S$ with respect to the\nsubsidiary system.\n\\subsection{A symmetric hyperbolic subsystem of evolution\n  equations}\n\\label{GaussGauge}\nIntroducing a timelike vector field $t^a$ not necessarily hypersurface\northogonal and its orthogonal projection tensor\n$h_{ab}:=g_{ab}-t_at_b\/(t_ct^c)$ allows to split the\nsystem of equations into two categories, the equations containing time\nderivatives and those containing no time derivatives (the\nconstraints). The equations with time derivatives provide a\nunder\/overdetermined system of evolution equations.\n\\\\\nThe system is\noverdetermined since for some quantities there are too many time\nevolution equations, e.g.\\ there are 12 time evolution equations from\n$\\N{R}_{abc}=0$\nfor 9 independent tensor components. 3 equations are a linear\ncombination of the other 9 equations and the constraints. An\nirreducible decomposition of the tensors $\\N{}$ is a systematic way to\nanalyze these dependencies. Since all the types of tensor index\nsymmetries appearing in the system have been thoroughly investigated\nin~\\cite{Fr91ot} I will only state which combinations are needed.\n\\\\\nThe system is underdetermined since there are 10 time evolution\nequations for the frame and the Ricci rotation coefficients missing.\nBy adding\n\\begin{equation}\n\\label{KoEichFrame}\n   e^{\\f{i}}{}_b \\, g^{\\f{j}\\f{k}} \\, e_{\\f{j}}{}^a \\left( \\nabla_a\n   e_{\\f{k}}{}^b \\right) = - F_{\\f{i}} =\n   \\gamma_{\\f{i}\\f{k}}{}^{\\f{k}}\n\\end{equation}\nand\n\\begin{equation}\n\\label{FrEichFrame}\n    \\partial_{\\f{k}} \\gamma^{\\f{i}\\f{k}\\f{j}}\n    + \\gamma^{\\f{i}\\f{k}\\f{j}} F_{\\f{k}}\n    + \\gamma^{\\f{l}\\f{k}\\f{i}} \\, \\gamma_{\\f{l}\\f{k}}{}^{\\f{j}}\n    - \\gamma^{\\f{l}\\f{k}\\f{j}} \\, \\gamma_{\\f{l}\\f{k}}{}^{\\f{i}}\n    = F^{\\f{i}\\f{j}},\n\\end{equation}\nthe system becomes complete. The freedom of giving ten functions\ncorresponds to the freedom of giving the lapse and the shift to\ndetermine the coordinates and the six parameters of the Lorentz group\nto determine the frame on every point. The gauge freedom is discussed\nin full detail in~\\cite{Fr85ot,Fr91ot}.\n\\\\\nA choice which makes the system especially simple for analytic\nconsiderations is a Gaussian coordinate and frame system defined as\nfollows. Give on the spacelike initial value surface $S$ coordinates\n$x^\\mu, \\mu = 1..3$, and 3 orthonormal vector fields $e_{\\f{i}}{}^a,\n\\f{i}=\\f{1}..\\f{3}$ in this hypersurface. The affine parameter of the\ngeodesics of the hypersurface orthonormal, timelike vector field\n$e_{\\f{0}}{}^a$ defines the\ntime coordinate $x^0=t$. The spacelike coordinates are transported by\nthese geodesics into a neighbourhood of the initial surface. By\ngeodesic transport of $e_{\\f{i}}{}^a, \\f{i}=\\f{1}\\ldots\\f{3}$, and\n$e_{\\f{0}}{}^a$  a frame is obtained in this neighbourhood. By\nconstruction we have\n$$\n  e_{\\f{0}}{}^0=1, \\quad e_{\\f{0}}{}^\\mu = 0 \\mbox{ for }\\mu = 1..3\n$$\nand\n$$\n  \\gamma^{\\f{i}}{}_{\\f{0}\\f{k}} = 0.\n$$\nIt is well known that Gaussian coordinates develop caustics if the\nenergy momentum tensor fulfills certain energy conditions, see\ne.g.~\\cite[lemma 9.2.1]{Wa84GR}. In the\nunphysical spacetime the $\\Omega{}$ terms provide a kind of unphysical\nenergy-momentum tensor.\nWhether this energy-momentum tensor fulfills the energy-momentum\nconditions is a difficult question and not known to the author.\nNevertheless the coordinates develop caustics as has been shown by\nnumerical calculations \\cite{Hu93nu}.\n\\\\\nThe following combinations give a symmetric hyperbolic\nsystem for the remaining variables as can be deduced from the\nconsiderations in~\\cite{Fr83cp}:\n\\begin{subequations}\n\\label{EvoSyst}\n\\begin{eqnarray}\n&& \\N{\\Omega}_{\\f{0}} = 0 , \\\\\n&& \\N{D\\Omega}_{\\f{0}b} = 0 , \\\\\n&& \\N{\\omega}_{\\f{0}} = 0 , \\\\\n&& \\N{e}^a{}_{b\\f{0}} = 0 , \\\\\n&& \\N{\\gamma}_{\\f{0}\\f{1}c}{}^d = 0 , \\\\\n&& g^{ab} \\N{R}_{\\f{i}ab} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3}, \\\\\n&& \\N{R}_{\\f{0}\\f{i}\\f{i}} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3}, \\\\\n&& \\N{R}_{\\f{0}\\f{i}\\f{j}} + \\N{R}_{\\f{0}\\f{j}\\f{i}} = 0 ,\n   \\qquad (\\f{i},\\f{j}) = (\\f{1},\\f{2}),(\\f{1},\\f{3}),(\\f{2},\\f{3}), \\\\\n&& \\N{d}_{\\f{2}\\f{1}\\f{2}} - \\N{d}_{\\f{3}\\f{1}\\f{3}} +\n   \\N{d}_{\\f{2}\\f{0}\\f{2}} - \\N{d}_{\\f{3}\\f{0}\\f{3}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{0}\\f{2}} + \\N{d}_{\\f{1}\\f{2}\\f{1}} = 0 , \\\\\n&& \\N{d}_{\\f{1}\\f{0}\\f{1}} = 0 , \\\\\n&& \\N{d}_{\\f{1}\\f{0}\\f{2}} + \\N{d}_{\\f{1}\\f{2}\\f{1}} = 0 , \\\\\n&& - \\N{d}_{\\f{2}\\f{1}\\f{2}} +\n   \\N{d}_{\\f{3}\\f{1}\\f{3}} + \\N{d}_{\\f{2}\\f{0}\\f{2}} -\n   \\N{d}_{\\f{3}\\f{0}\\f{3}} = 0 , \\\\\n&& \\N{d}_{\\f{2}\\f{1}\\f{3}} + \\N{d}_{\\f{3}\\f{1}\\f{2}} + \\N{d}_{\\f{2}\\f{0}\\f{3}}+\n   \\N{d}_{\\f{3}\\f{0}\\f{2}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{0}\\f{3}} + \\N{d}_{\\f{1}\\f{3}\\f{1}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{2}\\f{3}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{0}\\f{3}} - \\N{d}_{\\f{1}\\f{3}\\f{1}} = 0 , \\\\\n&& \\N{d}_{\\f{2}\\f{1}\\f{3}} + \\N{d}_{\\f{3}\\f{1}\\f{2}} -\n   \\N{d}_{\\f{2}\\f{0}\\f{3}} - \\N{d}_{\\f{3}\\f{0}\\f{2}} = 0 , \\\\\n&& \\N{\\phi}_{\\f{0}} = 0 , \\\\\n&& \\N{D\\phi}_{\\f{0}b} = 0 , \\\\\n&& g^{ab} \\N{DD\\phi}_{\\f{i}ab} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3},\n\\\\\n&& \\N{DD\\phi}_{\\f{0}\\f{i}\\f{i}} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3},\n\\\\\n&& \\N{DD\\phi}_{\\f{0}\\f{i}\\f{j}} + \\N{DD\\phi}_{\\f{0}\\f{j}\\f{i}} = 0\n, \\qquad (\\f{i},\\f{j}) = (\\f{1},\\f{2}),(\\f{1},\\f{3}),(\\f{2},\\f{3}), \\\\\n&& \\N{D\\Box\\phi}_{\\f{0}} = 0.  \\yesnumber\n\\end{eqnarray}\n\\end{subequations}\nTo see that the system is really symmetric hyperbolic one has to write\ndown the system explicitly. By an appropriate, in the explicit\nform of the system obvious definition of new variables, the system has\nthe structure\n\\begin{equation}\n\\label{symhypEvoSys}\n  \\underline{\\underline{A_t}}\\,\\partial_t \\underline{f} +\n  \\sum_{i=1}^3 \\underline{\\underline{A_{x^i}}}\\,\\partial_{x^i}\n  \\underline{f} +\n  \\underline{b}(\\underline{f},x^\\mu) = 0,\n\\end{equation}\nwith a diagonal matrix\n$\\underline{\\underline{A_t}}$, which is positive definite for\n$1-\\frac{1}{4}\\Omega^2\\phi^2>0$, and symmetric matrices\n$\\underline{\\underline{A_{x^i}}}$. $\\underline{f}$ is the vector build\nfrom the variables.\nAll the remaining equations are linear combinations of~(\\ref{EvoSyst})\nand constraints.  Since the explicit form of the constraints is not\nneeded I do not list them.\n\\\\\nAs the entries in $\\underline{\\underline{A_t}}$ coming from\n$\\N{R}=0$ and $\\N{d}=0$ vanish for $1-\\frac{1}{4}\\Omega^2\\phi^2=0$ the\nfollowing results apply only if $\\Omega^2\\phi^2<4$ everywhere on the\ninitial value surface $S$ and thus in a neighbourhood of $S$. The\nphysical Einstein equations have a corresponding singularity for\n$1-\\frac{1}{4}\\mbox{$\\tilde{\\phi}$}^2=0$ (see equation \\ref{EinstPhys}).\n\\subsection{A sufficient condition for the propagation of the\n  constraints}\nAccording to the analysis in~\\cite{Fr91ot}, involving the left hand\nside of the following identities, a symmetric\nhyperbolic system of evolution equations for the remaining null quantities\ncan be extracted from:\n\\begin{subequations}\n\\label{PropConstrSys}\n\\begin{equation}\n  \\nabla_{[a}\\N{\\Omega}_{b]} =\n  - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\Omega - \\N{D\\Omega}_{[ab]}\n\\end{equation}\n\n\\begin{eqnarray}\n  \\lefteqn{ \\nabla_{[a}\\N{D\\Omega}_{b]c} = }   \\nonumber   \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}_{abc}{}^d \\Omega_d\n  - \\frac{1}{2} T^d{}_{ab} \\nabla_d \\Omega_c\n  + \\frac{1}{2} \\Omega \\N{R}_{abc}\n  + \\frac{1}{2} \\mbox{$\\hat{R}$}_{c[b} \\N{\\Omega}_{a]}\n  - \\frac{3}{2} \\Omega^2 \\N{\\Omega}_{[a} \\mbox{$\\hat{T}$}_{b]c} \\nonumber \\\\ && \\qquad\n  - \\frac{1}{2} \\N{\\omega}_{[a} g_{b]c}\n  + \\frac{1}{3} \\Omega^2 m_{[a|d|}{}^d g_{b]c}\n  + \\frac{1}{3} \\Omega^2 \\N{m}_{[a|d|}{}^d g_{b]c}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n  \\lefteqn{ \\nabla_{[a}\\N{\\omega}_{b]} = }  \\nonumber          \\\\ && \\qquad\n  - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\omega\n  - \\frac{1}{24} \\Omega^3 T^c{}_{ab} \\nabla_c T\n  + \\frac{1}{24} \\N{\\Omega}_{[a} \\nabla_{b]} R\n  + \\frac{1}{8} \\Omega^2 \\N{\\Omega}_{[a} \\nabla_{b]} T  \\nonumber \\\\ && \\qquad\n  + \\frac{1}{2} ( \\mbox{$\\hat{R}$} _{c[b} - \\Omega^2 \\mbox{$\\hat{T}$}_{c[b} ) \\N{D\\Omega}_{a]}{}^c\n  + ( \\frac{1}{24} R + \\frac{1}{6} \\Omega^2 T ) \\N{D\\Omega}_{[ab]}\n  + \\frac{1}{2} \\Omega^c \\N{R}_{abc}             \\nonumber        \\\\ && \\qquad\n  + \\frac{1}{3} \\Omega \\, \\Omega_{[b} m_{a]c}{}^c\n  + \\frac{1}{2} \\Omega \\, \\Omega^c \\N{m}_{abc}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\label{PropNR}\n  \\lefteqn{ \\nabla_{[a} \\N{R}_{bc]d} = }        \\nonumber  \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}_{[abc]}{}^f \\mbox{$\\hat{R}$}_{fd}\n  + \\frac{1}{2} \\N{\\gamma}_{[ab|d]}{}^f \\mbox{$\\hat{R}$}_{c]f}\n  - \\frac{1}{2} T^f{}_{[ab} \\nabla_{|f|}\\mbox{$\\hat{R}$}_{c]d}\n  - \\frac{1}{24} T^f{}_{[ab} (\\nabla_{|f|} R ) g_{c]d} \\nonumber \\\\ && \\qquad\n  + \\N{D\\Omega}_{[a|f|} d_{bc]d}{}^f\n  + \\Omega^f ( \\N{d}_{[ca|d|} g_{b]f} - \\N{d}_{[ca|f|} g_{b]d} )\n  + \\N{\\Omega}_{[a} m_{bc]d}           \\nonumber                 \\\\ && \\qquad\n  - \\frac{2}{3} \\N{\\Omega}_{[a} m_{b|f|}{}^f g_{c]d}\n  - \\frac{1}{2} \\Omega^2 \\N{\\gamma}_{[abc]}{}^f \\mbox{$\\hat{T}$}_{fd}\n  - \\frac{1}{2} \\Omega^2 \\N{\\gamma}_{[ab|d|}{}^f \\mbox{$\\hat{T}$}_{c]f}\n  - \\frac{1}{2} \\Omega^2 T^f{}_{[ab} \\nabla_{|f|} \\mbox{$\\hat{T}$}_{c]d} \\nonumber\n  \\\\ &&\n\\qquad\n  - 3 \\Omega \\N{D\\Omega}_{[ab} \\mbox{$\\hat{T}$}_{c]d}\n  + \\Omega \\N{D\\Omega}_{[a}{}^f \\mbox{$\\hat{T}$}_{c|f|} g_{b]d}\n  + \\frac{1}{3} \\Omega \\N{D\\Omega}_{[ab} g_{c]d}\n  + \\frac{1}{12} \\Omega \\N{\\Omega}_{[a} ( \\nabla_b T ) g_{c]d}\n    \\nonumber  \\\\ && \\qquad\n  + \\frac{1}{12} \\Omega^2 T^f{}_{[ab} ( \\nabla_{|f|} T ) g_{c]d}\n  - 2 \\Omega_{[a} \\N{m}_{bc]d}\n  + \\Omega_f \\N{m}_{[ca}{}^f g_{b]d}      \\nonumber              \\\\ && \\qquad\n  - \\Omega \\nabla_{[a} \\N{m}_{bc]d}\n  + 2 \\Omega_{[a} m_{c|f|}{}^f g_{b]d}\n  - \\frac{2}{3} \\Omega ( \\nabla_{[a} m_{b|f|}{}^f ) g_{c]d}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n  \\lefteqn{ \\nabla^{c} \\N{d}_{abc} = }  \\nonumber     \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}^c{}_{da}{}^f d_{fbc}{}^d\n  + \\frac{1}{2} \\N{\\gamma}^c{}_{db}{}^f d_{afc}{}^d\n  + \\frac{1}{2} \\N{\\gamma}^c{}_{dc}{}^f d_{abf}{}^d\n  + \\frac{1}{2} \\N{\\gamma}^{cd}{}_{df} d_{abc}{}^f  \\nonumber   \\\\ && \\qquad\n  - \\frac{1}{2} T^{fcd} \\nabla_f d_{abcd}\n  - \\nabla^c m_{abc}\n  + \\frac{2}{3} \\nabla_{[b} m_{a]c}{}^c\n\\end{eqnarray}\n\n\\begin{equation}\n  \\nabla_{[a} \\N{e}^d{}_{bc]} =\n  \\N{\\gamma}_{[abc]}{}^d + \\N{e}^f{}_{[ab} \\N{e}^d{}_{c]f}\n\\end{equation}\n\n\\begin{eqnarray}\n\\label{dNgamma}\n  \\lefteqn{ \\nabla_{[f} \\N{\\gamma}_{ab]cd} = } \\nonumber \\\\ && \\qquad\n  - T^g{}_{[fa} R\\I{diff}_{b]gcd}\n  - \\N{\\Omega}_{[f} d_{ab]cd}\n  - \\Omega ( \\N{d}_{[bf|c|} g_{a]d} - \\N{d}_{[bf|d|} g_{a]c} )\n  \\nonumber \\\\ && \\qquad\n  - \\N{R}_{[fb|d|} g_{a]c}\n  + \\N{R}_{[fb|c|} g_{a]d}\n\\end{eqnarray}\n\n\\begin{equation}\n  \\nabla_{[a}\\N{\\phi}_{b]} =\n  - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\phi - \\N{D\\phi}_{[ab]}\n\\end{equation}\n\n\\begin{equation}\n  \\nabla_{[a}\\N{D\\phi}_{b]c} =\n  \\frac{1}{2} \\N{\\gamma}_{abc}{}^d \\phi_d\n  - \\frac{1}{2} T^d{}_{ab} \\nabla_d \\phi_c\n  - \\N{DD\\phi}_{abc}\n  - \\frac{1}{4} \\N{D\\Box\\phi}_{[a} g_{b]c}\n\\end{equation}\n\n\\begin{equation}\n  \\nabla_a \\N{\\Box\\phi} =\n  \\N{D\\Box\\phi}_a\n  - \\frac{1}{6} R \\N{\\phi}_a\n\\end{equation}\n\n\\begin{eqnarray}\n\\label{PropNDDp}\n  \\lefteqn{ \\nabla_{[a} \\N{DD\\phi}_{bc]d} = }   \\nonumber  \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}_{abc}{}^f \\hat{\\phi}_{fd}\n  + \\frac{1}{2} \\N{\\gamma}_{[ab|d|}{}^f \\hat{\\phi}_{c]f}\n  - \\frac{1}{2} T^f{}_{[ab} \\nabla_{|f|} \\hat{\\phi}_{c]d}\n  + \\frac{1}{6} \\N{\\phi}_{[a} ( \\nabla_b R ) g_{c]d}   \\nonumber \\\\ && \\qquad\n  - \\frac{1}{12} \\phi T^f{}_{[ab} g_{c]d} \\nabla_f R\n  + \\frac{1}{6} R \\N{D\\phi}_{[ab} g_{c]d}\n  + \\frac{1}{2} T^g{}_{[ab} R\\I{diff}_{c]gd}{}^f \\phi_f\n  - \\frac{1}{2} R_{[bc|d|}{}^f \\N{D\\phi}_{a]f}        \\nonumber \\\\ && \\qquad\n  + \\frac{1}{2} ( \\nabla_{[a} \\N{\\gamma}_{bc]d}{}^f ) \\phi_f,\n\\end{eqnarray}\nand\n\\begin{equation}\n  \\nabla_{[a} \\N{D\\Box\\phi}_{b]} =\n  - \\frac{1}{2} T^d{}_{ab} \\nabla_d \\phi_c{}^c\n  - \\frac{1}{6} \\N{\\phi}_{[a} \\nabla_{b]} R\n  - \\frac{1}{12} \\phi T^c{}_{ab} \\nabla_c R\n  - \\frac{1}{6} R \\N{D\\phi}_{[ab]}.\n\\end{equation}\n\\end{subequations}\nThe last term in (\\ref{PropNDDp}) is homogeneous in null quantities as\ncan be seen from (\\ref{dNgamma}),\nThe deviation of these equalities is even more lengthy than the\nequalities itself, but the essential ideas behind it can already be seen\nin the deviation of the first:\n\\begin{eqnarray*}\n  \\nabla_{[a}\\N{\\Omega}_{b]} & = &\n    \\nabla_{[a}  \\nabla_{b]}\\Omega - \\nabla_{[a}  \\Omega_{b]} \\\\\n  & = &\n    - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\Omega - \\N{D\\Omega}_{[ab]}\n    + \\frac{1}{2} \\mbox{$\\hat{R}$}_{[ab]} \\Omega\n    - \\frac{1}{2} \\Omega^3 \\mbox{$\\hat{T}$}_{[ab]}\n    - \\omega g_{[ab]},\n\\end{eqnarray*}\nwith the last three terms vanishing since the tensors are symmetric. Note\nthat vanishing of the torsion, $T^c{}_{ab}=0$,  and\n$2\\,\\nabla_{[a}\\nabla_{b]}\\omega_c=R_{abc}{}^d \\omega_d$ cannot be\nused since they only\nhold if both the time evolution and the constraint equations for the frame\n$e_{\\f{i}}{}^\\mu $ and the Ricci rotation coefficients\n$\\gamma^a{}_{\\f{i}\\f{j}}$ hold everywhere.\n\\\\\nA sufficient set of conditions for homogeneity of the system derived\nin the null quantities is\n\\begin{subequations}\n\\label{IntegrBed}\n\\begin{equation}\n\\label{IntegrBed1}\n  m_{ab}{}^b = 0 \\bmod \\N{},\n\\end{equation}\n\\begin{equation}\n\\label{IntegrBed2}\n  \\nabla^c m_{abc} + \\frac{2}{3} \\nabla_{[a} m_{b]c}{}^c = 0 \\bmod \\N{},\n\\end{equation}\n\\begin{equation}\n\\label{IntegrBed3}\n  \\nabla_{[a} m_{b]c}{}^c = 0 \\bmod \\N{},\n\\end{equation}\nand\n\\begin{equation}\n\\label{IntegrBed4}\n  \\Omega \\nabla_{[a} \\N{m}_{bc]d} =\n     f \\, \\nabla_{[a} \\N{R}_{bc]d} \\bmod \\N{}, \\quad f \\ne -1.\n\\end{equation}\n\\end{subequations}\nA straightforward but long calculation shows that these conditions are\nfulfilled by the conformally invariant scalar field with\n$f=-\\frac{1}{4}\\Omega^2\\phi^2$. Equation  (\\ref{PropNR})\nbecomes singular for $1-\\frac{1}{4}\\Omega^2\\phi^2=0$.\n\\\\\nThe very technical integrability conditions (\\ref{IntegrBed}) have a\nvery simply interpretation. Replacing $m_{abc}$ with $t_{abc}$ --- they\nonly differ by null quantities --- the conditions\n(\\ref{IntegrBed1}--\\ref{IntegrBed3}) reduce to $\\mbox{$\\tilde{\\nabla}$}^b \\mbox{$\\tilde{T}$}_{ab} = 0$\nand ${\\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\nabla}$}^c \\mbox{$\\tilde{T}$}_{ac} = 0}$. Condition (\\ref{IntegrBed4}) is only\nof technical nature, it gives the principal part a\nsimple block form.\n\\\\\n{}From the considerations in \\cite{Fr91ot} also follows that the domain\nof dependence of $S$ with respect to the evolution equation of the\nconstraints includes the domain of dependence of $S$ with respect to\nthe subsidiary system.\n\\section{The hyperboloidal initial value problem}\n\\label{HypInitValProblSec}\nSo far a system of equations $(\\N{}=0)$ has been derived which contains\nfor at least one choice of gauge a symmetric hyperbolic subsystem of\nevolution equations. The remaining equations in the system --- either\nconstraints or a combination of constraints and time evolution\nequations --- will be satisfied for a solution of the evolution\nequations, if the constraints are satisfied by the initial data. If\nboth, the time evolution and the constraints, are fulfilled, $(\\tilde\nM,\\mbox{$\\tilde{g}$}_{ab},\\mbox{$\\tilde{\\phi}$})$ is a weakly asymptotically flat solution of the Einstein\nequation. This follows from the way the system $\\N{}=0$ for the\nunphysical spacetime has been derived.\n\\\\\nThe essential points in the proofs of the theorems in\n\\cite[chapter~10]{Fr91ot} are the symmetric hyperbolicity of the\nsubsidiary system and the form (\\ref{NDO}) of the equations for\n$\\Omega{}$. Therefore the same techniques can be used and the proofs\nwill not be repeated. The difference to\nthe model treated here lies in the derivation of the subsidiary system\nand the proof of the propagation of the constraints,\nwhich has been done in the previous chapters.\n\\subsection{The initial value problem}\nWe consider the following initial value problem:\n\\begin{Def}\n\\label{HypInitValProbl}\nA ``{\\bf hyperboloidal initial data set for the conformally invariant\nscalar field}'' consists of a pair $(\\bar S,f_0)$ such that:\n\\begin{enumerate}\n\\item $\\bar S = S \\cap \\partial S$ is a smooth manifold with boundary\n  $\\partial S$ diffeomorphic to the closed unit ball in $\\Bbb{R}^3$.\n  As coordinates on $S$ the pull backs of the natural coordinates\n  on $\\Bbb{R}^3$ are used.\n\\item $f_0$ is the vector $\\underline f$ of functions in system\n  (\\ref{EvoSyst}) written in the form (\\ref{symhypEvoSys}) at initial\n  time $t_0$.\n\\item The fields provided by $f_0$ have uniformly continuous derivatives\n  with respect to the coordinates of $S$ to all orders\\footnote{The\n    assumption about the smoothness of the data can certainly be\n    weakened from $C^\\infty{}$ to $C^n$ for sufficiently large $n$ but\n    then more technical effort would be needed in the proofs.}.\n\\item On $S$: $\\Omega>0$. On $\\partial S$: $\\Omega=0$ and $\\nabla_a\n  \\Omega{}$ is a future directed null vector.\n\\item The fields provided by $f_0$ satisfy the constraints following\n  from $\\N{}=0$ (\\ref{quaSys} and \\ref{NM}) and the gauge conditions.\n\\end{enumerate}\n\\end{Def}\nA point which deserves special notice is the existence of a\nhyperboloidal initial data set. The proof that\nthose data exist has to overcome two problems.\n\\\\\nFirstly,  the regularity of the solution on $\\partial S$ which is\nthe consistency of the data with asymptotical flatness.\nFor scalar field data with compact support regularity\nconditions are given in~\\cite{AnCXXxx,AnCA92ot}, which are sufficient\nfor the existence of a solution of the constraints near $\\partial S$.\n\\\\\nSecondly there is a problem with a possible singularity of\nequations in $\\N{}=0$ at {$1-\\frac{1}{4}\\,\\Omega^2\\phi^2=0$}.\n\\\\\nL.~Anderson and\nP.~Chrusc\\'{\\i}el are preparing a paper analyzing both\nproblems \\cite{AnCXXxx}.\n\\subsection{Theorems}\nThe ``theorems'' will be given in a form not containing every\ntechnical detail, since these technical details would make them\nlengthy and\ncan\nbe easily deduced from the theorems in~\\cite{Fr91ot} by replacing the\nYang-Mills matter with the (conformally invariant) scalar field.\n\\\\\nSince the constraints of $\\N{}=0$ propagate we have:\n\\begin{Theorem}\n\\label{physUnphysequi}\n  Any (sufficiently smooth) solution of the subsidiary system satisfying the\n  constraints on a spacelike hypersurface $\\bar S$ and\n  $1-\\frac{1}{4}\\,\\Omega^2\\phi^2>0$  defines in the\n  domain of dependence with respect to $g_{ab}$ of $\\bar S$ a\n  solution to the unphysical system. Thus $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab},\\mbox{$\\tilde{\\phi}$})$ is\n  a weakly asymptotically flat solution of the Einstein equation.\n\\end{Theorem}\nSince the evolution equations are symmetric hyperbolic a unique\nsolution of the initial value problem exists for a finite time.\n{}From the combination with theorem (\\ref{physUnphysequi}) follows:\n\\begin{corollar}\n\\label{ExUni}\n  For every regular solution of the constraints\n  on $\\bar S$ with ${1-\\frac{1}{4}\\,\\Omega^2\\phi^2\\mid_{\\bar S} >0}$ exists\n  locally a unique, weakly asymptotically\n  flat solution of the Einstein equation.\n\\end{corollar}\nFor the Minkowski space we can extent $\\bar S$ and the solution of\nthe constraints beyond $\\partial S$ to $S'$ and\nget a solution in the unphysical spacetime which extents beyond\n$i^+$. The continuous dependence of the solution of symmetric\nhyperbolic systems on the data and the form of (\\ref{NO}),\n(\\ref{NDO}) and (\\ref{No}) (see the proof\nof theorem (10.2) in~\\cite{Fr91ot}) guarantees\nthat there is a solution covering the whole domain of dependence of\n$\\bar S$. Furthermore the proof there shows that\n$\\{p\\,|\\,\\Omega(p)=0\\}$ has an isolated critical point $i^+$, where\nall future directed timelike geodesics of $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$ end, thus:\n\\begin{Theorem}\n  For a sufficient small deviation of the data from Minkowskian data the\n  solution of theorem~\\ref{ExUni} possesses a regular future null\n  infinity and a regular future timelike infinity.\n\\end{Theorem}\n\\section{The conformal equivalence of the scalar fields}\n\\label{SkalarAequiv}\nThis section shortly reviews the equivalence transformation between\nspacetime models with scalar matter under the viewpoint of solving\nhyperboloidal initial value problems. Other\naspects of this equivalence transformation, especially the generation\nof exact solutions, have been studied\nin~\\cite{AcWA93ce,Be74es,Be75bh,Kl93sf,KlK93ie,Pa91mw,XaD92eg}.\n\\subsection{Local equivalence of solutions}\n\\label{GenEquivalence}\nSpacetime models $(\\tilde M,\\tilde g_{ab},\\tilde\\phi)$\nwith scalar matter $\\tilde\\phi$ described by the action\n\\begin{equation}\n\\label{ScalarAction}\n  \\tilde S = \\int_{\\tilde M} \\left[ A(\\tilde\\phi) R - B(\\tilde\\phi)\n                    (\\tilde\\nabla_a\\tilde\\phi)\n                    (\\tilde\\nabla^a\\tilde\\phi)\n             \\right] \\> (-\\tilde g)^{\\frac{1}{2}} \\> d^4\\tilde x\n\\end{equation}\nwill be considered. Boundary terms in the action have been omitted,\n$\\mbox{$\\tilde{g}$}{}$ is the determinant of $\\mbox{$\\tilde{g}$}_{\\mu\\nu}$.\n\\\\\nBy varying the action $\\tilde S$ with respect to $\\mbox{$\\tilde{\\phi}$}{}$ and\n$\\mbox{$\\tilde{g}$}_{ab}$ the following field equations result:\n\\begin{subequations}\n\\label{allgSys}\n\\begin{eqnarray}\n\\label{allgWell}\n  B(\\mbox{$\\tilde{\\phi}$}) \\, \\tilde{\\vphantom{\\phi}\\Box} \\,\\tilde\\phi\n  + \\frac{1}{2} \\, \\frac{dB}{d\\tilde\\phi} \\,\n    \\left( \\mbox{$\\tilde{\\nabla}$}^a\\tilde\\phi \\right) \\, \\left( \\mbox{$\\tilde{\\nabla}$}_a\\tilde\\phi \\right)\n  + \\frac{1}{2} \\, \\frac{dA}{d\\tilde\\phi} \\, \\tilde R\n  \\quad & =  & \\quad 0 \\qquad \\qquad\n\\\\\n  A(\\tilde\\phi) \\, \\left( \\tilde R_{ab} - \\frac{1}{2} \\, \\tilde R\n  \\, \\tilde g_{ab} \\right)\n  + B(\\tilde \\phi) \\, \\left( \\frac{1}{2} \\,\n    \\left(\\mbox{$\\tilde{\\nabla}$}^c\\tilde \\phi\\right) \\,\n               \\left(\\mbox{$\\tilde{\\nabla}$}_c\\tilde \\phi\\right) \\mbox{$\\tilde{g}$}_{ab}\n    - \\left(\\mbox{$\\tilde{\\nabla}$}_a\\tilde \\phi\\right) \\,\n       \\left(\\mbox{$\\tilde{\\nabla}$}_b\\tilde \\phi\\right) \\right)\n\\nonumber\n\\\\\n\\label{allgGeo}\n  - \\left( \\mbox{$\\tilde{\\nabla}$}_a\\mbox{$\\tilde{\\nabla}$}_b A(\\tilde\\phi) \\right)\n   + \\left( \\mbox{$\\tilde{\\nabla}$}^c\\mbox{$\\tilde{\\nabla}$}_c A(\\tilde\\phi) \\right) \\tilde g_{ab}\n  \\quad & = & \\quad 0.\n\\end{eqnarray}\n\\end{subequations}\n$A$ and $B$ are assumed to be $C^\\infty$ functions.\nFor $ B \\neq 0$ the principal part of (\\ref{allgWell}) does not vanish\nand thus (\\ref{allgWell}) is a wave equation. For that reason I assume\n$B(\\tilde\\phi) > \\epsilon > 0$ for every $\\tilde\\phi$. (\\ref{allgGeo})\nis a second order equation for the metric if $A(\\mbox{$\\tilde{\\phi}$})\\ne{}0$.\n\\\\\nIn the spacetime region $\\tilde H := \\left\\{x \\in \\tilde{M} | \\, {\\rm\n    sign}(A(\\tilde\\Phi))>0 \\quad \\forall\n  \\tilde\\Phi \\in [\\mbox{$\\tilde{\\phi}$}_0,\\mbox{$\\tilde{\\phi}$}(x)]\\right\\}$ the trans\\-for\\-ma\\-tion\\footnote{The\n    choice of the parameter $\\mbox{$\\tilde{\\phi}$}_0$ reflects gauge freedom. Models\n    where there is no $\\mbox{$\\tilde{\\phi}$}_0$ with $A(\\mbox{$\\tilde{\\phi}$}_0) > 0$ will not be\n    considered.}\n\\begin{subequations}\n\\label{Transf}\n\\begin{eqnarray}\n\\label{allgphiTrans}\n  \\tilde{\\bar\\phi} & = & \\int_{\\tilde\\phi_0}^{\\tilde\\phi} \\frac{1}{A}\n    \\sqrt{ \\frac{3}{2} \\,\n    \\left(\\frac{dA}{d\\phi} \\right)^2 + A\\,B } \\quad\n  d\\phi\n\\\\\n\\label{allgTrans}\n  \\tilde {\\bar g}_{ab} & = & A \\, \\tilde g_{ab}\n\\end{eqnarray}\n\\end{subequations}\ngives a solution of the system (\\ref{allgSys}) with a massless\nKlein-Gordon field\n$\\widetilde{\\bar\\phi}$ as matter model corresponding to the choice\n$(A,B)=(1,1)$ and the equations\n\\begin{subequations}\n\\label{KGGl}\n\\begin{eqnarray}\n  \\widetilde{\\bar{\\vphantom{\\phi}\\Box}}\\widetilde{\\bar\\phi} & = & 0\n\\\\\n  \\widetilde{\\bar{R}}_{ab} - \\frac{1}{2} \\, \\widetilde{\\bar{R}} \\,\n  \\widetilde{\\bar{g}}_{ab}\n  & = & \\widetilde{\\bar{T}}_{ab}[{\\widetilde{\\bar{\\phi}}}]\n\\end{eqnarray}\nwith energy momentum tensor\n\\begin{equation}\n  \\widetilde{\\bar{T}}_{ab}[{\\widetilde{\\bar{\\phi}}}] =\n  (\\widetilde{\\bar\\nabla}_a\\widetilde{\\bar{\\phi}}) \\,\n  (\\widetilde{\\bar\\nabla}_b\\widetilde{\\bar{\\phi}})\n  - \\frac{1}{2} \\, (\\widetilde{\\bar\\nabla}_c\n                          \\widetilde{\\bar{\\phi}}) \\,\n                         (\\widetilde{\\bar\\nabla}^c\n                          \\widetilde{\\bar{\\phi}})\n                          \\, \\widetilde{\\bar g}_{ab}.\n\\end{equation}\n\\end{subequations}\n\\\\\n{}From the assumptions about $A$ and $B$ follows that the corresponding\nKlein-Gordon field will be unbounded approaching the\npart of the boundary of $\\tilde H$ where $A(\\mbox{$\\tilde{\\phi}$})\\rightarrow 0$.\nThe singularity in the Klein-Gordon field\nshows up at least in a singularity of the equations for $(\\tilde M,\\tilde\ng_{ab}, \\tilde \\phi)$.\n\\\\\nFor two of the scalar fields in the above class the field equations\nare very special, the already mention massless Klein-Gordon field\n$\\tilde{\\bar\\phi{}}$\n(\\ref{KGGl}) and the conformally invariant scalar field $\\mbox{$\\tilde{\\phi}$}{}$,\n$(A,B)=(1-\\frac{1}{4}\\mbox{$\\tilde{\\phi}$}^2,\\frac{3}{2})$ ($\\mbox{$\\tilde{\\phi}$}{}$ can be\nrescaled by an arbitrary factor).\n\\\\\nThe first, because the set of equations in the physical spacetime\nbecomes remarkable simple and has been analyzed intensely with\nanalytical (e.g.~\\cite{Ch91tf}) and numerical\n(e.g.~\\cite{Ch92CB}) methods for spacetimes with spherical symmetry.\n\\\\\nThe second, yielding the equations (\\ref{model}),\nbecause the matter equations are invariant under rescalings $g_{ab} =\n\\Omega^2 \\, \\mbox{$\\tilde{g}$}_{ab}$ and $\\phi = \\Omega^{-1} \\mbox{$\\tilde{\\phi}$}{}$.\n\\\\\nThe transformation between the two special cases is\n\\begin{subequations}\n\\begin{eqnarray}\n  \\tilde{\\B{\\phi}}    & = &\n    \\sqrt{6} \\, \\mbox{arctanh} \\frac{\\mbox{$\\tilde{\\phi}$}}{2}\n  \\\\\n  \\tilde{\\B{g}}_{ab}  & = & ( 1-\\frac{1}{4}\\mbox{$\\tilde{\\phi}$}^2 ) \\, \\mbox{$\\tilde{g}$}_{ab}\n\\end{eqnarray}\n\\end{subequations}\nwhich is a bijective mapping from\n$\\mbox{$\\tilde{\\phi}$} \\in ]-2,2[$ to $\\tilde{\\bar{\\phi}}\\in ]-\\infty,\\infty[$.\n\\\\\nDue to the following diagram, illustrating the above described\nrelations,  it is evident that there is a variable transformation\nregularizing\nthe\nunphysical equations for the Klein-Gordon field:\n\\begin{center}\n\\unitlength1cm\n\\begin{picture}(15,6.5)\n\\addcontentsline{lof}{figure}{{\\string\\numberline\\space{}Konforme\n    \\protect\"Aquivalenz von Skalarfeldern}}\n  %\n \n  \\put(0,0){\\makebox(2.5,2)[l]{\\shortstack[l]{unphysical\\\\spacetime}}}\n  \\put(3.5,0){\\framebox(4,2){%\n              \\shortstack[l]{conformal field $\\phi$,\\\\reg.\\ equations}}}\n  \\put(11.5,0){\\framebox(4,2){%\n              \\shortstack[l]{KG field $\\bar{\\phi}$,\\\\sing.\\ equations}}}\n  %\n \n  \\put(5.5,3.75){\\vector(0,-1){1.5}}\n    \\put(6,2.75){\\shortstack[l]{$\\phi=\\mbox{$\\tilde{\\phi}$}\/\\Omega$\\\\$g_{ab}=\\Omega^2\\mbox{$\\tilde{g}$}_{ab}$}}\n  \\put(13.5,3.75){\\vector(0,-1){1.5}}\n    \\put(11,2.75){\\shortstack[r]{$\\bar{\\phi}=\\tilde{\\bar{\\phi}}\/\\bar{\\Omega}$\\\\\n                        $\\bar{g}_{ab}=\\bar{\\Omega}^2\\tilde{\\bar{g}}_{ab}$}}\n  %\n \n  \\put(0,4){\\makebox(3,2)[l]{\\shortstack[l]{physical\\\\spacetime}}}\n  \\put(3.5,4){\\framebox(4,2){\\shortstack{conformal field $\\mbox{$\\tilde{\\phi}$}$\\\\\n    $\\mbox{$\\tilde{\\phi}$} \\in ]{-2},{2}[$}}}\n  \\put(11.5,4){\\framebox(4,2){\\shortstack{KG field\n        $\\tilde{\\bar{\\phi}}$\\\\ $\\tilde{\\bar{\\phi}}\\in\n        ]-\\infty,\\infty[$}}}\n  %\n \n  \\put(7.75,5){\\vector(1,0){3.5}}\n  \\put(8.5,5.2){\\mbox{$\\bar\\phi=f(\\mbox{$\\tilde{\\phi}$})$}}\n  \\put(8.5,4.6){\\mbox{$\\bar g_{ab}=\\omega^2\\mbox{$\\tilde{g}$}_{ab}$}}\n  %\n\\end{picture}\n\\end{center}\nBy mapping an arbitrary scalar field $\\tilde{\\bar{\\bar \\phi}}$ with\naction (\\ref{ScalarAction}) to the Klein-Gordon field $\\tilde{\\bar\n  \\phi}$ and then to the conformally invariant scalar field $\\tilde\n\\phi $ regular equations for $\\bar{\\bar \\phi}$ are obtained.\n\\subsection{The hyperboloidal initial value problem}\nSince $A(\\phi)>0$ on $\\bar S$ all scalar field models connected\nby transformation (\\ref{Transf}) to a\nhyperboloidal initial value problem with a conformally invariant\nscalar field as matter source are weakly asymptotically flat.\n\\\\\nFor a massless KG model there is a one parameter gauge freedom in the\nscalar field. If $\\tilde{\\bar{\\phi}}$ is a solution, then so is\n$\\tilde{\\bar{\\phi}}+\\tilde{\\bar{\\phi}}_0$ with\n$\\tilde{\\bar{\\phi}}_0=\\mbox{const} {}$, as the energy momentum tensor depends on\nderivatives of $\\tilde{\\bar{\\phi}}$ only. This can also be seen by\nmapping a Klein-Gordon model to a Klein-Gordon model with\n$\\tilde{\\bar{\\phi}}_0\\ne{}0$ and (\\ref{Transf}). The analogue holds for\nevery considered scalar field model.\nFor the hyperboloidal initial value problem $\\mbox{$\\tilde{\\phi}$} = \\Omega\n\\phi{}$, therefore $\\mbox{$\\tilde{\\phi}$}{}$ vanishes at \\mbox{$\\cal J$}{}, fixing the gauge in\n$\\mbox{$\\tilde{\\phi}$}{}$.\n\\\\\nIn definition (\\ref{HypInitValProbl}) $1-\\frac{1}{4} \\Omega^2\n\\phi^2 \\mid_{\\bar S} >0$ was assumed. But with the Bekenstein black\nhole~\\cite{Be75bh} a weakly asymptotically flat\nsolution is known where $A(\\mbox{$\\tilde{\\phi}$})$ vanishes on a regular\npart of the spacetime. In this case the transformation gives a\npossible extension of a massless Klein-Gordon scalar field solution beyond a\nsingularity -- the Klein-Gordon field $\\widetilde{\\bar\\phi}$ and the\nmetric $\\widetilde{\\bar g}_{ab}$ degenerate there.\n\\vspace{0.5cm}\nIt is a pleasure for me to thank Helmut Friedrich, Bernd Schmidt, and\nJ\\\"urgen Ehlers for the very helpful discussions during the grow of\nthis work which is part of my Ph.~D.\\ thesis.\n\n\\begin{appendix}\n\\section{Notation}\nThe signature of the Lorentzian metric $g_{ab}$ is $(-,+,+,+)$.\n\\\\\nWhenever possible I use abstract indices as described\nin~\\cite[chapter~2]{PeR84SA}. Small Latin letters denote abstract\nindices, underlined\nsmall Latin letters are frame indices. For the components of a tensor with\nrespect to coordinates small Greek letters are used. The frame\n$\\left(\\frac{\\partial}{\\partial x^\\mu}\\right)^a$ is constructed from\nthe coordinates $x^\\mu $, $e_{\\f{i}}{}^a$ denotes an arbitrary frame.\nIn this notation $v_a$ is a covector, $v_{\\f{i}}$ a scalar, namely\n$v_a\\, e_{\\f{i}}{}^a$.\n\\\\\n$v(f)$ is defined to be the action of the vector $v^a$ on the function\n$f$, i.e.\\ for every covariant derivative $\\nabla_a$: $t(f)=t^a\\,\\nabla_a f$.\n\\\\\nThe transformation between abstract, coordinate, and frame indices is\ndone by contracting with $e_{\\f{i}}{}^a$ and $e_{\\f{i}}{}^\\mu $. All\nindices may be raised and lowered with the metric $g_{AB}$ and the\ninverse $g^{AB}$. $g^{AC}\\,g_{CB} = \\delta^A{}_B$, $A$ and $B$ are\narbitrary indices, e.g.\\  $e_{\\f{i}a}=g_{ab}\\,e_{\\f{i}}{}^b$ and\n$e^{\\f{i}}{}_a=g^{\\f{i}\\f{j}}\\,e_{\\f{i}a}$.\n\\\\\nFor a frame $e_{\\f{i}}{}^a$ and a covariant derivative\n$\\nabla_a$ the Ricci rotation coefficients\nare defined as\n\\begin{equation*}\n  \\gamma^a{}_{\\f{i}\\f{j}} := e_{\\f{i}}{}^b \\nabla_b e_{\\f{j}}{}^a.\n\\end{equation*}\n{}From this definition follows\n\\begin{equation*}\n  e_{\\f{i}}{}^a \\, e^{\\f{j}}{}_b \\, (\\nabla_a t^b) =\n  e_{\\f{i}}(t^{\\f{j}}) + \\gamma^{\\f{j}}{}_{\\f{i}\\f{k}} \\, t^{\\f{k}}.\n\\end{equation*}\n\\\\\nWith respect to a coordinate frame $e_{\\mu}{}^a \\equiv\n\\left(\\frac{\\partial}{\\partial x^\\mu}\\right)^a$  the components\n$\\gamma^\\lambda{}_{\\mu\\nu}$ are the Christoffel symbols\n$\\Gamma^\\lambda{}_{\\mu\\nu}$.\n\\\\\nThe torsion $T^a{}_{bc}$ is defined by\n\\begin{equation*}\n  \\nabla_a\\nabla_b f - \\nabla_b\\nabla_a f = - T^c{}_{ab} \\, \\nabla_c f,\n\\end{equation*}\nthe Riemann tensor $R_{abc}{}^d$ by\n\\begin{equation*}\n  \\nabla_a\\nabla_b\\omega_c - \\nabla_b\\nabla_a\\omega_c =\n  R_{abc}{}^d \\, \\omega_d  - T^{d}{}_{ab} \\, \\nabla_d \\omega_c.\n\\end{equation*}\nContraction gives the Ricci tensor,\n\\begin{equation*}\n  R_{ab} = R_{acb}{}^c,\n\\end{equation*}\nand the Ricci scalar\n\\begin{equation*}\n  R = R_{ab}\\, g^{ab}.\n\\end{equation*}\nThe Einstein tensor is given by\n\\begin{equation*}\n  G_{ab} = R_{ab} - \\frac{1}{2} \\, R \\, g_{ab}.\n\\end{equation*}\n\\\\\nThe speed of light $c$ is set to $1$ as the gravitational constant\n$\\kappa $ in $G_{ab}=\\kappa\\, T_{ab}$.\n\\end{appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}