diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfpub" "b/data_all_eng_slimpj/shuffled/split2/finalzzfpub" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfpub" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\noindent\nThe multilinear Kakeya problem was introduced in \\cite{B}, and its\nstudy began in earnest in \\cite{BCT}, where the natural conjecture was established \nup to the endpoint.\nWorking in $\\mathbb{R}^n$, we suppose that we are given $n$ transverse families \n$\\mathcal{T}_1, \\dots, \\mathcal{T}_n$ of $1$-tubes, which means that each $T \\in \\mathcal{T}_j$ \nis a $1$-neighbourhood of a doubly-infinite line in $\\mathbb{R}^n$ with direction \n$e(T) \\in \\mathbb{S}^{n-1}$, and that the directions $e(T)$ for $T \\in \\mathcal{T}_j$ all lie within a \nsmall fixed neighbourhood (depending only on the dimension $n$) of the $j$'th standard basis vector $e_j$. \n\n\\medskip\n\\noindent\nThe question is whether for each $q \\geq 1\/(n-1)$ we have the inequality\n\\begin{equation*}\\label{MKC}\n\\int_{\\mathbb{R}^n} \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\chi_{T_1}(x) \\dots \n\\sum_{T_{n} \\in \\mathcal{T}_n} a_{T_n} \\chi_{T_n}(x) \\right)^q \\; dx\n\\leq C_{n,q} \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\dots \n\\sum_{T_{n} \\in \\mathcal{T}_n} a_{T_n}\\right)^q\n\\end{equation*}\nfor nonnegative coefficients $a_{T_j}$. In \\cite{BCT} this was proved for each $q > 1\/(n-1)$ using a \nheat-flow technique which, because of certain error terms arising, did not apply at the endpoint \n$q = 1\/(n-1)$. (For further background on this problem consult \\cite{BCT}.)\n\n\\medskip\n\\noindent\nMore recently, Guth in \\cite{G} established the endpoint case $q = 1\/(n-1)$ using completely different \ntechniques motivated in part by the polynomial method used by Dvir \\cite{Dv} to solve the finite field \nKakeya set problem, but which also relied upon a fairly heavy dose of algebraic topology, and which were \ntherefore perhaps a little intimidating to the analyst or combinatorialist. In particular, Guth used the \ntechnology of cohomology classes, cup products and the Lusternik--Schnirelmann vanishing lemma in \nestablishing his result. We believe\nthat the endpoint multilinear Kakeya theorem is of such significance and importance that a proof free of \nthese techniques should be available, and so the purpose of this paper is to provide an argument leading to\nGuth's result which does not rely upon such sophisticated algebraic topology, but whose input is instead the \nBorsuk--Ulam theorem. It is hoped therefore that this paper might lead\nto further exploitation of Guth's techniques in a more flexible\nsetting. (For some recent works applying the multilinear\nKakeya perspective in other contexts, see \\cite{B1}, \\cite{B2} and \\cite{B3}.) \n\n\\medskip\n\\noindent\nThe Borsuk--Ulam theorem, while topological in nature, nevertheless has many proofs accessible \nto the analyst -- see for example \\cite{Mat}, and also \\cite{C} for a recent such proof. (See Section \n\\ref{sec:BU} below for its statement.) The use of the Borsuk--Ulam theorem in the context of Kakeya theorems is \nby now natural, as it can be considered as a topological analogue of the elementary linear-algebraic statement that \nthere are no linear injections $T: V \\to W$ if $V$ and $W$ are finite-dimensional vector spaces \nwith dim $V > $ dim $W$; this was a key element of Dvir's solution \\cite{Dv} of the finite field \nKakeya problem. It also features explicitly in Guth's warm-up discussion to the full result of \\cite{G}.\n\n\\medskip\n\\noindent\nIn order to proceed, \nwe place matters in a more general context which does not impose conditions on the directions \nof the tubes, nor requires the level of multilinearity to equal the dimension of the underlying \neuclidean space. Thus we now suppose that \nwe are given $d$ arbitrary families of $1$-tubes $\\mathcal{T}_1, \\dots, \\mathcal{T}_d$ in $\\mathbb{R}^n$, \nwhere $d \\leq n$. For $v_1, \\dots v_d \\in \\mathbb{R}^n$ let $v_1 \\wedge \\cdots \\wedge \nv_d$ denote the unsigned (i.e. nonnegative) $d$-dimensional volume of the parallelepiped whose sides are \ngiven by the vectors $v_1, \\dots, v_d$. \n\n\n\\begin{theorem}[The Multilinear Kakeya Theorem]\\label{main}\nLet $2 \\leq d \\leq n$. Then there exists a constant $C_{d,n}$ such that if $\\mathcal{T}_1, \\dots, \n\\mathcal{T}_d$ are families of $1$-tubes in $\\mathbb{R}^n$, we have\n\\begin{equation}\n\\begin{aligned}\n\\int_{\\mathbb{R}^n} & \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\chi_{T_1}(x) \\dots \n\\sum_{T_{d} \\in \\mathcal{T}_d} a_{T_d} \\chi_{T_d}(x) \\; e(T_1) \\wedge \\dots \\wedge e(T_d) \\right)^{1\/(d-1)} \n\\; dx \\\\\n& \\leq C_{d,n} \\left(\\sum_{T_{1} \\in \\mathcal{T}_1} a_{T_1} \\dots \n\\sum_{T_{d} \\in \\mathcal{T}_d} a_{T_d}\\right)^{1\/(d-1)}.\n\\end{aligned}\n\\end{equation}\n\\end{theorem}\n\n\\medskip\n\\noindent\n(The case $d=2$ is of course trivial.) \n\n\\medskip\n\\noindent\nThe situation where the level of multilinearity is less than the ambient euclidean dimension was already \naddressed in \\cite{BCT}, where once again the result was established up to the endpoint. The incorporation \nof the factor $e(T_1) \\wedge \\dots \\wedge e(T_d)$ on the left-hand side is natural in view of the\naffine-invariant formulation of the Loomis--Whitney inequality, and was considered \nin Section 7 of \\cite{BG}, where Theorem \\ref{main} was first proved. Indeed, when $d=n$, the statement of \nTheorem \\ref{main} is affine-invariant.\\footnote{The multilinear Kakeya theorem can also be cast in the following equivalent form when $d=n$. For a unit vector \n$\\omega \\in \\mathbb{R}^n$ let $\\Pi_\\omega$ denote the hyperplane in $\\mathbb{R}^n$ which is \nperpendicular to $\\omega$ and which contains the origin. Let $\\pi_\\omega : \\mathbb{R}^n \\to \\Pi_\\omega$\nbe the orthogonal projection map. Then for nonnegative $g_j$ we have\n$$ \\int_{\\mathbb{R}^n} \\left(\\int_{\\mathbb{S}^{n-1}} \\dots \\int_{\\mathbb{S}^{n-1}}\n\\left(\\prod_{j=1}^n g_j(\\omega_j, \\pi_{\\omega_j} x)\\right)\\omega_1 \\wedge \\dots \\wedge \\omega_n \\; d\\sigma(\\omega_1) \\dots d \\sigma(\\omega_n)\\right)^{1\/(n-1)} dx $$\n$$\\leq C_{n} \\prod_{j=1}^n \\left(\\int_{\\mathbb{S}^{n-1}} \\int_{\\Pi{_{\\omega_j}}} g_j(\\omega_j, \\xi) \nd\\xi d \\sigma(\\omega_j)\\right)^{1\/(n-1)}.$$} \nA variant of Theorem \\ref{main} \nwhere lines are replaced by algebraic curves of bounded degree was also proved in \\cite{BG} (and can \nlikewise be established by replacing Guth's original argument for Theorem \\ref{algtop0} below by \nthat of the current paper). On the other hand, the results of \\cite{BCT} have a somewhat more general scope\nin so far as they apply to $1$-neighbourhoods of $k$-planes for arbitrary $k$, rather than just\n$1$-neighbourhoods of lines, i.e. tubes, as in the present discussion.\n\n\\medskip\n\\noindent\nThe principal notion that Guth employs in proving the endpoint theorem is that of the \n{\\em visibility} vis$\\,(Z)$ of a hypersurface $Z \\subseteq \\mathbb{R}^n$ -- see Section \n\\ref{sect:vis} below for the definition, which differs from Guth's in\nso far as in our treatment it (roughly)\nscales as does $(n-1)$-dimensional Hausdorff measure $\\mathcal{H}_{n-1}$ -- \nand the centrepiece of Guth's argument is the following result:\n\n\\begin{theorem}\\label{algtop0}\nGiven a nonnegative function\n$M$ defined on the lattice $\\mathcal{Q}$ of unit cubes of $\\mathbb{R}^n$,\nthere exists a non-zero polynomial $p$ such that\n$$ {\\rm{deg }} \\; p \\leq C_n \\left(\\sum_{Q \\in \\mathcal{Q}} M(Q)^n\\right)^{1\/n}$$\nand such that if we set $Z = Z_p = \\{x \\in \\mathbb{R}^n \\; : \\; p(x) =\n0\\}$, then for all $Q \\in \\mathcal{Q}$ we have \n$$ {\\rm{vis }}\\; (Z \\cap Q) \\geq C_n M(Q).$$\n\\end{theorem}\n\n\\medskip\n\\noindent\nIt is in the proof of this result that Guth uses algebraic-topological techniques, and \nthe main contribution of the present paper is to provide a proof of Theorem \\ref{algtop0} which \ndoes not use such topological machinery, but is instead a consequence\nof the Borsuk--Ulam theorem. (In fact, in our proof of Theorem\n\\ref{algtop0}, we do not use the Borsuk--Ulam theorem {\\em per se} but\ninstead an equivalent Lusternik--Schnirelmann type covering statement. See Section \\ref{sec:BU}.)\nOn the other hand, we must acknowledge that many of the arguments and\nconstructions of the present paper are inspired by those of Guth's approach.\n\n\\medskip\n\\noindent\nIn view of the connection between visibility and $(n-1)$-dimensional Hausdorff measure, and as a warm-up \nto our proof of Theorem \\ref{algtop0}, we indicate how the Borsuk--Ulam theorem can be used to \nestablish the following morally weaker variant of Theorem \\ref{algtop0}.\n\n\\begin{proposition}\\label{warmup}\nGiven a finitely supported function $M$ defined on the lattice $\\mathcal{Q}$\nof unit cubes of $\\mathbb{R}^n$ and taking nonzero \nvalues\nin $[1, \\infty)$, there exists a\nnon-zero polynomial $p$ such that \n$$ {\\rm{deg }} \\; p \\leq C_n \\left(\\sum_{Q\\in\\mathcal{Q}} M(Q)^n\\right)^{1\/n}$$\nand such that for all $Q\\in\\mathcal{Q}$ \n$$ \\mathcal{H}_{n-1} \\; (Z \\cap Q) \\geq C_n M(Q).$$\n\\end{proposition}\n\n\\begin{proof} Break up each $Q$ into $\\sim M(Q)^n$ congruent subcubes $S$; note that \naltogether we have $\\sim \\sum_Q M(Q)^n$ small cubes $S$ of various sizes. Consider the map \n$$ F: p \\mapsto \\left\\{\\int_{\\{p > 0\\} \\cap S} 1 -\\int_{\\{p < 0\\} \\cap S} 1 \\right\\}_{S}$$\ndefined on the vector space $\\mathcal{P}_k$ of polynomials of degree at most $k$ in $n$ real\nvariables, which has dimension $\\sim k^n$. Clearly $F$ is continuous, homogeneous of degree \n$0$ and odd.\n\n\\medskip\n\\noindent\nSo we can think of $F$ as \n$$F : \\mathbb{S}^N\\rightarrow \\mathbb{R}^J $$\nwhere $N \\sim k^n$ and $J \\sim \\sum_Q M(Q)^n$.\n\n\n\\medskip\n\\noindent\nSo provided $N \\geq J$ -- which we can arrange if $k \\sim \\left(\\sum_Q\nM(Q)^n\\right)^{1\/n}$ -- the Borsuk--Ulam theorem tells us that \n$F$ vanishes at some $p$. This means that the zero set $Z$ of $p$ exactly bisects each $S$. \n\n\\medskip\n\\noindent\nNow if $S$ is a subcube of $Q$, $S$ will have volume $\\sim M(Q)^{-n}$ and\ndiameter $\\sim M(Q)^{-1}$ and hence any bisecting surface will meet\nit in a set of $(n-1)$-dimensional measure $\\gtrsim\nM(Q)^{-(n-1)}$. This will be true for each of the $M(Q)^n$ disjoint $S$'s whose union\nis $Q$, so $Z$ will meet $Q$ in a set of $(n-1)$-dimensional measure $\\gtrsim M(Q)^n \\times\nM(Q)^{-(n-1)} = M(Q)$, as was needed.\n\\end{proof}\n\n\\medskip\n\\noindent\nIn the proof we used the ``geometrically obvious'' fact that a\nhypersurface bisecting the unit cube must have large surface area\ninside the cube. For a discussion of this in the context of the unit\nball, see Lemma \\ref{bisect} in the Appendix. Note that in the statements of both Theorem \\ref{algtop0} \nand Proposition \\ref{warmup}, a polynomial has the desired properties if and only if\nany non-zero scalar multiple of it does; for this reason we may choose to search for a suitable \npolynomial within the unit sphere of the class of polynomials of a given degree. \n\n\\medskip\n\\noindent\nProposition \\ref{warmup} is morally weaker than Theorem \\ref{algtop0} because not only does it place stronger \nconditions on $M$, but more importantly, in many situations of interest, we have \n${\\rm{vis }}\\; (Z \\cap Q) \\leq C_n \\mathcal{H}_{n-1} \\; (Z \\cap Q)$ -- see \\eqref{geomxx} below.\n\n\\medskip\n\\noindent\nOn an informal level, the fundamental difference between the proof of Theorem \\ref{algtop0} and that of Proposition \n\\ref{warmup} is that, roughly speaking, we no longer chop each cube $Q$ into $\\sim M(Q)^n$ congruent \n{\\em subcubes}, but we instead select, for each $Q$, an {\\em ellipsoid} \n$E(Q)$ of volume $\\sim M(Q)^{-n}$, so that $\\sim M(Q)^n$ translates of $E(Q)$ essentially tessellate $Q$. \nHowever the {\\em shape} and {\\em orientation} of the ellipsoid $E(Q_0)$ will depend not only on the \nvalue of $M(Q_0)$ but on the whole ensemble $\\{M(Q)\\}_Q$, and is in effect an output of the \nBorsuk--Ulam theorem at the same time as it produces the desired \npolynomial. At the risk of over-simplifying matters, \nwe now give an informal example which illustrates why, if we want the broad thrust of the proof of Proposition \\ref{warmup}\nto work in the context of Theorem \\ref{algtop0}, the shape of the ellipsoid selected {\\em must} depend on the totality of \nthe function $M(Q)$. This example may be safely ignored on a first reading of the paper.\n\n\\medskip\n\\noindent\n{\\bf Informal example.} Let $n=2$ and consider the function $M(Q)$ which is supported on a row of \n$N$ unit cubes centred at $(k-1\/2, 1\/2)$ for $1 \\leq k \\leq N$, and takes\nthe value $N^{1\/2}$ on each of these cubes. \nThen $\\left(\\sum_Q M(Q)^2\\right)^{1\/2} = N$. Consider the polynomial \n$$p(x) = (x_1 - 1) \\dots (x_1 - N)\\cdot(x_2 - 1\/2N)(x_2 - 3\/2N) \\dots (x_2 - (2N-1)\/2N)$$\nwhich has degree $2N$, and let $Z$ denote its zero set. For a subset\n$Z' \\subseteq Z$ of $Z$, and for each $Q$\\footnote{In this example we assume that the right-hand edge of a rectangle\nbelongs to the rectangle while the left-hand edge does not.}\nin the support of $M$, consider the projections counted with \nmultiplicities of $Z' \\cap Q$, in the directions of the two standard basis vectors $e_1$ and $e_2$; let their\ntotal lengths be $a_1(Z' \\cap Q)$ and $a_2(Z' \\cap Q)$ respectively, and let \n$$W(Z' \\cap Q) := \\{a_1(Z' \\cap Q)a_2(Z' \\cap Q)\\}^{1\/2}$$ \nbe their geometric mean. Now it transpires that the quantity $W(Z' \\cap Q)$ is closely related to \nvis$(Z' \\cap Q)$ -- see Section \\ref{sect:vis} below -- and we shall\npretend (for the rest of this example) that $W$ \nreally is the visibility. Note that if $Z_1$ and $Z_2$ are disjoint subsets of $Z \\cap Q$ we \nhave\\footnote{The corresponding property for visibility fails.}\n$$W(Z_1 \\cup Z_2) \\geq W(Z_1) + W(Z_2).$$ \nNow $a_1(Z \\cap Q) = 1$ and $a_1(Z \\cap Q) = N$ so that $W(Z \\cap Q) \n= N^{1\/2}$. We consider \nwhether it is possible to break up each $Q$ into rectangles $R_j$ of area $1\/N$\nso that if $p$ bisects each rectangle then we can deduce that\n$N^{1\/2} = W(Z \\cap Q)$ by using\n$W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j)$. \n\n\\smallskip\n\\noindent\nFirstly, we could break up $Q$ into subcubes $R_j$ of side \n$N^{-1\/2}$. Now for all $R_j$ except those which meet the right-hand edge of $Q$ we shall have \n$W(Z \\cap R_j) = 0$, while for those which do meet the right hand edge of\n$Q$ we have $W(Z \\cap R_j) = N^{-1\/4}$, which only gives\n$W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j) = N^{1\/4}$; this is not adequate.\n\n\\smallskip\n\\noindent\nNext, we could try breaking up $Q$ into vertical rectangles $R_j$ of sides $1\/N \\times 1$. Only the \nrectangle $R_0$ meeting the right-hand edge of $Q$ will have a non-zero value of $W(Z \\cap R_j)$, \nand $W(Z \\cap R_0) = 1$, giving $W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j) = 1$, which is even worse.\n\n\\smallskip\n\\noindent\nFinally, we could try breaking up $Q$ into horizontal rectangles $R_j$ of sides $1 \\times 1\/N$. In this case\neach $W(Z \\cap R_j) = N^{-1\/2}$, resulting in the desired $W(Z \\cap Q) \\geq \\sum_j W(Z \\cap R_j) \\geq N^{1\/2}$.\n\n\\smallskip\n\\noindent\nSo only the third decomposition into horizontal rectangles is compatible with our needs. Once we have accepted \nthat the polynomial $p$ above is more or less\n``canonical'' for this $M$, we are essentially forced to break \nup each $Q$ into horizontal rectangles of sides $1 \\times 1\/N$ in order for our strategy to be successful. \nCrucially, observe that this \ndecomposition reflects the global shape of the function $M$: if the support of $M$ had been along the \n$x_2$-axis we would have had to instead decompose each $Q$ into\nvertical rectangles. The decomposition must therefore be aligned with the ``global\nprofile'' of $M$. \\hfill \\qedsymbol\n\n\\bigskip\n\\noindent\nTo simplify the constructions in the proof we actually stop just short of fully\ndeveloping the moral outline given above. In fact we do not spend any time\nconstructing ellipsoids at the scale $\\sim M(Q)^{-n}$ which would be needed\nto really nail down the zero set of the polynomial we get from\nthe Borsuk--Ulam theorem. Instead we let ourselves be satisfied with\nfinding a \\emph{good} polynomial $p$ with zero set $Z_p$ which satisfies\na given lower bound on the visibility\n$\\operatorname{vis}\\,(Z_p\\cap Q)$ for all cubes $Q$.\nThis we get by constructing ellipsoids for all \\emph{bad}\npolynomials,\nwhich are those polynomials whose zero sets\nhave visibility less than desired on some cube.\nUsing these ellipsoids we show that the bad polynomials cannot cover\nthe unit sphere in the space of polynomials and in this way we see that\nthere must be a good polynomial, which gives the lower bounds mentioned\nabove.\nHerein lies the reason for using the covering statement instead of the\nBorsuk--Ulam theorem itself.\nThis very informal outline is developed more fully in\nSection~\\ref{sec:outline}.\n\n\\medskip\n\\noindent\nFor completeness, we also indicate in the following sections how Theorem \\ref{main} \nfollows from Theorem \\ref{algtop0}, so that we give what is in essence a fully self-contained \nproof of Theorem \\ref{main} (subject to the appeal to the Borsuk--Ulam theorem).\nThroughout, $C$ and $c$ will denote generic constants which depend only \non the dimension $n$ and the degree of multilinearity $d \\leq n$; $P \\lesssim Q$ and $P \\gtrsim Q$\nmean $P \\leq C Q$ and $P \\geq C Q$ respectively, and $P \\sim Q$ means both $P \\lesssim Q$ and $P \\gtrsim Q$.\n\n\\medskip\n\\noindent\n{\\em {\\bf {\\em Acknowledgements:}}} Both authors would like to thank Jon Bennett and Jim Wright for many illuminating \nconversations on the topics of this paper, and helpful remarks on early drafts of it. The first author would like to thank Larry Guth \nfor sharing his insights into the philosophy behind the endpoint multilinear Kakeya theorem in Hyderabad in August 2010.\n\n\\section{A Preliminary reduction}\n\\medskip\n\\noindent\nRecall that we have collections $ \\mathcal{T}_j$, $1 \\leq j \\leq d$,\nof $1$-tubes $T$ in $\\mathbb{R}^n$ with directions $e(T) \\in \\mathbb{S}^{n-1}$. \nLet $\\mathcal{Q}$ denote the lattice of unit cubes in $\\mathbb{R}^n$.\n\n\n\\begin{proposition}\\label{MK}\nIn order to prove Theorem \\ref{main}, it suffices to establish the following assertion:\nfor every finitely supported nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$ satisfying\n$\\sum_Q M(Q)^n = 1$, there exist nonnegative functions \n$S_j : \\mathcal{Q} \\times \\mathcal{T}_j \\to \\mathbb{R}$ such that for all\n$T_j \\in \\mathcal{T}_j$ with $T_j \\cap Q \\neq \\emptyset$,\n\\begin{equation}\\label{need1}\ne(T_1)\\wedge \\dots \\wedge e(T_d) \\, M(Q)^n \\leq C S_1(Q,T_1) \\dots S_d(Q,T_d)\n\\end{equation}\nand, for all $j$ and all $T_j \\in \\mathcal{T}_j$ \n\\begin{equation}\\label{need2}\n\\sum_{Q \\,: \\, T_j \\cap Q \\neq \\emptyset} S_j(Q, T_j) \\leq C.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFirstly, if we can find $S_j$ as in the statement of the proposition, homogeneity dictates that\nfor every finitely supported nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$\nthere exist nonnegative functions \n$S_j : \\mathcal{Q} \\times \\mathcal{T}_j \\to \\mathbb{R}$ such that for all\n$T_j \\in \\mathcal{T}_j$ with $T_j \\cap Q \\neq \\emptyset$,\n\\begin{equation}\\label{tyu}\ne(T_1)\\wedge \\dots \\wedge e(T_d) \\, M(Q)^n \\leq C S_1(Q,T_1) \\dots S_d(Q,T_d) \\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/n}\n\\end{equation}\nand, for all $j$ and all $T_j \\in \\mathcal{T}_j$ \n\\begin{equation}\\label{uyt}\n\\sum_{Q \\,: \\, T_j \\cap Q \\neq \\emptyset} S_j(Q, T_j) \\leq C \\left( \\sum_Q M(Q)^n \\right)^{1\/n}.\n\\end{equation}\n\n\\medskip\n\\noindent\nSecondly, we note that by the $l^1$ nature of the right hand side in\nTheorem \\ref{main}, we may\nassume that the sets $\\mathcal{T}_j$ are finite and that all the coefficients $a_{T_j}$ are equal to $1$.\n\n\\medskip\n\\noindent\nFor a unit cube $Q$ let \n$$F(Q) = \\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \ne(T_1)\\wedge \\dots \\wedge e(T_d).$$\nIt then suffices to prove\n$$ \\sum_Q F(Q)^{1\/(d-1)} \\leq C \\left(\\# \\mathcal{T}_1 \\dots \\# \\mathcal{T}_d \\right)^{1\/(d-1)}.$$\nLet $M(Q)^n = F(Q)^{1\/(d-1)} = F(Q)^{(1\/(d-1) - 1\/d)d}$, so that \n\\begin{eqnarray*}\n\\begin{aligned}\n&\\sum_Q F(Q)^{1\/(d-1)}\n= \\sum_Q F(Q)^{1\/d} M(Q)^{n\/d} \\\\\n= & \\sum_Q \\left(\\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \ne(T_1)\\wedge \\dots \\wedge e(T_d)\\right)^{1\/d}M(Q)^{n\/d} \\\\\n= &\\sum_Q \\left(\\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \ne(T_1)\\wedge \\dots \\wedge e(T_d) \\; M(Q)^n \\right)^{1\/d} \\\\\n\\leq C & \\sum_Q \\left(\\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \nS_1(Q,T_1) \\dots S_d(Q,T_d) \\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/n} \\right)^{1\/d} \\\\\n= C & \\sum_Q \\left(\\prod_{j=1}^d \\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \nS_j(Q,T_j)\\right)^{1\/d}\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn} \\\\\n\\leq C & \\prod_{j=1}^d \\left(\\sum_Q \\sum_{T_j \\in \\mathcal{T}_j \\mbox{ with }T_j \\cap Q \\neq \\emptyset} \nS_j(Q,T_j)\\right)^{1\/d}\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn} \\\\\n= C & \\prod_{j=1}^d \\left(\\sum_{T_j \\in \\mathcal{T}_j} \\sum_{Q \\mbox{ with }T_j \\cap Q \\neq \\emptyset}\nS_j(Q,T_j)\\right)^{1\/d}\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn} \\\\\n\\leq C & \\prod_{j=1}^d \\left( \\# \\mathcal{T}_j\\right)^{1\/d} \\left( \\sum_Q M(Q)^n \\right)^{1\/n}\n\\left(\\sum_Q M(Q)^n\\right)^{(n-d)\/dn}\\\\\n= C & \\prod_{j=1}^d \\left( \\# \\mathcal{T}_j\\right)^{1\/d} \\left( \\sum_Q F(Q)^{1\/(d-1)} \\right)^{1\/d}\n\\end{aligned}\n\\end{eqnarray*}\nwhere the inequalities follow from \\eqref{tyu}, H\\\"older's inequality and \\eqref{uyt} respectively.\nRearranging, we obtain\n$$ \\left(\\sum_Q F(Q)^{1\/(d-1)}\\right)^{(d-1)\/d} \n\\leq C \\prod_{j=1}^d \\left( \\# \\mathcal{T}_j\\right)^{1\/d},$$\nfrom which the result follows.\n\\end{proof}\n\n\\medskip\n\\noindent\nInterestingly, the line of argument here can be reversed in certain\ncircumstances: assuming that the special case of the multilinear \nKakeya theorem for transverse families of tubes $\\mathcal{T}_j$\nholds, it follows that for all $M$ one can \nfind $S_j$ satisfying \\eqref{tyu} and \n\\eqref{uyt}. See \\cite{CV} for more details.\n\n\\section{Directional surface area and visibility}\\label{sect:vis}\n\n\\medskip\n\\noindent\nWe follow Guth \\cite{G} and Bourgain--Guth \\cite{BG} in defining the functions $S_j$ and \nestablishing their desired properties \\eqref{need1} and \\eqref{need2}. In order to do this \nsome geometric notions are required. We first recall the notion of directional surface area \n(termed ``directed volume'' by Guth) of a hypersurface $Z \\subseteq \\mathbb{R}^n$ in \nthe direction of a unit vector $e$. If the element of surface area of $Z$ is denoted by \n$dS = d \\mathcal{H}_{n-1}|_S$, and $e$ is a unit vector, the element of the component of \nsurface area of $Z$ perpendicular to $e$ is $ |e \\cdot n(x)| dS(x)$ where $n(x)$ is the \nunit normal at $x$ (which is assumed to make sense for $\\mathcal{H}_{n-1}$--almost every \n$x \\in Z$).\nThus the {\\bf directional surface area of $Z$ in the direction $e \\in \\mathbb{S}^{n-1}$} is defined\nas\n$$ \\mbox{ surf}_e(Z) = \\int_Z |e \\cdot n(x)|\\; dS(x).$$\nIf $Z$ is given by the graph of a function $\\Gamma : \\Omega \\subseteq \\mathbb{R}^{n-1} \\to \n\\mathbb{R}$ above the hyperplane $x_n = 0$, then its directional surface area in the direction \n$e_n$ is simply the $(n-1)$-dimensional area of $\\Omega$. If $Z$ is given by disjoint graphs of functions\nabove the hyperplane $x_n = 0$ then its directional surface area in the direction $e_n$ is just \n$\\int_{\\mathbb{R}^{n-1}} J(y) \\; dy$ where $J(y)$ is the number of times the line through $y$ parallel \nto $e_n$ passes through $Z$. These considerations lead immediately to Guth's ``cylinder estimate'':\n\n\\begin{lemma}[Guth's cylinder estimate]\\label{cylinder}\nIf $T$ is a $1$-tube in $\\mathbb{R}^n$ and $Z = \\{x \\; : \\; p(x) = 0 \\}$ is the zero hypersurface \nof a non-zero polynomial $p$ of degree at most $k$, then \n$$ {\\rm{ surf}}_{e(T)}(Z \\cap T) \\leq C k.$$\n\\end{lemma}\n\n\n\\medskip\n\\noindent\nSecondly, we associate a fundamental centrally-symmetric convex body $K(Z)$ to a hypersurface $Z$. Indeed, with $\\mathbb{B}$ \ndenoting the unit ball of $\\mathbb{R}^n$, defin\n\\begin{equation}\\label{Kdefn}\nK(Z) := \\{ u \\in \\mathbb{B} \\; : \\; \\mbox{ surf}_{\\widehat{u}}(Z) \\leq 1\/|u|\\}.\n\\end{equation}\nHere $\\widehat{u}$ is the unit vector in the direction of $u$. (Notice that if $Z$ is such that \nsurf$_e(Z) \\geq 1$ for all unit vectors $e$, then the requirement that $u$ lie in $\\mathbb{B}$ is \nsuperfluous.) It is clear that $K(Z)$ is symmetric. To see that it is in fact convex, note that $u$ satisfies \n$\\mbox{ surf}_{\\widehat{u}}(Z) \\leq 1\/|u|$\nif and only if $\\int_Z | u \\cdot n | \\; dS \\leq 1$; this condition is clearly retained under \nconvex combinations of $u$'s. We then define \\footnote{Guth's definition of visibility is \nthe $n$'th power of the one given here, but we find the current definition more natural for \nthree reasons: firstly it allows us to emphasise the ``$L^n$''-aspect of the statement of \nTheorem \\ref{algtop0} which, at least when $n=d$, is no coincidence, and is a reflection of \nthe fact that the optimal $L^p$ estimate for the linear Kakeya problem in $\\mathbb{R}^n$ \nis conjectured to be at $p=n$; secondly it scales roughly as does $(n-1)$-dimensional Hausdorff \nmeasure which permits the comparison with Proposition \\ref{warmup}; and thirdly, in the theory of \nfinite-dimensional Banach spaces, if $K$ is a convex body in isotropic position,\nthe quantity $\\left(\\mbox{vol } K\\right)^{-1\/n}$ arises naturally as its isotropic constant.} the \n{\\bf visibility} of $Z$ as\n$$ \\mbox{ vis}(Z) := \\left(\\mbox{vol } K(Z)\\right)^{-1\/n}.$$\nNote that since $K(Z) \\subseteq \\mathbb{B}$ we always have $\\mbox{ vis}(Z) \\geq C$.\n\n\\medskip\n\\noindent\nThe next lemma allows us to relate visibilty to geometric means of directional surface areas.\n\n\\begin{lemma}\\label{linalg}\nSuppose that for all unit vectors $e \\in \\mathbb{R}^n$ we have $ 1 \\lesssim {\\rm{ surf}}_e (Z) \\lesssim D$.\nIf $v_1, \\dots , v_d,$ $( 1 \\leq d\\leq n )$ are unit vectors, then\n$$ \\left(v_1 \\wedge \\dots \\wedge v_d\\right)^{1\/n} {\\rm vis }(Z) \n\\leq C D^{(n-d)\/n} \\left({\\rm surf }_{v_1}(Z) \\dots \\, {\\rm surf }_{v_d}(Z)\\right)^{1\/n}.$$ \n\\end{lemma}\n\n\\begin{proof}\nWe may assume that $\\{v_1, \\dots , v_d\\}$ is linearly independent and we extend it to a \nbasis $\\{v_1, \\dots, v_n\\}$ where $v_{d+1}, \\dots , v_n$ are mutually orthogonal unit vectors \nwhich are also orthogonal to the span of $\\{v_1, \\dots, v_d\\}$. \n\n\\medskip\n\\noindent\nSince surf$_e(Z) \\gtrsim 1$ for all $e$, we have that \n$\\pm c v_j\/\\operatorname{surf}_{v_j}(Z) \\in K(Z)$ for all $j$, so that by convexity of $K(Z)$\n\\begin{eqnarray*}\n\\begin{aligned}\n\\mbox{vol }K(Z) &\\geq C \\; v_1 \\wedge \\dots \\wedge v_n \\; \\prod_{j=1}^n \\mbox{surf }_{v_j}(Z)^{-1} \\\\\n&= C \\; v_1 \\wedge \\dots \\wedge v_d \\; \\prod_{j=1}^d \\mbox{surf }_{v_j}(Z)^{-1} \n\\prod_{j=d+1}^n \\mbox{surf }_{v_j}(Z)^{-1} \\\\\n&\\geq C D^{-(n-d)} \\; v_1 \\wedge \\dots \\wedge v_d \\; \\prod_{j=1}^d \\mbox{surf }_{v_j}(Z)^{-1},\n\\end{aligned}\n\\end{eqnarray*}\nfrom which the result follows.\n\\end{proof}\n\n\n\\medskip\n\\noindent\nIt is not hard to show that under the assumption that surf$_e(Z) \\gtrsim 1$ for all $e$,\n$$ \\mbox{ vis}(Z) \\sim \\prod_{j=1}^n \\mbox{ surf}_{e_j}(Z)^{1\/n}$$\nwhere $e_1, \\dots, e_n$ are the principal directions of the John\nellipsoid associated to $K(Z)$ (i.e. the ellipsoid of maximal volume contained in $K(Z)$ --\nsee \\cite{John}) and hence \n\\begin{equation}\\label{geomxx}\n\\mbox{vis}(Z) \\sim \\inf_{\\{f_j\\} \\mbox{ approx orthonormal }} \n\\prod_{j=1}^n \\mbox{ surf}_{f_j}(Z)^{1\/n}\n\\end{equation}\nwhere we say that the unit vectors $f_1, \\dots , \nf_n$ are ``approximately orthonormal'' if their wedge product\nsatisfies $f_1 \\wedge \\dots \\wedge f_n \\geq c_n$ for a suitable\ndimensional constant $c_n$. By the arithmetic-geometric mean\ninequality the right-hand side of\n\\eqref{geomxx} is in turn dominated by $\\mathcal{H}_{n-1}(Z)$. This \nshows in particular that Theorem \\ref{algtop0} is morally stronger than\nProposition \\ref{warmup}. \n\n\\medskip\n\\noindent\nThe John ellipsoid $E$ of a symmetric convex body $K$ satisfies $E \\subseteq K \\subseteq\nn^{1\/2}E$, and combining the latter inclusion with Lemma \\ref{cylinder} we obtain:\n\n\\begin{lemma}\\label{three}\nLet $p$ be a non-zero polynomial such that for some unit vector $e$,\n$\\operatorname{surf}_{e}(Z_{p}\\cap Q) \\lesssim 1.$ Then \n$$\\operatorname{vis}(Z_{p} \\cap Q)^{n\/(n-1)} \\leq C \\, {\\rm { deg }}\\; p.$$\n\n\n\\end{lemma}\n\\begin{proof}\n Let $E$ be the John ellipsoid associated to $K(Z_{p}\\cap Q)$ and let\n $l_1\\geq l_2\\geq \\dots \\geq l_n$ be the lengths of the principal\n axes of $E$. Let $A = \\operatorname{vis}(Z_{p} \\cap Q)$. By\n hypothesis and the fact that $K(Z_{p}\\cap Q)\\subseteq n^{1\/2} E$, we have $l_1\\gtrsim 1$.\n Moreover we have $(l_1\\dots l_n)^{-1\/n}\\sim A$, so $l_2\\dots l_n \\lesssim A^{-n}$\n and therefore $l_n\\lesssim A^{-n\/(n-1)}$. So if $e_n$ is the direction with which $l_n$ is associated,\n we have that $\\operatorname{surf}_{e_n} (Z_{p}\\cap Q)\\gtrsim A^{n\/(n-1)}$.\n The cylinder estimate now gives $\\operatorname{deg}p \\gtrsim A^{n\/(n-1)}$. \n\\end{proof}\n\n\n\n\\medskip\n\\noindent\nIn order to deal with a continuity issue later in the argument (in Lemma \\ref{convergence} of \nSection \\ref{Antipodes}), we need (as does Guth) to define variants of \nthe directional surface area and visibility which are continuous functionals of \n$Z = Z_p$ when the polynomial $p$ is allowed to vary. In view of the fact that the class\nof polynomials with the desired properties for Theorem \\ref{algtop0} is invariant under \nmultiplication by non-zero scalars, it is natural to consider the unit sphere of the class\n$\\mathcal{P}_k$ of polynomials of degree at most $k$ in $n$ real variables. \nIndeed, $\\mathcal{P}_k$ is a vector space of dimension $\\sim k^n$, and so its unit sphere \n$\\mathcal{P}_k^\\ast$ is homeomorphic to $\\mathbb{S}^N$ where $N = N(k) \\sim k^n$. So with \n$k$ fixed, we allow $p$ to vary within $\\mathcal{P}_k^\\ast$.\\footnote{From now on we shall use \nthe notations $\\mathcal{P}_k^\\ast$ and $\\mathbb{S}^N$ where $N= N(k)$ interchangeably, the former \nwhen we are thinking of individual polynomials, the latter when continuity and topological \nconsiderations are foremost.} \nThe continuity property needed\nis most simply achieved by replacing $\\mbox{surf}_e(Z)$ for surfaces of the form\n$Z=Z_p\\cap U$ (where $p\\in \\mathbb{S}^N$ and $U$ is open in $\\mathbb{R}^n$) by\n$\\operatorname{surf}_{e,\\varepsilon}(Z)$ which we define as the average of\n$\\mbox{surf}_e(Z')$ with $Z'=Z_{p'}\\cap U$ over $p'$\nin a ball of radius $\\varepsilon$ centred at $p$ in $\\mathbb{S}^N$.\nFrom this we define $K_\\varepsilon(Z)$ and\n$\\operatorname{vis}_\\varepsilon(Z)$ in analogy to $K(Z)$ and $\\operatorname{vis}(Z)$.\nIn the argument we will have to choose $\\varepsilon$ sufficiently small so that these\nentities behave in certain ways similarly to the unmollified versions.\n\n\\medskip\n\\noindent\nIt is a routine matter to verify that $K_\\varepsilon(Z)$ is convex and that the\nthree lemmas of this section hold with these mollified variants. To be precise, \nfixing $k$ and the associated definitions of $\\operatorname{surf}_{e,\\varepsilon}(Z)$,\n$K_\\varepsilon(Z)$ and $\\operatorname{vis}_\\varepsilon(Z)$ as above\nfor $\\mathcal{P}_k^\\ast$, we have, (with implicit constants independent of $\\varepsilon > 0$):\n\n\\begin{lemma}\\label{cylinder-moll} \nIf $T$ is a $1$-tube in $\\mathbb{R}^n$ and $Z = \\{x \\; : \\; p(x) = 0 \\}$ is the zero hypersurface \nof a polynomial $ p \\in \\mathcal{P}_k^\\ast$, then for all $\\varepsilon>0$,\n$$ {\\rm{ surf}}_{e(T), \\varepsilon}(Z \\cap T) \\leq C k.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{linalg-moll}\nSuppose that $p \\in \\mathcal{P}_k^\\ast$ and that $Z=Z_p\\cap U$ as above. \nAlso suppose that for some $\\varepsilon>0$ and\nall unit vectors $e \\in \\mathbb{R}^n$ we have \n$ 1 \\lesssim {\\rm{ surf}}_{e, \\varepsilon} (Z) \\lesssim D$.\nIf $v_1, \\dots , v_d,$ $( 1 \\leq d\\leq n )$ are unit vectors, then,\n$$ \\left(v_1 \\wedge \\dots \\wedge v_d\\right)^{1\/n} {\\rm vis_{\\varepsilon} }(Z) \n\\leq C D^{(n-d)\/n} \\left({\\rm surf }_{v_1, \\varepsilon}(Z) \\dots \\, {\\rm surf }_{v_d, \\varepsilon}(Z)\\right)^{1\/n}.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{three-moll}\nSuppose $p \\in \\mathcal{P}_k^\\ast$ is such that for some $\\varepsilon >0$ and some unit vector $e$, \n$\\operatorname{surf}_{e, \\varepsilon}(Z_{p}\\cap Q) \\lesssim 1.$ Then \n$$\\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q)^{n\/(n-1)}\\leq Ck.$$\n\\end{lemma}\n\n\\medskip\n\\noindent\nThe reader may wish to proceed with the unmollified variants in mind \non a first reading.\n\n\\section{Application of the main result to multilinear Kakeya}\n\\noindent\nThe version of Theorem \\ref{algtop0} that we will actually need is:\n\n\\begin{theorem}\\label{algtop}\nGiven a nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$,\nthere exists a nonnegative integer $k$, a polynomial $p \\in \\mathcal{P}_k^\\ast$ and an $\\varepsilon>0$ \nsuch that \n$$ k \\leq C \\left(\\sum_Q M(Q)^n\\right)^{1\/n}$$\nand such that for all $Q \\in \\mathcal{Q}$ \n$$ \\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\geq C M(Q).$$\n\\end{theorem}\n\\noindent\n(Note that since it is always the case that $\\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\gtrsim 1$, the \nlatter condition has content only for those $Q$ with $M(Q) \\gtrsim 1$.)\n\n\\medskip\n\\noindent\nIn this section we show how this result implies the conditions of Proposition \\ref{MK}, that is, \ngiven a finitely supported nonnegative function $M : \\mathcal{Q} \\to \\mathbb{R}$ satisfying\n$\\sum_Q M(Q)^n = 1$, there exist nonnegative functions \n$S_j : \\mathcal{Q} \\times \\mathcal{T}_j \\to \\mathbb{R}$ such that \\eqref{need1} and \\eqref{need2} hold.\n\n\n\n\\medskip\n\\noindent\nGiven such a finitely supported nonnegative function $M(Q)$ with $\\sum_Q M(Q)^n = 1$, we define $M_0(Q) = \\lambda M(Q)$ for \nsome $\\lambda \\gg 1$ which is required to satisfy $\\lambda \\geq M(Q)^{-n}$ for all $Q$ in the support of $M$. \nApply Theorem \\ref{algtop} with data $M_0$ to obtain a $k$, a $p \\in \\mathcal{P}_k^\\ast$ and an \n$\\varepsilon > 0$ such that\n\\begin{equation}\n \\label{deg} k \\leq C \\lambda\n\\end{equation}\nand\n\\begin{equation}\n \\label{wer}\n \\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q) \\geq C \\lambda M(Q).\n\\end{equation}\n\n\n\\medskip\n\\noindent\nUsing \\eqref{deg}, \\eqref{wer} and our requirement on $\\lambda$, we have\n\\begin{eqnarray*}\n\\begin{aligned}\nk &\\leq C \\lambda = C \\lambda^{n\/(n-1)} \\lambda^{-1\/(n-1)} \\\\\n&\\leq C \\left(\\frac{\\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q)}{M(Q)}\\right)^{n\/(n-1)}\n\\left(M(Q)^{-n}\\right)^{-1\/(n-1)}\\\\\n&= C \\operatorname{vis}_\\varepsilon(Z_{p} \\cap Q)^{n\/(n-1)}.\n\\end{aligned}\n\\end{eqnarray*}\n\n\\medskip\n\\noindent\nUsing Lemma~\\ref{three-moll} we deduce that for all cubes $Q$ in the support of $M$ and all unit vectors $e$ we have \n$\\operatorname{surf}_{e,\\varepsilon}(Z_{p}\\cap Q) \\gtrsim 1$.\\footnote{Another way of achieving this is to \nmultiply the polynomial which Theorem~\\ref{algtop}\nproduces with a polynomial whose zero set consists of\nhyperplanes parallel to the coordinate hyperplanes which pass through the cubes in the\nsupport of $M$. This has an insignificant effect on the degree of the polynomial provided $\\lambda$ is large enough.\nHowever, some care must be taken when considering how this augmentation interacts with the mollification.} This in turn will permit us to apply Lemma~\\ref{linalg-moll} (relating visibility to geometric means of directional surface areas) below.\n\n\n\\medskip\n\\noindent\nWe turn to the verification of \\eqref{need2}. By Lemma~\\ref{cylinder-moll} and \\eqref{deg} we have, for all $e \\in \\mathbb{S}^{n-1}$,\n\\begin{equation}\\label{awq}\n{\\rm surf }_{e,\\varepsilon}(Z_p \\cap Q) \\leq C \\lambda\n\\end{equation} \nand moreover\n\\begin{equation}\\label{qwa}\n\\sum_{Q \\, : \\, Q \\cap T_j \\neq \\emptyset} \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap Q)\n\\leq C \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap \\tilde{T}_j) \n\\leq C k \\leq C \\lambda , \n\\end{equation}\n(where $\\tilde{T}$ denotes the expansion of a tube $T$ about its axis by a dimensional factor).\n\n\\medskip\n\\noindent\nWe now define\n$$S_j(Q, T_j) : = \\lambda^{-1} \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap Q)$$ \nand observe that \\eqref{qwa} immediately implies\n$$ \\sum_{Q \\, : \\, Q \\cap T_j \\neq \\emptyset} S_j(Q, T_j) \\leq C, $$\nwhich establishes \\eqref{need2}.\n\n\\medskip\n\\noindent\nOn the other hand, to see that \\eqref{need1} is satisfied, note that \n\\eqref{wer} and \\eqref{awq} together with Lemma \\ref{linalg-moll} give\n$$ C \\lambda M(Q) \\leq {\\rm vis_\\varepsilon }(Z_p \\cap Q) \\leq \n\\frac{C \\lambda^{(n-d)\/n} \\left({\\rm surf }_{e(T_1),\\varepsilon}(Z_p \\cap Q) \n\\dots \\, {\\rm surf }_{e(T_d),\\varepsilon}(Z_p \\cap Q)\\right)^{1\/n}}\n{\\left(e(T_1) \\wedge \\dots \\wedge e(T_d)\\right)^{1\/n}},$$\nand so\n\n$$ S_1(Q,T_1) \\dots S_d(Q,T_d) = \\lambda^{-d}\\prod_{j=1}^d \\mbox{ surf}_{e(T_j),\\varepsilon}({Z_p} \\cap Q)$$\nis at least\n$$ C \\lambda^{-d}\\lambda^n M(Q)^n \\lambda^{d-n} e(T_1)\\wedge \\dots \\wedge e(T_d) \n= C M(Q)^n e(T_1)\\wedge \\dots \\wedge e(T_d),$$\nand thus \\eqref{need1} is established.\n\n\\medskip\n\\noindent\nConsequently, the multilinear Kakeya theorem is reduced to proving Theorem \\ref{algtop}.\n\n\\section{The Borsuk--Ulam theorem and a covering lemma}\\label{sec:BU}\n\n\\noindent\nThe Borsuk--Ulam theorem is as follows:\n\n\\begin{theorem}[Borsuk--Ulam]\nSuppose that $N \\geq J$ and that $F: \\mathbb{S}^N \\to \\mathbb{R}^J$ is continuous and satisfies \n$F(-x) = - F(x)$ for all $x \\in \\mathbb{S}^N$. Then there is an $x \\in \\mathbb{S}^N$ such that \n$F(x) = 0$.\n\\end{theorem}\n\\noindent\n\n\\medskip\n\\noindent\nFor a delightful discussion of this theorem and its applications, see \\cite{Mat}. See also \\cite{C}\nfor a recent proof of the Borsuk--Ulam theorem using only point-set topology and Stokes' theorem. \nIncluded in \\cite{Mat} there is a discussion of various equivalent forms of this theorem, some of which \n(known as Lusternik--Schnirelmann results) take the form of covering statements for \nthe sphere. In this section we formulate another such equivalent covering statement which we shall use \nin our proof of Theorem \\ref{algtop}.\n\n\n\\begin{lemma}\\label{covering}\nSuppose that $A_i \\subseteq \\mathbb{S}^N$ for $ 1 \\leq i \\leq J$, and suppose that for each $i$,\n$A_i \\cap (\\overline{-A_i}) = \\emptyset$. \nIf $J \\leq N$, then the $2J$ sets $A_i$ and $-A_i$ do not cover $\\mathbb{S}^N$. \n\\end{lemma}\n\n\\medskip\n\\noindent\nNote that no topological hypothesis on the sets $A_i$ is needed.\n \n\\begin{proof}\nLet $F_i(x) = d(x, -A_i) - d(x, A_i)$ for $1 \\leq i \\leq J$. Then, with $F: \\mathbb{S}^N \\to \\mathbb{R}^J$\ndefined by $F(x) = (F_1(x), \\dots , F_J(x))$, we have that $F$ is continuous and $F(-x) = -F(x)$ for all $x$, \nso by the Borsuk--Ulam theorem there is an $x$ with $F(x) = 0$. We claim that this $x$ does not belong \nto any $A_i$ or to $-A_i$. For if $x \\in A_i$ we have\n$d(x, A_i) = 0$, hence $d(x, -A_i) = 0$, hence $x \\in\n\\overline{-A_i}$, a contradiction. Likewise, since by hypothesis $\n\\overline{A_i} \\cap (-A_i) = \\emptyset$, \n$x \\in -A_i$ gives $x \\in \\overline{A_i}$, another contradiction.\n\\end{proof}\n\n\\medskip\n\\noindent\n{\\bf{Remark.}} The converse argument also holds: if we assume the assertion of the lemma, \n{\\em{but only for open sets}} $U_i$, we can recover the Borsuk--Ulam theorem. Indeed, suppose \n$F: \\mathbb{S}^N \\rightarrow \\mathbb{R}^J$ is continuous, $F(-x) = -F(x)$ and $N \\geq J$. Let\n$U_i$ be the open set $\\{x \\; : \\; F_i(x) > 0\\}$. Then $-U_i = \\{x \\; : \\; F_i(x) < 0\\}$, so that\n$\\overline{- U_i} \\subseteq \\{x \\, : \\, F_i(x) \\leq 0\\}$ and so\n$U_i \\cap (\\overline{-U_i}) = \\emptyset$.\nBy assumption there is an $x$ which is not in any of the $U_i$ or $-U_i$. So $F_i(x) \\leq 0$ for \nall $i$ and $F_i(x) \\geq 0$ for all $i$. Hence $F_i(x) = 0$ for all $i$, that is, $F(x) = 0$.\\footnote{That\nthe assertion of Lemma \\ref{covering} for open sets $U_i$ logically implies the same \nstatement for sets $A_i$ with no topological restrictions can easily be seen directly from the fact that the \nmetric space $\\mathbb{S}^N$ satisfies appropriate separation axioms. More precisely, if two subsets $A$ and $B$ \nof a metric space are separated in the sense that the closure of either does not meet the other, then there are \nopen sets $U$ and $V$ with the same property such that $A \\subseteq U$ and $B \\subseteq V$.} \n\n\n\\section{Outline of the proof of Theorem \\ref{algtop}} \n\\label{sec:outline}\n\n\\noindent\nWe now describe the scheme of the proof of Theorem \\ref{algtop}. The function $M$ is given, \nand we will be working with the class $\\mathcal{P}_k^\\ast = \\mathbb{S}^N$\nof normalised polynomials \n$p : \\mathbb{R}^n \\to \\mathbb{R}$ of degree bounded by some $k \\in \\mathbb{N}$. Recall that $N \\sim k^n$.\nFor each such polynomial $p$, its zero set is the algebraic hypersurface \n$Z_p = \\{x \\, : \\, p(x) = 0 \\}$, \nand we let \n$$S(Q) = \\{ p \\in \\mathcal{P}_k^\\ast \\; : \\; \\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\leq M(Q)\\}.$$ \nFollowing Guth \\cite{G}, the aim is to show that if we take a suitable \n$k \\sim (\\sum_Q M(Q)^n)^{1\/n}$, and a suitable $\\varepsilon >0$, then we can find a \npolynomial in $\\mathcal{P}_k^\\ast$ which is not \nin any of the $S(Q)$. (Note that $S(Q) = \\emptyset$ \nfor those $Q$ such that $M(Q) \\lesssim 1$.) Our method to \nestablish this diverges somewhat from that of Guth, but there are of course many points of contact \nbetween the two lines of argument.\n\n\\medskip\n\\noindent\nLet, for $r \\geq 0$,\n$$S^{(r)}(Q) = \\{ p \\in \\mathcal{P}_k^\\ast \\; : \\; \\operatorname{vis}_\\varepsilon(Z_p \\cap Q) \\sim 2^{-r} M(Q)\\}.$$ \nThen \n$$ S(Q) = \\bigcup_{1 \\lesssim 2^r \\lesssim M(Q)}S^{(r)}(Q)$$ \nsince $S^{(r)}(Q) = \\emptyset$ for $r$ such that $2^r \\gtrsim M(Q)$.\n \n\\medskip\n\\noindent\nWe shall introduce a collection $\\mathcal{C}$ of ``colours'' $\\Theta$ whose cardinality is bounded by $C$. \nFor each colour $\\Theta$ we shall define subsets $S^{(r), \\Theta}(Q)$ of $S^{(r)}(Q)$\nwhich have the property that\n\\begin{equation}\\label{colours}\nS^{(r)}(Q) = \\bigcup_{\\Theta \\in \\mathcal{C}} S^{(r), \\Theta}(Q).\n\\end{equation}\n\n\\medskip\n\\noindent\nFor each fixed $Q$ and $r$ such that $1 \\lesssim 2^r \\lesssim M(Q)$ we will \ndefine an indexing set $\\mathcal{A}_{Q, r}$ of cardinality\n$C 2^{-rn} M(Q)^n$, and for each $\\alpha \\in \\mathcal{A}_{Q, r}$, \na subset $S^{(r), \\Theta}_\\alpha (Q)$ of $S^{(r), \\Theta}(Q)$ such that \n\\begin{equation}\\label{translates}\nS^{(r), \\Theta}(Q) = \\bigcup_{\\alpha \\in \\mathcal{A}_{Q, r}}\nS^{(r), \\Theta}_\\alpha (Q).\n\\end{equation}\nTo ensure that this decomposition is well-defined, it will transpire that $\\varepsilon$ must be taken to be small.\n\n\n\n\\medskip\n\\noindent\nFinally we shall decompose each $S^{(r), \\Theta}_\\alpha (Q)$ as\n$$ S^{(r), \\Theta}_\\alpha (Q) = S^{(r), \\Theta + }_\\alpha (Q) \\cup S^{(r), \\Theta - }_\\alpha (Q),$$\nwhere \n\\begin{equation}\\label{opposite}\nS^{(r), \\Theta -}_\\alpha(Q) = - S^{(r), \\Theta +}_\\alpha(Q)\n\\end{equation} \nin such a way that for all $Q, r, \\Theta$ and $\\alpha$,\n\\begin{equation}\\label{qw}\nS^{(r), \\Theta + }_\\alpha (Q) \\cap \\overline{S^{(r), \\Theta - }_\\alpha (Q)} = \\emptyset.\n\\end{equation}\n\n\\medskip\n\\noindent\n(The closure here refers to the natural topology of $\\mathcal{P}_k^\\ast = \\mathbb{S}^N$.)\n\n\\medskip\n\\noindent\nThe reason for the introduction of colours is to ensure that\nthere is sufficient separation between the sets $S^{(r), \\Theta \\pm\n}_\\alpha (Q)$ and their antipodes for \\eqref{qw} to hold.\n\n\\medskip\n\\noindent\nIn summary then, \n\\begin{equation}\\label{union}\n\\bigcup_Q S(Q) = \\bigcup_Q \\bigcup_{1 \\lesssim 2^r \\lesssim M(Q)}\\bigcup_{\\Theta \\in \\mathcal{C}} \n\\bigcup_{\\alpha \\in \\mathcal{A}_{Q, r}} \\left(S^{(r), \\Theta + }_\\alpha (Q) \n\\cup S^{(r), \\Theta - }_\\alpha (Q)\\right),\n\\end{equation}\nwhere $S^{(r), \\Theta + }_\\alpha (Q)$ and $S^{(r), \\Theta - }_\\alpha (Q)$\nsatisfy \\eqref{opposite} and \\eqref{qw}. \n\n\\medskip\n\\noindent\nLemma \\ref{covering} then implies that if the cardinality of the set indexing the union on the \nright hand side of \\eqref{union} is less than or equal to $N$,\nthen the sets in the union cannot cover $\\mathbb{S}^N = \\mathcal{P}_k^\\ast$. \n\n\\medskip\n\\noindent\nNow the number of terms indexing the union is at most\n$$ C \\sum_Q \\sum_{r \\geq 0} \\sum_{\\Theta \\in \\mathcal{C}} \\, \\sum_{\\alpha \\in \\mathcal{A}_{Q, r}} \n1 \\leq C \\sum_Q \\sum_{r \\geq 0} 2^{-rn} M(Q)^n \\leq C \\sum_Q M(Q)^n.$$\nSo provided that $N \\gtrsim \\sum_Q M(Q)^n$, amongst the polynomials in $\\mathbb{S}^N =\n\\mathcal{P}_k^\\ast$, there will exist one which is not in any of the $S(Q)$. Since \n$N \\sim k^n$, we can therefore take $k$ with $k \\sim \\left(\\sum_Q M(Q)^n \\right)^{1\/n}$\nand a $p \\in \\mathcal{P}_k^\\ast$ which, for suitable $\\varepsilon > 0$, satisfies \n$\\operatorname{vis}_\\varepsilon(Z_p \\cap Q) > M(Q)$ for all $Q$, as was needed.\n\n\\medskip\n\\noindent\nIt remains now to define the various decompositions introduced above,\nand establish the assertions we have made concerning them.\n\n\n\n\\section{Colours}\n\\noindent\nIn this section we describe how to establish \\eqref{colours} in such a way that the indexing set \n$\\mathcal{C}$ has cardinality at most $C$.\n\n\\medskip\n\\noindent\nLet $\\mathcal{E}$ denote the class of centred ellipsoids in $\\mathbb{R}^n$,\nthat is images of the unit ball $\\mathbb{B}$\nby affine linear maps $A$. Each ellipsoid $A (\\mathbb{B})$ is determined by an orthonormal basis of \nprincipal axes or directions given by the normalised eigenvectors of $A^tA$, and corresponding semiaxes, \nthe squares of whose lengths are the eigenvalues of $A^tA$.\nThus $\\mathcal{E}$ is a manifold of dimension\n$n(n-1)\/2 + n = n(n+1)\/2$. \n\n\\medskip\n\\noindent\nLet $\\mathcal{K}$ denote the class of centrally symmetric convex bodies in $\\mathbb{R}^n$.\nBy the John ellipsoid theorem \\cite{John}, every member $K$ of $\\mathcal{K}$ is close to some ellipsoid $E$ \nin the sense that $n^{-1\/4} E \\subseteq K \\subseteq n^{1\/4} E$. \n\n\\medskip\n\\noindent\nThere is a natural metric (the Banach--Mazur metric) to put on the class $\\mathcal{K}$, given by\n$$d(K,L) = \\log \\inf \\{ \\alpha \\geq 1 \\, : \\, \\alpha^{-1}K \\subseteq L \\subseteq \\alpha K\\}.$$ \nThe John ellipsoid theorem asserts that every convex body is distant at most \n$(\\log n)\/4 \\lesssim 1$ from some ellipsoid. An ellipsoid with semiaxes of lengths $2^{k_1}, \\dots , 2^{k_n}$ where\n$k_1 + \\dots + k_n =0$ will be distant $ \\lesssim \\max |k_j| $ from the unit ball. Two congruent ellipses \nin $\\mathbb{R}^2$ with semiaxes of lengths $1$ and $N$ and principal directions differing by $\\theta$ will be \ndistant $\\lesssim \\theta N $ apart.\n\n\n\n\\medskip\n\\noindent\nTo set the scene for the covering property of ellipsoids that we need, note that in $\\mathbb{R}^N$, \ngiven a scale $\\rho > 0$ and a pre-assigned number $\\gamma > 1$, \nwe can find a set $\\mathcal{X}$ of $\\rho$-separated points $x_i \\in \\mathbb{R}^N$ such that every \npoint of $\\mathbb{R}^N$ is in some $B(x_i,\\rho)$, and such that $\\mathcal{X}$ can be partitioned \ninto $O_N(\\gamma^N)$ families (colours), so that points of $\\mathcal{X}$ of the same colour are \ndistant at least $ \\gamma \\rho$ from each other. This property expresses the idea that the \ndimensionality of $\\mathbb{R}^N$ as a metric space is $N$. We can then assign to each $x \\in \\mathbb{R}^N$ \none or more colours according to whether $d(x, x_i) < \\rho$ for some $x_i \\in \\mathcal{X}$ of that \nparticular colour. \n\n\\medskip\n\\noindent\nSimilarly it is not hard to verify that given $\\rho >0$ and $\\gamma >1$, there exists a $\\rho$-separated \nsubset $\\mathcal{E}_0$ of $\\mathcal{E}$ such that $\\{B(E, \\rho) \\, : \\, E \\in \\mathcal{E}_0\\}$ \ncovers $\\mathcal{E}$ and such that we can partition $\\mathcal{E}_0$ into at most \n$O_n( \\gamma^{n(n+1)\/2})$ families (colours) such that any two ellipsoids in $\\mathcal{E}_0$ \nof the same colour are distant at least $\\gamma \\rho$ from each other. \n\n\\medskip\n\\noindent\nChoosing $\\rho = 1$ and $\\gamma$ sufficiently large depending only on the dimension $n$, and using the \nJohn ellipsoid theorem, we obtain the following: \n\n\\begin{lemma}\\label{ellipsoid}\nSupose $\\alpha_n > 1 $ is sufficiently large.\nThen there exists a set $\\mathcal{E}_0 \\subseteq \\mathcal{E}$ with the property that for every \n$K \\in \\mathcal{K}$ there is an $E \\in \\mathcal{E}_0$ such that\n\\begin{equation}\\label{close}\n\\alpha_n^{-1} K \\subseteq E \\subseteq \\alpha_n K\n\\end{equation}\nand such that the set $\\mathcal{E}_0$ can be partitioned into at most $C = C(n, \\alpha_n)$\ncolours in such a way that every $K \\in \\mathcal{K}$ satisfies \\eqref{close} for at most one \n$E \\in \\mathcal{E}_0$ {\\em of a given colour}.\n\\end{lemma}\n\n\n\\medskip\n\\noindent\nGiven $n$ we now fix $\\alpha_n$ sufficiently large, and fix our palette $\\mathcal{C}$ \nconsisting of at most $C(n, \\alpha_n)$ colours once and for all so that the conclusion of \nLemma \\ref{ellipsoid} holds. We say that two convex bodies $E$ and $K$ are {\\bf close} if \n\\eqref{close} holds. So every $K \\in \\mathcal{K}$ is close to some member of $\\mathcal{E}_0$, \nbut there is at most one $E \\in \\mathcal{E}_0$ of a given colour to which it is close. \nFor a colour $\\Theta \\in \\mathcal{C}$ let \n$$\\mathcal{E}^{\\Theta}_0 = \\{ E \\in \\mathcal{E}_0 \\; : \\; E \\mbox{ is of colour } \\Theta\\}.$$\n\n\\medskip\n\\noindent\nFinally, given $Q$ and $r \\geq 0$,\nlet\n$$S^{(r), \\Theta} (Q) = \\{ p \\in S^{(r)}(Q) \\, : \\, K_\\varepsilon(Z_p \\cap Q) {\\mbox{ is close to a member of }}\n\\mathcal{E}^{\\Theta}_0\\};$$\nthen we have\n$$S^{(r)}(Q) = \\bigcup_{\\Theta \\in \\mathcal{C}} S^{(r), \\Theta}(Q),$$\nand \\eqref{colours} is established.\n\n\n\\section{Translates}\\label{sec:translates}\n\\noindent\nWe now fix $Q$, $r \\geq 0$ and a colour $\\Theta \\in \\mathcal{C}$. In this section we establish \n\\eqref{translates} for suitable subsets $S^{(r), \\Theta}_\\alpha(Q) \\subseteq \nS^{(r), \\Theta}(Q)$ which are indexed by $\\alpha \\in \\mathcal{A}_{Q, r}$, where $\\mathcal{A}_{Q, r}$\nhas cardinality $\\sim 2^{-rn} M(Q)^n$. We can assume that $S^{(r), \\Theta}(Q) \\neq \\emptyset$.\n\n\\medskip\n\\noindent\nIf $p \\in S^{(r), \\Theta}(Q)$, the convex body $K_\\varepsilon(Z_p \\cap Q) \\subseteq \\mathbb{B}$ \nhas volume $\\sim 2^{rn}\/M(Q)^{n} \\lesssim 1$, and it is close to a unique member $E(p)$ of $\\mathcal{E}^{\\Theta}_0$ \nof comparable volume. Hence we can fit $\\sim 2^{-rn} M(Q)^{n}$ disjoint parallel translates of $E(p)$\ninside $Q$, with the translations along the principal directions of $E(p)$. Likewise,\nif $\\eta < 1$ is a numerical scaling factor, we can fit $\\sim \\eta^{-n}2^{-rn} M(Q)^{n}$ disjoint parallel \ntranslates of $\\eta E(p)$ inside $Q$, with the translations again along the principal directions of $E(p)$.\nIndeed, if the lengths of the semiaxes of $E(p)$ are $l_1, \\dots, l_n\n\\leq c$, and the principal directions are \n$e_1, \\dots, e_n$, we can place the centres of the translated copies of $\\eta E(p)$ at the points \n$x_Q + \\eta \\sum_j m_j l_j e_j$ for $m_j \\in 2 \\mathbb{Z}$ and $|m_j| \\leq c \\eta^{-1}l_j^{-1}$; \nhere $x_Q$ is the centre of $Q$.\nIn this construction\nthe number of translated copies equals the product\n\\begin{equation}\n \\label{numtranslates}\n c\\eta^{-n}(l_1\\dots l_n)^{-1}=c \\eta^{-n}2^{-rn} M(Q)^{n}.\n\\end{equation}\n\n\\begin{lemma}\n \\label{manybisections}\nThere is a dimensional constant $C_n$ such that if $p \\in S^{(r), \\Theta}(Q)$\nand $\\eta < 1$, then $Z_p$ bisects \nat most $ C_n \\eta ^{-(n-1)} 2^{-rn} M(Q)^{n}$ disjoint translates of $\\eta E(p)$ in $Q$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $E(p)$ has principal directions $\\{e_j\\}$ and corresponding semiaxes with lengths $\\{l_j\\}$. \nIf $Z_p$ bisects a translate $\\eta E(p) + \\xi$ of $\\eta E(p)$, then for at least one $j$ we will \nhave\\footnote{Here and in the next two displayed equations we are using the {\\em unmollified} \nnotion of directional surface area $\\mbox{surf}_e$.}\n$$ \\mbox{ surf}_{e_j}(Z_p \\cap (\\eta E(p) + \\xi)) \\geq C_n \\operatorname{vol}(\\eta E(p) + \\xi) \/ \\eta l_j\n= C_n \\eta ^{n-1} \\operatorname{vol}E(p)\/l_j.$$\nThis is just the affine-invariant formulation of the fact that a hypersurface which bisects the \nunit ball must have large $(n-1)$-dimensional Hausdorff measure inside\nthe ball -- see Lemma \\ref{bisect} in the Appendix.\nSo \n$$\\sum_{j=1}^n l_j \\operatorname{surf}_{e_j}(Z_p \\cap (\\eta E(p) + \\xi))\n \\geq C_n \\eta ^{n-1} \\operatorname{vol}E(p).$$\nIf now $Z_p$ bisects as many as $ A 2^{-rn} M(Q)^{n}$ disjoint copies of $\\eta E(p)$ in $Q$, \nwe will have\n$$\\sum_{j=1}^n l_j \\operatorname{surf}_{e_j}(Z_p \\cap Q) \\geq \nC_n A \\eta^{n-1} \\operatorname{vol}E(p)\\,\\, 2^{-rn} M(Q)^{n}.$$\n\n\\medskip\n\\noindent\nIf $p'$ is another polynomial in $\\mathbb{S}^N$ sufficiently close to $p$,\nand if $p$ bisects an ellispoid $E$, we can conclude that $p'$ approximately bisects $E$ in the sense that\n$p'$ is positive on at least $40\\%$ of $E$ and negative on at least $40\\%$ of $E$.\nSince we are only interested in a finite number of ellipsoids here, namely the translates $\\eta E+\\xi$\nfor $E\\in\\mathcal{E}_0$ with $M(Q)^{-n}\\lesssim\\operatorname{vol}E\\lesssim 1$ for the relevant $Q$,\nthen by choosing $\\varepsilon$ small enough we will have this approximate bisection property for all\npolynomials which affect the value of $\\operatorname{surf}_{e,\\varepsilon}$ in the expressions above.\nTherefore we have the estimate\n$$\\sum_{j=1}^n l_j \\operatorname{surf}_{e_j,\\varepsilon}(Z_p \\cap Q) \\geq \nC_n A \\eta^{n-1} \\operatorname{vol}E(p)\\,\\, 2^{-rn} M(Q)^{n}.$$\n\n\\medskip\n\\noindent\nThe definition (cf. \\eqref{Kdefn}) of $K_\\varepsilon(Z_p \\cap Q)$ together with the fact that\n$K_\\varepsilon(Z_p \\cap Q)$ and $E(p)$ are close \nimplies that \n$$ l_j \\operatorname{surf}_{e_j,\\varepsilon}(Z_p \\cap Q) \\leq C_n$$\nfor each $j$, and moreover, as $p \\in S^{(r), \\Theta}(Q)$,\n$$\\operatorname{vol}E(p) \\sim 2^{rn}\/M(Q)^{n}.$$ \nTherefore $A$ must satisfy $A \\leq C_n \\eta^{-(n-1)}$.\n\\end{proof}\n\n\\medskip\n\\noindent\nChoose $\\eta$ sufficiently small so that $C_n \\eta 0\\} \\; \\cap \\; E_\\alpha) > \\mbox{ vol}(\\{p<0 \\}\n\\; \\cap \\; E_\\alpha) $$\n(in which case we say that $p \\in S^{(r), \\Theta +}_\\alpha(Q)$), or \n$$ \\mbox{ vol }(\\{p>0\\} \\; \\cap \\; E_\\alpha) < \\mbox{ vol}(\\{p<0 \\}\n\\; \\cap \\; E_\\alpha), $$\n(in which case we say that $p \\in S^{(r), \\Theta -}_\\alpha(Q)$).\n\n\\medskip\n\\noindent\nThen\n$$ S^{(r), \\Theta}_\\alpha(Q) = S^{(r), \\Theta +}_\\alpha(Q) \\cup S^{(r), \\Theta -}_\\alpha(Q).$$ \nMoreover \n$$ S^{(r), \\Theta -}_\\alpha(Q) = - S^{(r), \\Theta +}_\\alpha(Q),$$ \nand so to establish \\eqref{qw} we wish to show that for all $\\alpha$, \n$$S^{(r), \\Theta +}_\n\\alpha(Q) \\cap \\overline{S^{(r), \\Theta -}_\\alpha(Q)} = \\emptyset.\n$$\n\n\\medskip\n\\noindent\nTo see this, suppose for a contradiction that for some\n$\\alpha \\in \\mathcal{A}_{Q, r}$ there is a $p \\in S^{(r),\n\\Theta +}_\\alpha(Q)$ and a sequence of $p_m \\in S^{(r), \\Theta -}_\\alpha(Q)$\nwhich converges to $p$ in $\\mathbb{S}^N$. That is, we suppose that \n\\begin{equation}\\label{last}\n\\mbox{ vol }(\\{p>0\\} \\; \\cap \\; E_\\alpha (p)) > \\mbox{ vol}(\\{p<0 \\}\n\\; \\cap \\; E_\\alpha(p)) \n\\end{equation}\nand\n$$ \\mbox{ vol }(\\{p_m> 0\\} \\; \\cap \\; E_\\alpha (p_m)) < \\mbox{ vol}(\\{p_m <0 \\}\n\\; \\cap \\; E_\\alpha(p_m))$$\nwhere $p_m$ converges to $p$ in $\\mathbb{S}^N$.\n\n\\begin{lemma}\\label{convergence}\nFix $Q$, $r$ and $\\Theta$. Suppose that $p \\in S^{(r), \\Theta}(Q)$, $p_m \\in S^{(r), \\Theta}(Q)$ \nfor $m \\in \\mathbb{N}$ and that $p_m$ converges to $p$ in $\\mathbb{S}^N$. Then for all sufficiently \nlarge $m$ we have $E(p_m) = E(p)$. If $\\alpha \\in \\mathcal{A}_{Q,r}$ and in addition \n$p, p_m \\in S^{(r), \\Theta}_\\alpha(Q)$, then for $m$ sufficiently large, \n$E_\\alpha(p_m) = E_\\alpha(p)$\n\\end{lemma}\n\n\\begin{proof}\nSince we are using the mollified version of the directional surface area and quantities defined in terms of it, \nthe convergence of $p_m$ to $p$ in $\\mathbb{S}^N$ implies that the convex bodies $K_\\varepsilon(Z_{p_m} \\cap Q)$\nconverge to $K_\\varepsilon(Z_{p} \\cap Q)$ as $m \\to \\infty$\\footnote{in the sense that there is a sequence $\\gamma_m \\geq 1$ \nwith $\\gamma_m \\to 1$ such that\n$\\gamma_m^{-1} K_\\varepsilon (Z_{p_m} \\cap Q) \\subseteq K_\\varepsilon(Z_{p} \\cap Q) \\subseteq \\gamma_m \nK_\\varepsilon(Z_{p_m} \\cap Q)$} and in particular $K_\\varepsilon(Z_{p_m} \\cap Q)$ and $K_\\varepsilon(Z_{p} \\cap Q)$\nare close for $m$ sufficiently large.\nSince $p$ and $p_m$ are members of $S^{(r), \\Theta}(Q)$ then\n$K_\\varepsilon(Z_{p_m} \\cap Q)$ and $K_\\varepsilon(Z_p \\cap Q)$\nmust be close to \\emph{some} member of $\\mathcal{E}_0^\\Theta$\nand\nthus, for $m$ sufficiently large, \nthey\nare close to the {\\em same} member of $\\mathcal{E}^{\\Theta}_0$, which \nmust be $E(p)$. In particular, for $m$ sufficiently large, we have $E(p_m) = E(p)$ and consquently \n$E_\\alpha(p_m) = E_\\alpha(p)$ for all $\\alpha$.\n\\end{proof}\n\\noindent\n(It is at the end of the proof of this lemma, and in the construction of the sets $S^{(r), \\Theta}_\\alpha(Q)$, \nwhere the relevance of the earlier decomposition into colours becomes clear.)\n\n\\medskip\n\\noindent\nSo, for $m$ sufficiently large we have\n$$ \\mbox{ vol }(\\{p_m> 0\\} \\; \\cap \\; E_\\alpha (p)) < \\mbox{ vol}(\\{p_m <0 \\}\n\\; \\cap \\; E_\\alpha(p))$$\nwhich, upon taking limits and using the fact that vol $(\\{ p = 0\\}) = 0$ as $p$ is non-zero,\nimplies\n$$ \\mbox{ vol }(\\{p > 0\\} \\; \\cap \\; E_\\alpha (p)) \\leq \\mbox{ vol}(\\{p < 0 \\}\n\\; \\cap \\; E_\\alpha(p)),$$\nwhich is in contradiction with \\eqref{last}. Hence\n$$S^{(r), \\Theta +}_\\alpha(Q) \\cap \\overline{S^{(r), \\Theta -}_\\alpha(Q)} = \\emptyset,$$\nand we are therefore finished.\n\n\\section{Appendix -- Bisecting balls}\\label{app}\n\n\\medskip\n\\noindent\nIn this appendix we indicate a simple proof of the (geometrically obvious) fact that a hypersurface\nwhich bisects the unit ball must have large surface area inside the\nball. Let $\\mathbb{B}$ be the closed unit ball in $\\mathbb{R}^n$ and suppose $p: \\mathbb{R}^n \\to\n\\mathbb{R}$ is a polynomial. Let\n$$E = \\{x \\in \\mathbb{B} \\, : \\, p(x) \\leq 0 \\}$$\nand \n$$F = \\{x \\in \\mathbb{B} \\, : \\, p(x) \\geq 0 \\}.$$\n\n\\begin{lemma}\\label{bisect}\nIf $ {\\rm vol} \\, (E) = a \\, {\\rm vol} \\, (\\mathbb{B})$ and\n$ {\\rm vol} \\, (F) = b \\, {\\rm vol} \\, (\\mathbb{B})$ \nwhere $a + b = 1$, then \n$$\\mathcal{H}_{n-1} (\\{ x \\in \\mathbb{B} \\, : \\, p(x) = 0 \\}) \n> \\frac{1}{2}\\left(a^{(n-1)\/n} + b^{(n-1)\/n} -1 \\right)\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1}).$$\n\\end{lemma}\n\n\\begin{proof}\nIt is easy to see that \n$ \\partial E \\cup \\partial F = \\mathbb{S}^{n-1} \\cup (E \\cap F)$\nand\n$ \\partial E \\cap \\partial F = E \\cap F.$\nSince\n$$ \\mathcal{H}_{n-1}( \\partial E \\cup \\partial F) = \n\\mathcal{H}_{n-1}( \\partial E) + \\mathcal{H}_{n-1}(\\partial F) \n- \\mathcal{H}_{n-1}( \\partial E \\cap \\partial F)$$\nand \n$$ \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1} \\cup (E \\cap F)) =\n\\mathcal{H}_{n-1} (\\mathbb{S}^{n-1}) + \\mathcal{H}_{n-1} (E \\cap F)\n- \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1} \\cap E \\cap F)$$\nwe have\n$$ 2 \\mathcal{H}_{n-1} (E \\cap F)\n= \\mathcal{H}_{n-1}( \\partial E) +\n\\mathcal{H}_{n-1}(\\partial F) - \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1})\n+ \\mathcal{H}_{n-1}\n(\\mathbb{S}^{n-1} \\cap E \\cap F),$$\nand so\n$$\\mathcal{H}_{n-1} (E \\cap F) \n\\geq \\frac{1}{2}\\left(\\mathcal{H}_{n-1}( \\partial E) +\n\\mathcal{H}_{n-1}(\\partial F) - \\mathcal{H}_{n-1} (\\mathbb{S}^{n-1})\\right).$$\n\n\\medskip\n\\noindent\nBy the isoperimetric inequality we have\n$$\\mathcal{H}_{n-1}( \\partial E) \\geq a^{(n-1)\/n}\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1})$$\nand \n$$\\mathcal{H}_{n-1}( \\partial F) \\geq b^{(n-1)\/n}\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1}),$$\n(with strict inequality in at least one place) so that \n$$\\mathcal{H}_{n-1} (E \\cap F) \n> \\frac{1}{2}\\left(a^{(n-1)\/n} + b^{(n-1)\/n} -1 \\right)\n\\mathcal{H}_{n-1}(\\mathbb{S}^{n-1})$$\nas required. \n\\end{proof}\n\n\\medskip\n\\noindent\nSince for $n \\geq 2$ and $0 < a, b < 1$ with $a+b =1$ we have $ (a+b)^{(n-1)\/n} < a^{(n-1)\/n} + b^{(n-1)\/n}$,\nthis establishes the desired bound.\n\\medskip\n\\noindent\n \n\\medskip\n\\noindent\nThe following question may be of interest. \nLet $K \\subseteq \\mathbb{R}^n$ be a symmetric convex body, which we can normalise so that its John ellipsoid \nis the unit ball. Within the class of polynomial hypersurfaces $Z$ which cut $K$ in the proportions $a:b$, \nhow do we minimise the surface area of $Z \\cap K$?\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nLet $n,k\\in \\mathbb{N}$ and let $\\brho=(\\rho_1,\\dots,\\rho_k)$ with $1\\leq \\rho_\\ell\\leq n\\leq k$ for $\\ell\\in [k]:=\\{1,2,\\dots, k\\}$ such that $\\sum_{\\ell\\in [k]} \\rho_\\ell=n^2$. A {\\it $\\brho$-latin square} $L$ of order $n$ is an $n\\times n$ array filled with $k$ different {\\it symbols}, $[k]$, each occurring at most once in each row and at most once in each column, such that each symbol $\\ell$ occurs exactly $\\rho_\\ell$ times in $L$ for $\\ell\\in [k]$. An $r \\times s$ {\\it $\\brho$-latin rectangle} on the set $[k]$ of symbols is an $r\\times s$ array in which each symbol in $[k]$ occurs at most once in each row and in each column, and in which each symbol $\\ell$ occurs at most $\\rho_\\ell$ times for $\\ell\\in [k]$. A {\\it latin square} (or {\\it rectangle}) is a $\\brho$-latin square (or {\\it rectangle}) with $\\brho=(n,\\dots,n)$. \n\nWe are interested in conditions that ensure that an $r \\times s$ $\\brho$-latin rectangle can be extended to a $\\brho$-latin square. The cases $(\\brho,s)=((n,\\dots,n),n)$, and $\\brho=(n,\\dots,n)$ were settled by Hall \\cite{MR13111}, and Ryser \\cite{MR42361}, respectively. Most recently, the case $s=n$ was solved by Goldwasser, Hilton, Hoffman, and \\\"{O}zkan \\cite{MR3280683} (See Theorem \\ref{rhohallthmgoldetal}). For an old yet excellent survey on embedding latin squares, we refer the reader to Lindner \\cite{MR1096296}. \n\nWe need a few pieces of notation before we can state our main result. The complement of a set $S$ is denoted by $\\bar S$, $x \\dotdiv y:=\\max\\{0, x-y\\}$, and we write $x-y\\dotdiv z$ for $(x-y)\\dotdiv z$. If $L$ is an $r\\times s$ $\\brho$-latin rectangle, $e_\\ell$ will denote the number of occurrences of symbol $\\ell$ in $L$. We will always assume that $n>\\min \\{r,s\\}$, and that $[r], [s]$, and $[k]$ are the set of rows, the set of columns, and the set of symbols of $L$. \nLet $i$ be a row in $[r]$, $j$ be a column in $[s]$, $\\ell$ be a symbol in $[k]$, and let $I\\subseteq [r], J\\subseteq [s], K\\subseteq [k]$. Then $\\mu_K(i), \\mu_K(j), \\mu_I(\\ell)$, and $\\mu_J(\\ell)$ are the number of symbols in $K$ that are missing in row $i$, the number of symbols in $K$ that are missing in column $j$, the number of rows in $I$ where symbol $\\ell$ is missing, and the number of columns in $J$ where symbol $\\ell$ is missing, respectively, and \n\\begin{align*}\n \\mu_K(I)&:=\\sum_{i\\in I}\\mu_K(i), & \\mu_I(K)&:=\\sum_{\\ell\\in K} \\mu_I(\\ell),\\\\\n \\mu_K(J)&:=\\sum_{j\\in J}\\mu_K(j), & \\mu_J(K)&:=\\sum_{\\ell\\in K} \\mu_J(\\ell).\n\\end{align*}\nObserve that \n\\begin{align*}\n &\\mu_K(I)= \\mu_I(K), &\\mu_{I}(\\ell)+\\mu_{\\bar I}(\\ell)=r-e_\\ell,\\\\\n &\\mu_K(J)= \\mu_J(K), &\\mu_{J}(\\ell)+\\mu_{\\bar J}(\\ell)=s-e_\\ell.\n\\end{align*}\n\nUsing our notation, the main result of \\cite{MR3280683} can be stated as follows. \n\\begin{theorem} \\cite{MR3280683} \\label{rhohallthmgoldetal}\nAn $r\\times n$ $\\brho$-latin rectangle $L$ can be completed to an $n\\times n$ $\\brho$-latin square if and only if $\\rho_\\ell - e_\\ell\\leq n-r$ for $\\ell\\in [k]$, and either one of the following conditions hold.\n\\begin{align}\n |J|(n-r) &\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{\\rho_\\ell-e_\\ell, \\mu_J(\\ell)\\Big\\} &\\forall J\\subseteq [n],\\label{orenec}\\\\\n |J|(n-r) &\\geq \\sum_{\\ell\\in [k]} \\Big( \\rho_\\ell-e_\\ell\\dotdiv \\mu_{\\bar J}(\\ell)\\Big) &\\forall J\\subseteq [n]. \\label{hoffmanc}\n\\end{align}\n\\end{theorem}\nRewriting \\eqref{orenec} for $\\bar J$ will result in \\eqref{hoffmanc}, so conditions \\eqref{orenec} and \\eqref{hoffmanc} are essentially the same (See Remark \\ref{ordenewcond}). In \\cite{MR3280683}, \\eqref{orenec} is being referred to as Hall's refined condition, which is a special case of the more general Hall's constrained condition. If you think of a set $J$ of columns of $L$ as an $r\\times |J|$ subrectangle of $L$, Hall's constrained condition checks an inequality similar to that of Hall's refined condition for all subsets of cells, rather than those subsets that form an $r\\times |J|$ rectangle. \n\n\n\n\n\nWe provide a short proof of Theorem \\ref{rhohallthmgoldetal} (See Theorem \\ref{rhohallthm}). We also generalize Theorem \\ref{rhohallthmgoldetal} further by establishing an analogue of Ryser's Theorem for $\\brho$-latin rectangles. \n\nSuppose that an $r\\times s$ $\\brho$-latin rectangle $L$ is extended to an $n\\times n$ $\\brho$-latin square, and let us fix a symbol $\\ell\\in [k]$. On the one hand, there are $\\rho_\\ell-e_\\ell$ occurrences of symbol $\\ell$ outside the top left $r\\times s$ subsquare. On the other hand, there are at most $n-r$ occurrences of symbol $\\ell$ in the last $n-r$ rows and at most $n-s$ occurrences of symbol $\\ell$ in the last $n-s$ columns. Therefore, we must have\n\\begin{align} \\label{mainryserineq}\ne_\\ell \\geq r+s+\\rho_\\ell - 2n\\quad \\forall \\ell\\in [k].\n\\end{align}\nIt turns out that condition \\eqref{mainryserineq} is a special case of a more general necessary condition. Let \n\\begin{align*}\n P_t=\\{\\ell \\in [k] \\ |\\ \\rho_\\ell-e_\\ell > n-t \\}\\quad \\mbox{for } t\\in \\{r,s\\}.\n\\end{align*}\nNote that $P_{\\min \\{r,s\\}}\\subseteq P_{\\max \\{r,s\\}}$. \n\n\n\\begin{definition} \\label{fitdef} A sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ is said to be {\\it fitting} if $a_\\ell,b_\\ell \\in\\mathbb{N}\\cup \\{0\\}$ for $\\ell\\in [k]$, and the following conditions are satisfied.\n\\begin{align*}\n\\begin{cases}\n \\displaystyle\\sum_{\\ell\\in [k]} a_\\ell=r(n-s)+|P_r|(n-r)-\\sum_{\\ell\\in P_r}(\\rho_\\ell-e_\\ell), \\\\[20pt]\n \\displaystyle\\sum_{\\ell\\in [k]} b_\\ell=s(n-r)+|P_s|(n-s)-\\sum_{\\ell\\in P_s}(\\rho_\\ell-e_\\ell),\\\\[20pt]\n a_\\ell+b_\\ell\\leq \n \\begin{cases}\n 2n- r-s+ e_\\ell-\\rho_\\ell& \\mbox {if } \\ell \\in P_{\\min \\{r,s\\}},\\\\ \n n- \\max \\{r,s\\} & \\mbox {if } \\ell \\in P_{\\max \\{r,s\\}}\\backslash P_{\\min \\{r,s\\}},\\\\\n \\rho_\\ell -e_\\ell & \\mbox {if } \\ell \\in \\bar P_{\\max \\{r,s\\}}.\n \\end{cases}\n \\end{cases}\n\\end{align*} \n\\end{definition}\nHere is our main result. \n\\begin{theorem} \\label{rhoryserthmfullversion}\nAn $r\\times s$ $\\brho$-latin rectangle $L$ can be completed to an $n\\times n$ $\\brho$-latin square if and only if\n there exists a fitting sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ \nsuch that any of the following conditions\n\\begin{align*}\n |I|(n-s) &\\leq \\sum_{\\ell\\in P_r} \\min \\Big\\{a_\\ell+\\rho_\\ell-e_\\ell-n+r, \\mu_I(\\ell)\\Big\\}+ \\sum_{\\ell\\in \\bar P_r} \\min \\Big\\{a_\\ell, \\mu_I(\\ell)\\Big\\} &\\forall I\\subseteq [r],\\\\\n \\sum_{\\ell\\in (K\\cap P_r)} (\\rho_\\ell-e_\\ell) &\\leq \\sum_{i\\in [r]} \\min \\Big\\{n-s, \\mu_K(i)\\Big\\}- \\sum_{\\ell\\in K} a_\\ell +|K\\cap P_r|(n-r) &\\forall K\\subseteq [k],\\\\\n |I|(n-s) &\\geq \\sum_{\\ell\\in P_r} \\Big( a_\\ell+\\rho_\\ell+\\mu_I(\\ell)\\dotdiv n\\Big)+\\sum_{\\ell\\in \\bar P_r} \\Big( a_\\ell\\dotdiv \\mu_{\\bar I}(\\ell)\\Big) &\\forall I\\subseteq [r],\\\\ \n \\sum_{\\ell\\in (K\\cap P_r)}(\\rho_\\ell-e_\\ell) &\\geq \\sum_{i\\in [r]} \\Big (n-s\\dotdiv \\mu_{\\bar K}(i)\\Big) + |K\\cap P_r|(n-r) - \\sum_{\\ell\\in K} a_\\ell &\\forall K\\subseteq [k],\\\\\n |I|(n-s) &\\leq \\sum_{\\ell\\in (K\\cap P_r)} (\\rho_\\ell-e_\\ell) +\\sum_{\\ell\\in K} a_\\ell+ \\mu_I(\\bar K) -|K\\cap P_r|(n-r) &\\forall I\\subseteq [r], K\\subseteq [k], \\\\\n \\sum_{\\ell\\in (K\\cap P_r)} (\\rho_\\ell-e_\\ell) &\\leq |I|(n-s)+|K\\cap P_r|(n-r) + \\mu_K(\\bar I)- \\sum_{\\ell\\in K} a_\\ell &\\forall I\\subseteq [r], K\\subseteq [k],\n\\end{align*}\ntogether with any of the following conditions hold.\n\\begin{align*}\n |J|(n-r) &\\leq \\sum_{\\ell\\in P_s} \\min \\Big\\{b_\\ell+\\rho_\\ell-e_\\ell-n+s, \\mu_J(\\ell)\\Big\\} + \\sum_{\\ell\\in \\bar P_s} \\min \\Big\\{b_\\ell, \\mu_J(\\ell)\\Big\\} &\\forall J\\subseteq [s],\\\\\n \\sum_{\\ell\\in (K\\cap P_s)} (\\rho_\\ell-e_\\ell) &\\leq \\sum_{j\\in [s]} \\min \\Big\\{n-r, \\mu_K(j)\\Big\\}- \\sum_{\\ell\\in K} b_\\ell +|K\\cap P_s|(n-s) &\\forall K\\subseteq [k],\\\\\n |J|(n-r) &\\geq \\sum_{\\ell\\in [k]} \\Big( b_\\ell+\\rho_\\ell+\\mu_J(\\ell)\\dotdiv n\\Big) +\\sum_{\\ell\\in \\bar P_s} \\Big( b_\\ell\\dotdiv \\mu_{\\bar J}(\\ell)\\Big) &\\forall J\\subseteq [s],\\\\ \n \\sum_{\\ell\\in (K\\cap P_s)}(\\rho_\\ell-e_\\ell) &\\geq \\sum_{j\\in [s]} \\Big (n-r\\dotdiv \\mu_{\\bar K}(j)\\Big) + |K\\cap P_s|(n-s) - \\sum_{\\ell\\in K} b_\\ell &\\forall K\\subseteq [k],\\\\\n |J|(n-r) &\\leq \\sum_{\\ell\\in (K\\cap P_s)} (\\rho_\\ell-e_\\ell) + \\sum_{\\ell\\in K} b_\\ell+ \\mu_J(\\bar K)-|K\\cap P_s|(n-s) &\\forall J\\subseteq [s], K\\subseteq [k], \\\\\n \\sum_{\\ell\\in (K\\cap P_s)} (\\rho_\\ell-e_\\ell) &\\leq |J|(n-r)+|K\\cap P_s|(n-s) + \\mu_K(\\bar J) - \\sum_{\\ell\\in K} b_\\ell &\\forall J\\subseteq [s], K\\subseteq [k].\n\\end{align*}\n\\end{theorem}\n\n\nFor a much simpler generalization of Ryser's Theorem, see Corollary \\ref{cor2}.\n\\section{Tools}\nFor $x,y\\in \\mathbb{R}$, $x\\approx y$ means $\\lfloor y \\rfloor \\leq x\\leq \\lceil y \\rceil$. For a real-valued function $f$ on a domain $D$ and $S\\subseteq D$, $f(S) :=\\sum_{x\\in S}f(x)$. \n\n\nAll graphs under consideration are loopless, but they may have parallel edges. For a graph $G=(V,E)$, $u,v\\in V$ and $S,T\\subseteq V$ with $S\\cap T$, $\\dg_G(u)$, $\\mult_G(uv)$, $\\mult_G(uS)$, and $\\mult_G(ST)$ denote the number of edges incident with $u$, the number of edges whose endpoints are $u$ and $v$, the number of edges between $u$ and $S$, and the number of edges between $S$ and $T$, respectively. If the edges of $G$ are colored with $k$ colors (the set of colors is always $[k]$), then $G(\\ell)$ is the color class $\\ell$ of $G$ for $\\ell\\in [k]$. A bigraph $G$ with bipartition $\\{X, Y \\}$ will be denoted by $G[X, Y ]$, and for $S\\subseteq X$, $\\bar S:=X\\backslash S$. \n\n\nLet $G$ be a graph whose edges are colored, and let $\\alpha\\in V(G)$. By {\\it splitting} $\\alpha$ into $\\alpha_1,\\dots,\\alpha_p$, we obtain a new graph $F$ whose vertex set is $\\left(V(G)\\backslash \\{\\alpha\\}\\right) \\cup \\{\\alpha_1,\\dots, \\alpha_p\\}$ so that each edge $\\alpha u$ in $G$ becomes $\\alpha_i u$ for some $i\\in [p]$ in $F$. Intuitively speaking, when we {\\it split} a vertex $\\alpha$ into $\\alpha_1,\\dots,\\alpha_p$, we share the edges incident with $\\alpha$ among $\\alpha_1\\dots,\\alpha_p$. In this manner, $F$ is a {\\it detachment} of $G$, and $G$ is an {\\it amalgamation} of $F$ obtained by {\\it identifying} $\\alpha_1,\\dots, \\alpha_p$ by $\\alpha$. The following detachment lemma will be crucial in the proof of our main result. \n\\begin{lemma} \n\\label{amalgambahrod} \\cite{MR2946077}\n Let $G$ be a graph whose edges are colored with $k$ colors, and let $\\alpha,\\beta$ be two vertices of $G$. There exists a graph $F$ obtained by splitting $\\alpha$ and $\\beta$ into $\\alpha_1,\\dots,\\alpha_p$, and $\\beta_1,\\dots,\\beta_q$, respectively, such that the following conditions hold.\n\\begin{enumerate}\n \\item [\\textup{(i)}] $\\dg_{F(\\ell)}(\\alpha_i)\\approx \\dg_{G(\\ell)}(\\alpha)\/p$ for $i\\in [p],\\ell\\in [k]$;\n \\item [\\textup{(ii)}] $\\dg_{F(\\ell)}(\\beta_j)\\approx \\dg_{G(\\ell)}(\\beta)\/q$ for $j\\in [q],\\ell\\in [k]$;\n \\item [\\textup{(iii)}] $\\mult_F(\\alpha_i u)\\approx \\mult_G(\\alpha u)\/p$ for $i\\in[p],u\\in V(G)\\backslash \\{\\alpha,\\beta\\}$;\n \\item [\\textup{(iv)}] $\\mult_F(\\beta_j u)\\approx \\mult_G(\\beta u)\/q$ for $j\\in[q],u\\in V(G)\\backslash \\{\\alpha,\\beta\\}$;\n \\item [\\textup{(v)}] $\\mult_F(\\alpha_i \\beta_j)\\approx \\mult_G(\\alpha\\beta)\/(pq)$ for $i\\in[p],j\\in[q]$.\n\\end{enumerate} \n \n \n\\end{lemma}\nFor a hypergraph analogue, we refer the reader to \\cite{MR2942724}. To give the reader an idea about the usefulness of this detachment lemma, let us show how to construct $\\brho$-latin squares. Theorem \\ref{rholatconstthm} can be viewed as an immediate consequence of Theorem \\ref{rhohallthmgoldetal} (See \\cite[Theorem 5.1]{MR3280683}). \n\\begin{theorem} \\label{rholatconstthm} For every $n,k, \\brho:=(\\rho_1,\\dots,\\rho_k)$ with $1\\leq \\rho_1,\\dots,\\rho_k\\leq n\\leq k$ and $\\sum_{\\ell\\in [k]} \\rho_\\ell=n^2$, there exists a $\\brho$-latin square of order $n$.\n\\end{theorem}\n\\begin{proof}\nLet $G[\\{\\alpha\\}, \\{\\beta\\}]$ be a $k$-edge-colored bigraph with $\\mult_G(\\alpha \\beta)=n^2$, $\\mult_{G(\\ell)}(\\alpha \\beta)=\\rho_\\ell$ for $\\ell\\in [k]$. \nApplying the detachment lemma with $p=q=n$ yields the complete bigraph $F\\cong K_{n,n}$ whose colored edges corresponds to symbols in the desired $\\brho$-latin square of order $n$. \n\\end{proof}\n Let $f,g$ be integer functions on the vertex set of a graph $G$ such that $0\\leq g(x)\\leq f(x)$ for all $x$. A {\\it $(g,f)$-factor} is a spanning subgraph $F$ of $G$ with the property that $g(x)\\leq \\dg_F(x)\\leq f(x)$ for each $x$, and an {\\it $f$-factor} is an $(f,f)$-factor. We need the following result which is known as Ore's Theorem. For far reaching generalizations of Ore's Theorem, we refer the reader to Lov\\'{a}sz's seminal paper \\cite{MR325464}. \n\\begin{theorem}\\cite{MR83725} \\label{OresThm}\nThe bipgraph $G[X,Y]$ has an $f$-factor if and only if $f(X)=f(Y)$ and either one of the following conditions hold.\n\\begin{align*}\n f(A)&\\leq \\sum_{u\\in Y} \\min\\Big\\{f(u), \\mult_G(uA)\\Big\\} &\\forall A\\subseteq X,\\\\\n f(A) &\\leq f(B) + \\mult_G(A\\bar B) &\\forall A\\subseteq X, B\\subseteq Y.\n\\end{align*}\n\\end{theorem} \n\\begin{remark} \\label{ordenewcond} \\textup{\nLet us fix $A\\subseteq X$. For $u\\in Y$, we have\n\\begin{align*}\n \\kappa (u)&:=\\min\\Big\\{f(u), \\mult_G(uA)\\Big\\}+ \\Big (f(u)\\dotdiv \\mult_G(u A)\\Big)\\\\\n &=\n \\begin{cases}\n \\mult_G(uA)+ \\Big (f(u)- \\mult_G(u A)\\Big) & \\mbox{if } f(u)\\geq \\mult_G(uA) \\\\\n f(u)+0 & \\mbox{if } f(u)< \\mult_G(uA)\n \\end{cases}\\\\\n &=f(u).\n\\end{align*}\nSo, $\\sum_{u\\in Y} \\kappa(u)=f(Y)$, and the first condition in Theorem \\ref{OresThm} is equivalent to\n\\begin{align*}\n f(A)&\\leq f(Y)-\\sum_{u\\in Y} \\Big (f(u)\\dotdiv \\mult_G(u A)\\Big),\n\\end{align*}\nwhich is equivalent to \n\\begin{align*}\n f(\\bar A)\\geq \\sum_{u\\in Y} \\Big (f(u)\\dotdiv \\mult_G(u A)\\Big).\n\\end{align*}\n}\\end{remark}\n\n\n\\section{{\\normalfont Hall's Theorem for }\\protect\\boldmath$\\rho${\\normalfont-latin Rectangles} }\nHall \\cite{MR13111} showed that any $r\\times n$ latin rectangle can be extended to an $n\\times n$ latin square. Here we give a short proof of the main result of \\cite{MR3280683} which generalizes Hall's theorem to $\\brho$-latin rectangles. \n\n\n\n\\begin{theorem} \\label{rhohallthm}\nAn $r\\times n$ $\\brho$-latin rectangle $L$ can be completed to an $n\\times n$ $\\brho$-latin square if and only if $\\rho_\\ell - e_\\ell\\leq n-r$ for $\\ell\\in [k]$, and any of the following conditions hold.\n\\begin{align*}\n |J|(n-r) &\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{\\rho_\\ell-e_\\ell, \\mu_J(\\ell)\\Big\\} &\\forall J\\subseteq [n],\\\\\n \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell) &\\leq \\sum_{j\\in [n]} \\min \\Big\\{n-r, \\mu_K(j)\\Big\\} &\\forall K\\subseteq [k],\\\\%\\label{orenec1equiv}\\\\\n |J|(n-r) &\\geq \\sum_{\\ell\\in [k]} \\Big( \\rho_\\ell-e_\\ell\\dotdiv \\mu_{\\bar J}(\\ell)\\Big) &\\forall J\\subseteq [n],\\\\ \n \\sum_{\\ell\\in K}(\\rho_\\ell-e_\\ell) &\\geq \\sum_{j\\in [n]} \\Big (n-r\\dotdiv \\mu_{\\bar K}(j)\\Big) &\\forall K\\subseteq [k],\\\\%\\label{anothercondequi}\\\\\n |J|(n-r) &\\leq \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell) + \\mu_J(\\bar K) &\\forall J\\subseteq [n], K\\subseteq [k], \\\\%\\label{orecond2}\\\\\n \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell) &\\leq |J|(n-r) + \\mu_K(\\bar J) &\\forall J\\subseteq [n], K\\subseteq [k]\n\\end{align*}\n\\end{theorem}\n \\begin{proof} Recall that the set of symbols used is $[k]$. The necessity of \n\\begin{align} \\label{ryscond1}\n \\rho_\\ell - e_\\ell&\\leq n-r\\quad \\forall \\ell\\in [k]\n\\end{align}\nis immediate from ~\\eqref{mainryserineq}. The necessity of the remaining conditions will be evident at the end of the proof. For now, let us assume that \\eqref{ryscond1} is satisfied. Let $F[\\tilde X, Y]$ be the complete bigraph $K_{n,n}$ where $\\tilde X:=\\{x_1,\\dots,x_n\\}, Y:=\\{y_1,\\dots, y_n\\}$, and let $X=\\{x_1,\\dots, x_r\\}$. The edge $x_i y_{j}$ of $F$ is colored $\\ell$ if $L_{ij}=\\ell$ for $i\\in[r],j\\in[n]$. We have $\\dg_{F(\\ell)}(u)\\leq 1$ for $u\\in X\\cup Y,$ $e_\\ell=|E(F(\\ell))|\\leq \\rho_\\ell$ for $\\ell\\in [k]$, and $\\sum_{\\ell\\in [k]}e_\\ell=rn$. \nLet $G$ be the bigraph obtained from $F$ by amalgamating $x_{r+1},\\dots, x_n$ into a single vertex $\\alpha$, so $\\mult_G(\\alpha y_j)=n-r$ for $j\\in [n]$. Let $\\Gamma [Y, [k]]$ be the simple bigraph whose edge set is $$\\{u \\ell\\ \\big|\\ u\\in Y, \\ell\\in [k], \\dg_{F(\\ell)}(u)=0\\}.$$ \nFor $j\\in [n]$, $\\sum_{\\ell\\in [k]}\\dg_{F(\\ell)}(y_j)=r$, and for $\\ell\\in [k]$, there are $e_\\ell$ edges in $F(\\ell)$. Therefore, \n\\begin{align*\n\\begin{cases}\n\\dg_\\Gamma(y_j)=k-r & \\mbox{if } j\\in [n], \\\\\n\\dg_{\\Gamma}(\\ell)=n-e_\\ell & \\mbox{if } \\ell\\in [k].\n\\end{cases}\n\\end{align*}\nObserve that $L$ can be completed if and only if the uncolored edges of $F$ can be colored so that\n\\begin{align}\\label{colorcon1s}\n\\forall \\ell\\in [k], \\quad \\quad \\begin{cases}\n\\dg_{F(\\ell)}(u)\\leq 1 & \\mbox{if } u\\in \\tilde X\\cup Y, \\\\\n|E(F(\\ell))|=\\rho_\\ell.\n\\end{cases}\n\\end{align}\nWe show that the coloring of $F$ can be completed such that \\eqref{colorcon1s} holds if and only if the coloring of $G$ can be completed so that \n\\begin{align}\\label{colorcon2s}\n\\forall \\ell\\in [k],\\quad \\quad \\begin{cases}\n\\dg_{G(\\ell)}(u) \\leq 1 & \\mbox{if } u \\in X \\cup Y, \\\\\n\\dg_{G(\\ell)}(\\alpha) \\leq n-r, \\\\\n|E(G(\\ell))|= \\rho_\\ell.\n\\end{cases}\n\\end{align}\nTo see this, first assume that the coloring of $F$ can be completed such that \\eqref{colorcon1s} holds. Identifying all the vertices in $\\tilde X\\backslash X$ of $F$ by $\\alpha$ we will get the graph $G$ satisfying \\eqref{colorcon2s}. Conversely, suppose that we have a coloring of $G$\nsuch that \\eqref{colorcon2s} holds. Applying Lemma \\ref{amalgambahrod} to $G$, we get a graph $F'$ obtained by splitting $\\alpha$ into $\\alpha_1,\\dots,\\alpha_{n-r}$ such that the following hold.\n\\begin{enumerate}\n \\item [(i)] $\\dg_{F'(\\ell)}(\\alpha_i)\\approx\\dg_{G(\\ell)}(\\alpha)\/(n-r)\\leq 1$ for $i\\in [n-r],\\ell\\in [k]$;\n \\item [(ii)] $\\mult_{F'}(\\alpha_i u)= \\mult_G(\\alpha u)\/(n-r)=1$ for $i\\in[n-r],u\\in Y$.\n\\end{enumerate}\nSince $F'\\cong F$ and the coloring of $F'$ satisfies \\eqref{colorcon1s}, we are done. \n\nNow, we show that the coloring of $G$ can be completed such that \\eqref{colorcon2s} is satisfied if and only if there exists a subgraph $\\Theta$ of $\\Gamma$ with $n(k-r)$ edges so that \n\\begin{align}\\label{colorcon3s}\n\\begin{cases}\n\\dg_\\Theta (y_j)=n-r & \\mbox{if } j\\in [n], \\\\\n\\dg_\\Theta (\\ell)=\\rho_\\ell-e_\\ell & \\mbox{if } \\ell \\in [k].\n\\end{cases}\n\\end{align}\nTo prove this, suppose that the coloring of $G$ can be completed such that \\eqref{colorcon2s} holds. Let $\\Theta [ Y, [k]]$ be the bigraph whose edge set is \n$$\\{u\\ell\\ \\big|\\ u\\in Y, \\ell\\in [k], \\mbox{ and }\\alpha u\\in E(G(\\ell)) \\}.$$ \nObserve that $\\Theta \\subseteq \\Gamma$. Moreover, $\\dg_\\Theta(y_j)=\\mult_G(\\alpha y_j)=n-r$ for $j\\in [n]$, and $\\dg_{\\Theta}(\\ell)= |E(G(\\ell))|-e_\\ell=\\rho_\\ell-e_\\ell$ for $\\ell\\in [k]$, and so \\eqref{colorcon3s} holds. Conversely, suppose that $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3s} exists. For each $\\ell\\in [k]$, if $\\ell y_j \\in E(\\Theta)$ for some $j\\in [n]$, we color an $\\alpha y_j$-edge in $G$ with $\\ell$. Since $\\dg_\\Theta(y_j)=n-r$ for $j\\in [n]$, all the edges between $\\alpha$ and $Y$ can be colored this way. Since $\\Theta$ is simple, $d_{G(\\ell)}(u)\\leq 1$ for $\\ell\\in [k]$ and $u\\in Y$. It is also clear that $|E(G(\\ell))|= \\rho_\\ell-e_\\ell+e_\\ell=\\rho_\\ell$, and by ~\\eqref{ryscond1} \n$\\dg_{G(\\ell)}(\\alpha)=\\dg_\\Theta(\\ell)=\\rho_\\ell-e_\\ell \\leq n-r$ for $\\ell\\in [k]$. \n\n\nLet \n\\begin{align*}\n \\begin{cases}\n f: V(\\Gamma)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n f(y_j)=n-r & \\mbox{for } j\\in [n],\\\\\n f(\\ell)=\\rho_\\ell-e_\\ell & \\mbox{for }\\ell\\in [k]. \n \\end{cases}\n\\end{align*}\n Since $f(Y)= f([k])$, by Ore's Theorem and Remark \\ref{ordenewcond}, $\\Gamma$ has an $f$-factor (and so $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3s} exists) if and only if any of the following conditions holds. \n\\begin{align*}\n f(J)&\\leq \\sum_{\\ell\\in [k]} \\min\\Big\\{f(\\ell), \\mult_\\Gamma(\\ell J)\\Big\\} &\\forall J\\subseteq Y,\\\\\n f(K)&\\leq \\sum_{u\\in Y} \\min\\Big\\{f(u), \\mult_\\Gamma(uK)\\Big\\} &\\forall K\\subseteq [k],\\\\\n f(J)&\\geq \\sum_{\\ell\\in [k]} \\Big (f(\\ell)\\dotdiv \\mult_\\Gamma(\\ell \\bar J)\\Big) &\\forall J\\subseteq Y,\\\\\n f(K)&\\geq \\sum_{u\\in Y} \\Big (f(u)\\dotdiv \\mult_\\Gamma(u \\bar K)\\Big) &\\forall K\\subseteq [k],\\\\\n f(J) &\\leq f(K) + \\mult_\\Gamma(J\\bar K) &\\forall K\\subseteq [k], J\\subseteq Y,\\\\ f(K) &\\leq f(J) + \\mult_\\Gamma(K\\bar J) &\\forall K\\subseteq [k], J\\subseteq Y.\n\\end{align*}\nBut these conditions are equivalent to those of Theorem \\ref{rhohallthm}. \n\\end{proof}\n\n\\section{{\\normalfont Ryser's Theorem for }\\protect\\boldmath$\\rho${\\normalfont-latin Rectangles} } \\label{rhorythmprfsec}\nSeventy years ago, Ryser \\cite{MR42361} showed that any $r\\times s$ latin rectangle can be extended to an $n\\times n$ latin square if and only if $e_\\ell\\geq r+s-n$ for $\\ell\\in [n]$. We provide a generalization of this result to $\\brho$-latin rectangles by proving the following Theorem \\ref{rhoryserthmfullversion}. For a simpler generalization of Ryser's Theorem, see Corollary \\ref{cor2}.\n\n\n\\noindent \\textit{Proof of Theorem \\ref{rhoryserthmfullversion}.}\nRecall that the set of symbols used is $[k]$. The existence of a fitting sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ implies ~\\eqref{mainryserineq}. To see this, observe that by Definition \\ref{fitdef},\n$$\n0\\leq a_\\ell+b_\\ell\\leq 2n- r-s+ e_\\ell-\\rho_\\ell \\quad \\mbox {for } \\ell\\in P_{\\min\\{r,s\\}}. \n$$\nMoreover, using the definition of $P_{\\min\\{r,s\\}}$, we have\n$$\\rho_\\ell-e_\\ell \\leq n- \\min\\{r,s\\}\\leq 2n-r-s \\quad \\mbox{for } \\ell\\in \\bar P_{\\min\\{r,s\\}}.$$ \nThe necessity of all conditions will be evident at the end of the proof. For now, let us assume that ~\\eqref{mainryserineq} --- whose necessity was established in the introduction --- holds. \nLet $F[\\tilde X,\\tilde Y]$ be the complete bigraph $K_{n,n}$ where $\\tilde X:=\\{x_1,\\dots,x_n\\}, \\tilde Y:=\\{y_1,\\dots, y_n\\}$, and let $X=\\{x_1,\\dots, x_r\\},Y=\\{y_1,\\dots,y_s\\}$. The edge $x_i y_j$ of $F$ is colored $\\ell$ if $L_{ij}=\\ell$ for $i\\in[r],j\\in[s]$. We have $\\dg_{F(\\ell)}(u)\\leq 1$ for $u\\in X\\cup Y, \\ell\\in [k]$, $e_\\ell=|E(F(\\ell))|\\leq \\rho_\\ell$ for $\\ell\\in [k]$, and $\\sum_{\\ell\\in [k]}e_\\ell=rs$ (See Figure \\ref{fig1rhoryser} (b)). \nLet $G$ be the bigraph obtained from $F$ by amalgamating $x_{r+1},\\dots, x_n$ into a single vertex $\\alpha$, and amalgamating $x_{s+1},\\dots, y_n$ into a single vertex $\\beta$, so $\\mult_G(\\alpha y_j)=n-r$ for $j\\in [s]$, $\\mult_G(\\beta x_i)=n-s$ for $ i\\in [r]$, and $\\mult_G(\\alpha \\beta)=(n-r)(n-s)$ (See Figure \\ref{fig1rhoryser} (c)). Let $\\Gamma [X\\cup Y, [k]]$ be the simple bigraph whose edge set is \n$$\\{u\\ell\\ \\big |\\ u\\in X\\cup Y, \\ell\\in [k], \\dg_{F(\\ell)}(u)=0\\}.$$ \nFor $i\\in [r]$, since $\\sum_{\\ell\\in [k]}\\dg_{F(\\ell)}(x_i)=s$, we have $\\dg_\\Gamma(x_i)=k-s$. Similarly, for $j\\in [s]$, $\\sum_{\\ell\\in [k]}\n\\dg_{F(\\ell)}(y_j)=r$, and so we have $\\dg_\\Gamma(y_j)=k-r$. For $\\ell\\in [k]$, there are $e_\\ell$ edges in $F(\\ell)$, and so $\\mult_{\\Gamma}(\\ell X)=r-e_\\ell$ and $\\mult_{\\Gamma}(\\ell Y)=s-e_\\ell$ (Recall that by the latin property of $L$, $e_\\ell \\leq \\min \\{r,s\\}$). Therefore, $\\dg_\\Gamma(\\ell)=r+s-2e_\\ell$ for $\\ell\\in [k]$. In short, $\\Gamma$ meets the following properties (See Figure \\ref{fig1rhoryser} (d)).\n\\begin{align*\n\\begin{cases}\n\\dg_\\Gamma(x_i)=k-s & \\mbox{if } i\\in [r], \\\\\n\\dg_\\Gamma(y_j)=k-r & \\mbox{if } j\\in [s], \\\\\n\\mult_{\\Gamma}(\\ell X)=r-e_\\ell & \\mbox{if } \\ell \\in [k],\\\\\n\\mult_{\\Gamma}(\\ell Y)=s-e_\\ell & \\mbox{if } \\ell \\in [k],\\\\\n\\dg_\\Gamma(\\ell)=r+s-2e_\\ell & \\mbox{if } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\n\\begin{figure}[p] \n \\begin{adjustbox}{addcode={\\begin{minipage}{\\width}}{\\caption{\n A $\\rho$-latin rectangle and three associated auxiliary bigraphs\n }\\end{minipage}},rotate=90,center} \n \\includegraphics[scale=1.2]{fig1} \\end{adjustbox}\n \\label{fig1rhoryser} \n\\end{figure}\n\n\n\n\n\n\n\nObserve that $L$ can be completed if and only if the uncolored edges of $F$ can be colored so that\n\\begin{align}\\label{colorcon1}\n\\forall \\ell\\in [k],\\quad \\quad \\begin{cases}\n\\dg_{F(\\ell)}(u)\\leq 1 & \\mbox{if } u\\in \\tilde X \\cup \\tilde Y, \\\\\n|E(F(\\ell))|=\\rho_\\ell.\n\\end{cases}\n\\end{align}\n\nWe show that the coloring of $F$ can be completed such that \\eqref{colorcon1} is satisfied if and only if the coloring of $G$ can be completed such that,\n\\begin{align}\\label{colorcon2}\n\\forall \\ell\\in [k],\\quad \\quad \\begin{cases}\n\\dg_{G(\\ell)}(u) \\leq 1 & \\mbox{if } u \\in X \\cup Y, \\\\\n\\dg_{G(\\ell)}(\\alpha) \\leq n-r, \\\\\n\\dg_{G(\\ell)}(\\beta) \\leq n-s, \\\\\n|E(G(\\ell))|= \\rho_\\ell.\n\\end{cases}\n\\end{align}\nTo see this, first assume that the coloring of $F$ can be completed so that \\eqref{colorcon1} holds. Identifying all the vertices in $\\tilde X\\backslash X$ by $\\alpha$ and all the vertices in $\\tilde Y\\backslash Y$ by $\\beta$, we will get the graph $G$ satisfying \\eqref{colorcon2}. Conversely, suppose that we have a coloring of $G$\nsuch that \\eqref{colorcon2} holds. Applying the detachment lemma to $G$, we get a graph $F'$ obtained by splitting $\\alpha$ and $\\beta$ into $\\alpha_1,\\dots,\\alpha_{n-r}$, and $\\beta_1\\dots,\\beta_{n-s}$, respectively, such that the following hold.\n\\begin{enumerate}\n\\item [(i)] $\\dg_{F'(\\ell)}(\\alpha_i)\\approx\\dg_{G(\\ell)}(\\alpha)\/(n-r)\\leq 1$ for $i\\in [n-r],\\ell\\in [k]$;\n\\item [(ii)] $\\dg_{F'(\\ell)}(\\beta_j)\\approx \\dg_{G(\\ell)}(\\beta)\/(n-s)\\leq 1$ for $j\\in [n-s],\\ell\\in [k]$;\n\\item [(iii)] $\\mult_{F'}(\\alpha_i u)= \\mult_G(\\alpha u)\/(n-r)=1$ for $i\\in[n-r],u\\in Y$;\n\\item [(iv)] $\\mult_{F'}(\\beta_j u)= \\mult_G(\\beta u)\/(n-s)=1$ for $j\\in[n-s],u\\in X$;\n\\item [(v)] $\\mult_{F'}(\\alpha_i \\beta_j)= \\mult_G(\\alpha\\beta)\/\\big((n-r)(n-s)\\big)=1$ for $i\\in[n-r],j\\in[n-s]$.\n\\end{enumerate}\n Since $F'\\cong F$ and the coloring of $F'$ satisfies \\eqref{colorcon1}, we are done. \n\nFor $\\ell\\in [k]$, since $\\rho_\\ell\\leq n$, we have that $\\rho_\\ell-e_\\ell-n+r\\leq r-e_\\ell$ and $\\rho_\\ell-e_\\ell-n+s\\leq s-e_\\ell$. We show that the coloring of $G$ can be completed such that \\eqref{colorcon2} is satisfied if and only if there exists a subgraph $\\Theta$ of $\\Gamma$ with $n(r+s)-2rs$ edges so that \n\\begin{align}\\label{colorcon3}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if } i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r & \\mbox{if } j\\in [s], \\\\\n\\dg_\\Theta (\\ell)\\leq \\rho_\\ell-e_\\ell & \\mbox{if } \\ell \\in [k],\\\\\n\\mult_\\Theta (\\ell X)\\geq \\rho_\\ell-e_\\ell-n+r & \\mbox{if } \\ell \\in [k],\\\\\n\\mult_\\Theta (\\ell Y)\\geq \\rho_\\ell-e_\\ell-n+s & \\mbox{if } \\ell \\in [k].\n\\end{cases}\n\\end{align}\n\n\nTo prove this, suppose that the coloring $G$ can completed such that \\eqref{colorcon2} is satisfied. Let $\\Theta [X\\cup Y, [k]]$ be the bigraph whose edge set is \n$$\\{u \\ell\\ \\big |\\ u\\in X\\cup Y, \\ell\\in [k], \\mbox{ and either }\\alpha u\\in E(G(\\ell)) \\mbox{ or } \\beta u \\in E(G(\\ell)) \\}.$$ \nIt is clear that $\\Theta \\subseteq \\Gamma$. \nMoreover, \n $\\dg_\\Theta(x_i)=\\mult_G(\\beta x_i)=n-s$ for $i\\in [r]$, $\\dg_\\Theta(y_j)=\\mult_G(\\alpha y_j)=n-r$ for $j\\in [s]$, and $\\dg_{\\Theta}(\\ell)\\leq |E(G(\\ell))|-e_\\ell=\\rho_\\ell-e_\\ell$ for $\\ell\\in [k]$. Notice that \n $$\\rho_\\ell=|E(G(\\ell))|=e_\\ell+\\mult_{G(\\ell)}(\\alpha \\beta) +\\mult_{G(\\ell)} (\\beta X) + \\mult_{G(\\ell)} (\\alpha Y)\\quad \\mbox {for }\\ell\\in [k].$$\nThus, for $\\ell\\in [k]$ we have \n \\begin{align*}\n\\mult_\\Theta (\\ell X) &=\\mult_{G(\\ell)} (\\beta X) \\\\\n& = \\rho_\\ell-e_\\ell-\\mult_{G(\\ell)}(\\alpha \\beta) - \\mult_{G(\\ell)} (\\alpha Y)\\\\\n &= \\rho_\\ell-e_\\ell-\\dg_{G(\\ell)}(\\alpha) \\\\\n& \\geq \\rho_\\ell-e_\\ell-n+r,\\\\\n\\mult_\\Theta (\\ell Y)& =\\mult_{G(\\ell)} (\\alpha Y) \\\\\n &= \\rho_\\ell-e_\\ell-\\mult_{G(\\ell)}(\\alpha \\beta) - \\mult_{G(\\ell)} (\\beta X)\\\\\n& = \\rho_\\ell-e_\\ell-\\dg_{G(\\ell)}(\\beta) \\\\\n& \\geq \\rho_\\ell-e_\\ell-n+s,\n\\end{align*}\nand so \\eqref{colorcon3} holds. Observe that $\\rho_\\ell-e_\\ell \\geq \\dg_\\Theta (\\ell)= \\mult_\\Theta (\\ell X)+ \\mult_\\Theta (\\ell Y)\\geq \\rho_\\ell-e_\\ell-n+r + \\rho_\\ell-e_\\ell-n+s$; this is consistent with ~\\eqref{mainryserineq}. \n\n\n\nConversely, suppose that $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists. For each $\\ell\\in [k]$, if $\\ell x_i \\in E(\\Theta)$ for some $i\\in [r]$, we color a $\\beta x_i$-edge in $G$ with $\\ell$, and if $\\ell y_j \\in E(\\Theta)$ for some $j\\in [s]$, we color an $\\alpha y_j$-edge in $G$ with $\\ell$. Since $\\dg_\\Theta(x_i)=n-s$ for $i\\in [r]$, and $\\dg_\\Theta(y_j)=n-r$ for $j\\in [s]$, all the edges between $\\beta$ and $X$, and all the edges between $\\alpha$ and $Y$ can be colored this way. Since $\\Theta$ is simple, $d_{G(\\ell)}(u)\\leq 1$ for $\\ell\\in [k]$ and $u\\in X\\cup Y$. Then, we color the $\\alpha \\beta$-edges so that\n$$\\mult_{G(\\ell)}(\\alpha \\beta) = \\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\quad \\mbox {for }\\ell\\in [k].$$\nThis can be done, because $ \\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\geq 0$ for $\\ell\\in [k]$, and \n\\begin{align*}\n \\sum_{\\ell\\in [k]}\\mult_{G(\\ell)}(\\alpha \\beta)&=n^2-rs-n(r+s)+2rs\\\\\n &=(n-r)(n-s)\\\\&=\\mult_G(\\alpha \\beta).\n\\end{align*}\nFor $\\ell\\in [k]$, we have\n \\begin{align*}\n\\dg_{G(\\ell)} (\\alpha) &=\\mult_{G(\\ell)} (\\alpha Y)+ \\mult_{G(\\ell)} (\\alpha \\beta) \\\\\n &= \\mult_\\Theta (\\ell Y)+\\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\\\\n &= \\rho_\\ell-e_\\ell-\\mult_\\Theta (\\ell X)\\\\\n& \\leq n-r,\\\\\n\\dg_{G(\\ell)} (\\beta) &=\\mult_{G(\\ell)} (\\beta X)+ \\mult_{G(\\ell)} (\\alpha \\beta) \\\\\n& = \\mult_\\Theta (\\ell X)+\\rho_\\ell-e_\\ell-\\dg_\\Theta (\\ell)\\\\\n& = \\rho_\\ell-e_\\ell-\\mult_\\Theta (\\ell Y)\\\\\n& \\leq n-s,\n\\end{align*}\nand so \\eqref{colorcon2} is satisfied. \n\n\nLet $\\Gamma_1, \\Gamma_2$ be the subgraphs of $\\Gamma$ induced by $X\\cup [k]$, and $Y\\cup [k]$, respectively. The existence of $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} is equivalent to the existence of $\\Theta_1\\subseteq \\Gamma_1, \\Theta_2\\subseteq \\Gamma_2$ satisfying the following condition.\n\\begin{align*\n\\begin{cases}\n\\dg_{\\Theta_1} (x_i)=n-s & \\mbox{if } i\\in [r], \\\\\n\\dg_{\\Theta_1} (\\ell)\\geq \\rho_\\ell-e_\\ell-n+r & \\mbox{if } \\ell \\in [k],\\\\\n\\dg_{\\Theta_2} (y_j)=n-r & \\mbox{if } j\\in [s], \\\\\n\\dg_{\\Theta_2} (\\ell)\\geq \\rho_\\ell-e_\\ell-n+s & \\mbox{if } \\ell \\in [k],\\\\\n \\dg_{\\Theta_1} (\\ell)+\\dg_{\\Theta_2} (\\ell)\\leq \\rho_\\ell-e_\\ell & \\mbox{if } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\nThe three inequalities of this condition are equivalent to the existence of a sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ with $a_\\ell,b_\\ell\\in \\mathbb{N} \\cup \\{0\\}$ for $\\ell\\in [k]$ such that the following three properties are satisfied (Recall that $P_{\\min \\{r,s\\}}\\subseteq P_{\\max \\{r,s\\}}$).\n\n\\begin{align*}\n\\dg_{\\Theta_1} (\\ell)&=\\begin{cases}\na_\\ell+\\rho_\\ell-e_\\ell-n+r & \\mbox{if } \\ell \\in P_r,\\\\\na_\\ell & \\mbox{if } \\ell \\in \\bar P_r,\n\\end{cases}\\\\\n\\dg_{\\Theta_2} (\\ell)&=\\begin{cases}\nb_\\ell+\\rho_\\ell-e_\\ell-n+s & \\mbox{if } \\ell \\in P_s,\\\\\nb_\\ell & \\mbox{if } \\ell \\in \\bar P_s,\n\\end{cases}\\\\\n\\rho_\\ell-e_\\ell&\\geq\n\\begin{cases}\na_\\ell+b_\\ell+2\\rho_\\ell-2e_\\ell-2n+r+s & \\mbox{if } \\ell \\in P_{\\min \\{r,s\\}},\\\\\na_\\ell+b_\\ell+\\rho_\\ell-e_\\ell-n+\\max \\{r,s\\} & \\mbox{if } \\ell \\in P_{\\max \\{r,s\\}}\\backslash P_{\\min \\{r,s\\}},\\\\\na_\\ell+b_\\ell & \\mbox{if } \\ell \\in \\bar P_{\\max \\{r,s\\}},\n\\end{cases}\n\\end{align*}\nwhere the last property simplifies to the following.\n\\begin{align} \\label{abseqineqpro}\n a_\\ell+b_\\ell\\leq \\begin{cases}\n 2n- r-s+ e_\\ell-\\rho_\\ell& \\mbox {if } \\ell \\in P_{\\min \\{r,s\\}},\\\\\n n- \\max \\{r,s\\} & \\mbox {if } \\ell\\in \\ell \\in P_{\\max \\{r,s\\}}\\backslash P_{\\min \\{r,s\\}},\\\\\n \\rho_\\ell -e_\\ell & \\mbox {if } \\ell \\in \\bar P_{\\max \\{r,s\\}}.\n \\end{cases}\n\\end{align} \nWe define $f_1,f_2$ as follows. \n\\begin{align*}\n \\begin{cases}\n f_1: V(\\Gamma_1)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n f_1(x_i)=n-s & \\mbox{if } i\\in [r],\\\\\n f_1(\\ell)=a_\\ell+\\rho_\\ell-e_\\ell-n+r & \\mbox{if }\\ell\\in P_r, \\\\ \n f_1(\\ell)=a_\\ell & \\mbox{if }\\ell\\in \\bar P_r, \n \\end{cases}\n\\end{align*}\n\\begin{align*}\n \\begin{cases}\n f_2: V(\\Gamma_2)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n f_2(y_j)=n-r & \\mbox{for } i\\in [s],\\\\\n f_2(\\ell)=b_\\ell+\\rho_\\ell-e_\\ell-n+s & \\mbox{for }\\ell\\in P_s, \\\\ \n f_2(\\ell)=b_\\ell & \\mbox{for }\\ell\\in \\bar P_s. \n \\end{cases}\n\\end{align*}\n\n\n\nBy the preceding paragraph, $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists if and only if there exists a sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ satisfying \\eqref{abseqineqpro} such that $\\Gamma_1$ and $\\Gamma_2$ have an $f_1$-factor and $f_2$-factor, respectively. \nObserve that $f_1(X)=f_1([k])$ if and only if\n\\begin{align*}\n r(n-s)=\\sum_{\\ell\\in P_r} (a_\\ell+\\rho_\\ell-e_\\ell-n+r) + \\sum_{\\ell\\in \\bar P_r} a_\\ell,\n\\end{align*}\nor equivalently, \n\\begin{align} \\label{abseqsum1}\n \\sum_{\\ell\\in [k]} a_\\ell=r(n-s)+|P_r|(n-r)-\\sum_{\\ell\\in P_r} (\\rho_\\ell-e_\\ell).\n\\end{align}\nSimilarly, $f_2(Y)=f_2([k])$ if and only if\n\\begin{align*}\n s(n-r)=\\sum_{\\ell\\in P_s} (b_\\ell+\\rho_\\ell-e_\\ell-n+s) + \\sum_{\\ell\\in \\bar P_s} b_\\ell,\n\\end{align*}\nor equivalently, \n\\begin{align}\\label{abseqsum2}\n \\sum_{\\ell\\in [k]} b_\\ell=s(n-r)+|P_s|(n-s)-\\sum_{\\ell\\in P_s} (\\rho_\\ell-e_\\ell).\n\\end{align}\n\n\n\nBy Ore's Theorem, $\\Gamma_1$ has an $f_1$-factor if and only if \\eqref{abseqsum1} together with any of the following conditions hold.\n\\begin{align*}\n f_1(I)&\\leq \\sum_{\\ell\\in [k]} \\min\\Big\\{f_1(\\ell), \\mult_{\\Gamma_1}(\\ell I)\\Big\\} &\\forall I\\subseteq X,\\\\\n f_1(K)&\\leq \\sum_{u\\in X} \\min\\Big\\{f_1(u), \\mult_{\\Gamma_1}(uK)\\Big\\} &\\forall K\\subseteq [k],\\\\\n f_1(I)&\\geq \\sum_{\\ell\\in [k]} \\Big (f_1(\\ell)\\dotdiv \\mult_{\\Gamma_1}(\\ell \\bar I)\\Big) &\\forall I\\subseteq X,\\\\\n f_1(K)&\\geq \\sum_{u\\in X} \\Big (f_1(u)\\dotdiv \\mult_{\\Gamma_1}(u \\bar K)\\Big) &\\forall K\\subseteq [k],\\\\\n f_1(I) &\\leq f_1(K) + \\mult_{\\Gamma_1}(I\\bar K) &\\forall K\\subseteq [k], I\\subseteq X,\\\\\n f_1(K) &\\leq f_1(I) + \\mult_{\\Gamma_1}(K\\bar I) &\\forall K\\subseteq [k], I\\subseteq X,\n\\end{align*} \nSimilarly, $\\Gamma_2$ has an $f_2$-factor if and only if \\eqref{abseqsum2} and any of the following conditions hold.\n\\begin{align*}\n f_2(J)&\\leq \\sum_{\\ell\\in [k]} \\min\\Big\\{f_2(\\ell), \\mult_{\\Gamma_2}(\\ell J)\\Big\\} &\\forall J\\subseteq Y,\\\\\n f_2(K)&\\leq \\sum_{u\\in Y} \\min\\Big\\{f_2(u), \\mult_{\\Gamma_2}(uK)\\Big\\} &\\forall K\\subseteq [k],\\\\\n f_2(J)&\\geq \\sum_{\\ell\\in [k]} \\Big (f_2(\\ell)\\dotdiv \\mult_{\\Gamma_2}(\\ell \\bar J)\\Big) &\\forall J\\subseteq Y,\\\\\n f_2(K)&\\geq \\sum_{u\\in Y} \\Big (f_2(u)\\dotdiv \\mult_{\\Gamma_2}(u \\bar K)\\Big) &\\forall K\\subseteq [k],\\\\\n f_2(J) &\\leq f_2(K) + \\mult_{\\Gamma_2}(J\\bar K) &\\forall K\\subseteq [k], J\\subseteq Y,\\\\\n f_2(K) &\\leq f_2(J) + \\mult_{\\Gamma_2}(K\\bar J) &\\forall K\\subseteq [k], J\\subseteq Y.\n\\end{align*} \nThese conditions are equivalent to those of Theorem \\ref{rhoryserthmfullversion}.\n\\qed\n\\begin{remark} \\textup{\n\\begin{itemize}\n \\item [(a)] Let us verify that Theorem \\ref{rhoryserthmfullversion} implies Theorem \\ref{rhohallthm}. In Theorem \\ref{rhoryserthmfullversion}, let $s=n$. Recall that the existence of the fitting sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ implies that $\\rho_\\ell-e_\\ell \\leq n-r$ for $\\ell\\in [k]$. Therefore, $P_r=\\emptyset, P_n=\\{\\ell\\in [k] \\ |\\ \\rho_\\ell>e_\\ell\\}$. We have\n \\begin{align*}\n \\begin{cases}\n \\displaystyle\\sum_{\\ell\\in [k]} a_\\ell=0,\\\\ \\\\\n \\displaystyle\\sum_{\\ell\\in [k]} b_\\ell=n(n-r)-\\sum_{\\ell\\in P_n}(\\rho_\\ell-e_\\ell)=n(n-r)-\\sum_{\\ell\\in [k]}(\\rho_\\ell-e_\\ell)=0,\\\\ \n a_\\ell+b_\\ell\\leq \n \\begin{cases}\n n- r+ e_\\ell-\\rho_\\ell& \\mbox {if } \\ell \\in \\emptyset,\\\\\n 0 & \\mbox {if } \\ell \\in P_{n},\\\\\n \\rho_\\ell -e_\\ell=0 & \\mbox {if } \\ell \\in \\bar P_{n}.\n \\end{cases}\n \\end{cases}\n \\end{align*} \n Hence, there is a unique fitting sequence, namely, $a_\\ell=b_\\ell=0$ for $\\ell\\in [k]$.\nConsequently, the first six conditions of Theorem \\ref{rhoryserthmfullversion} will be trivial, and the remaining six conditions will be identical to those of Theorem \\ref{rhohallthm}. \n\\item [(b)] Now, we show that Ryser's Theorem is immediate from Theorem \\ref{rhoryserthmfullversion}. In Theorem \\ref{rhoryserthmfullversion}, if we let $k=n$ (so $L$ is a latin rectangle), then the existence of $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} is trivial. In fact, in this case, $\\Theta=\\Gamma$. \n\\end{itemize}\n}\\end{remark}\n\\begin{remark} \\textup{\nWe can restate Theorem \\ref{rhoryserthmfullversion} without introducing the sets $P_r$ and $P_s$. To do so, let us rewrite $f_1(\\ell)=a_\\ell+(\\rho_\\ell-e_\\ell+r\\dotdiv n)$, and $f_2(\\ell)=b_\\ell+(\\rho_\\ell-e_\\ell+s\\dotdiv n)$. The main condition in the definition of fitting sequence can be written in the following manner.\n\\begin{align*}\n\\begin{cases}\n \\displaystyle\\sum_{\\ell\\in [k]} a_\\ell=r(n-s)-\\sum_{\\ell\\in [k]}(\\rho_\\ell-e_\\ell+r\\dotdiv n),\\\\[20pt] \n \\displaystyle\\sum_{\\ell\\in [k]} b_\\ell=s(n-r)-\\sum_{\\ell\\in [k]}(\\rho_\\ell-e_\\ell+s\\dotdiv n),\\\\[20pt] \n a_\\ell+b_\\ell\\leq \\rho_\\ell-e_\\ell-(\\rho_\\ell-e_\\ell+r\\dotdiv n)-(\\rho_\\ell-e_\\ell+s\\dotdiv n).\n \\end{cases}\n\\end{align*}\nThe first six conditions of Theorem \\ref{rhoryserthmfullversion} can be restated as follows.\n}\\end{remark}\n\\begin{align*}\n |I|(n-s) &\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{a_\\ell+(\\rho_\\ell-e_\\ell+r\\dotdiv n), \\mu_I(\\ell)\\Big\\} &\\forall I\\subseteq [r],\\\\\n\\sum_{i\\in [r]} \\min \\Big\\{n-s, \\mu_K(i)\\Big\\} &\\geq \\sum_{\\ell\\in K} \\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r\\dotdiv n)\\Big)&\\forall K\\subseteq [k],\\\\\n |I|(n-s) &\\geq \\sum_{\\ell\\in [k]} \\Big( a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\dotdiv \\mu_{\\bar I}(\\ell)\\Big) &\\forall I\\subseteq [r],\\\\ \n\\sum_{i\\in [r]} \\Big (n-s\\dotdiv \\mu_{\\bar K}(i)\\Big) &\\leq \\sum_{\\ell\\in K}\\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\Big) &\\forall K\\subseteq [k],\\\\\n |I|(n-s) &\\leq \\sum_{\\ell\\in K}\\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\Big) + \\mu_I(\\bar K) &\\forall I\\subseteq [r], K\\subseteq [k], \\\\\n |I|(n-s) &\\geq \\sum_{\\ell\\in K}\\Big(a_\\ell+(\\rho_\\ell-e_\\ell+r \\dotdiv n)\\Big) - \\mu_K(\\bar I) &\\forall I\\subseteq [r], K\\subseteq [k].\n\\end{align*}\nSubstituting $r$ by $s$, $s$ by $r$, $I$ by $J$, and $a_\\ell$ by $b_\\ell$ in these conditions, one can obtain the remaining six conditions. \n\\section{Corollaries} \nThroughout this section, we will use the same notation as in \nSection \\ref{rhorythmprfsec}. Recall that in order to embed an $r\\times s$ $\\brho$-latin rectangle into an $n\\times n$ $\\brho$-latin square, it is necessary that\n$$\ne_\\ell\\geq r+s+\\rho_\\ell-2n \\quad \\forall \\ell\\in [k].\n$$\nWe show that imposing slightly stronger assumptions will lead to much simpler conditions than those of Theorem \\ref{rhoryserthmfullversion}. Most importantly, the sequence $\\{(a_\\ell,b_\\ell)\\}_{\\ell=1}^k$ will not be needed if we allow each $e_\\ell$ to be slightly bigger than what is necessary. \n\n\n\nWe will make use of the following $(g,f)$-factor theorem. Here, $N_G(A)$ in the neighborhood of $A$ in $G$. \n\\begin{theorem} \\cite[Theorem 5]{MR3564794}, \\cite[Theorem 1]{MR1081839} \\label{gffacthmcomb}\nThe bipgraph $G[X,Y]$ has a $(g,f)$-factor if and only if either one of the following two conditions hold.\n\\begin{align*}\n g(A)&\\leq \\sum_{u\\in N_G(A)} \\min \\Big\\{ f(u), \\mult_G(uA)\\Big\\}\n &\\forall A\\subseteq X, A\\subseteq Y,\\\\\nf(A)&\\geq \\sum_{u\\notin A} \\Big( g(u) \\dotdiv \\dg_{G-A}(u) \\Big) &\\forall A\\subseteq X\\cup Y.\n\\end{align*}\n\\end{theorem} \nSlight modification to the proof of \\cite[Theorem 1]{MR1081839} leads to the following simpler criteria for the case when $g(y)=0$ for $y\\in Y$: The bigraph $G[X,Y]$ has a $(g,f)$-factor with $g(y)=0$ for $y\\in Y$ if and only if\n\\begin{equation} \\label{refinedgfthmmain}\n f(B)\\geq \\sum_{x\\in A} \\Big( g(x) \\dotdiv \\dg_{G-B}(x) \\Big) \\quad \\forall A\\subseteq X, B\\subseteq Y.\n\\end{equation}\n\n\nIn our first application of Theorem \\ref{rhoryserthmfullversion}, we assume that $e_\\ell\\geq \\rho_\\ell-n+\\max\\{r,s\\}$ for $\\ell\\in [k]$. This, in particular implies that\n$$\nrs=\\sum_{\\ell\\in [k]}e_\\ell\\geq \\sum_{\\ell\\in [k]} \\big( \\rho_\\ell-n+\\max\\{r,s\\}\\big)=n^2-kn + k \\max\\{r,s\\},\n$$\nwhich leads to the following lower bound for the number of symbols.\n$$k\\geq \\frac{n^2-rs}{n-\\max\\{r,s\\}}\\geq n+\\max\\{r,s\\}.$$\n\n\n\\begin{corollary} \\label{corkbig}\nAn $r\\times s$ $\\brho$-latin rectangle with $e_\\ell\\geq \\rho_\\ell-n+\\max\\{r,s\\}$ for $\\ell\\in [k]$ can be completed to an $n\\times n$ $\\brho$-latin square if and only if any of the following conditions hold.\n\\begin{align*}\n |I|(n-s)+ |J|(n-r)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ \\rho_\\ell-e_\\ell, \\mu_I(\\ell) + \\mu_J(\\ell)\\Big\\}\n &\\forall I\\subseteq [r], J\\subseteq [s],\\\\\n\\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell)&\\geq \\sum_{i\\in I} \\Big( n-s \\dotdiv \\mu_{\\bar K}(i) \\Big)+\n \\sum_{j\\in J} \\big( n-r \\dotdiv \\mu_{\\bar K}(j) \\Big) &\\forall I\\subseteq [r], J\\subseteq [s], K\\subseteq [k].\n\\end{align*}\n\\end{corollary}\n\\begin{proof}\nSuppose that $e_\\ell\\geq \\rho_\\ell-n+\\max\\{r,s\\}$ for $\\ell\\in [k]$. Then \\eqref{mainryserineq} holds, and neither $\\rho_\\ell-e_\\ell-n+s$ nor $\\rho_\\ell-e_\\ell-n+r$ is positive for $\\ell\\in [k]$. Thus, \\eqref{colorcon3} will be simplified to the following.\n\\begin{align*}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if } i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r & \\mbox{if } j\\in [s], \\\\\n\\dg_\\Theta (\\ell)\\leq \\rho_\\ell-e_\\ell & \\mbox{if } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\n Let \n\\begin{align*}\n \\begin{cases}\n g,f: V(\\Gamma)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n g(x_i)=f(x_i)=n-s & \\mbox {for } i\\in [r],\\\\\n g(y_j)=f(y_j)=n-r & \\mbox {for } j\\in [s],\\\\\n g(\\ell)=0 & \\mbox {for } \\ell\\in [k],\\\\\n f(\\ell)=\\rho_\\ell-e_\\ell & \\mbox {for } \\ell\\in [k].\n \\end{cases}\n\\end{align*} \nBy Theorem \\ref{gffacthmcomb}, $\\Gamma [X\\cup Y, [k]]$ has a $(g,f)$-factor (and so $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists, and consequently, $L$ can be completed) if and only if the following conditions hold.\n\\begin{align*}\n g(I)+ g(J)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ f(\\ell), \\mult_\\Gamma(\\ell I)+ \\mult_\\Gamma(\\ell J)\\Big\\}\n &\\forall I\\subseteq X, J\\subseteq Y,\\\\\n g(K)&\\leq \\sum_{u\\in X\\cup Y} \\min \\Big\\{ f(u), \\mult_\\Gamma(u K)\\Big\\}\n &\\forall K\\subseteq [k].\n\\end{align*}\nThe first condition is equivalent to the first condition of this corollary. The second condition is trivial for $g(K)=0$ for $K\\subseteq [k]$. By \\eqref{refinedgfthmmain} $\\Gamma$ has a $(g,f)$-factor if and only if\n\\begin{equation*}\n f(K)\\geq \\sum_{u\\in U} \\Big( g(u) \\dotdiv \\dg_{\\Gamma-K}(u) \\Big) \\quad \\forall U\\subseteq (X\\cup Y), K\\subseteq [k].\n\\end{equation*}\nThis is equivalent to the second condition of this corollary.\n\\end{proof}\n\n\nIn our next application, we assume that $e_\\ell\\geq r+s-\\rho _\\ell$ for $\\ell\\in [k]$, which implies the following.\n$$\nrs=\\sum_{\\ell\\in [k]}e_\\ell\\geq \\sum_{\\ell\\in [k]} \\big( r+s-\\rho _\\ell\\big)=k(r+s)-n^2,\n$$\nThis leads to the following upper bound for the number of symbols.\n$$k\\leq \\frac{n^2+rs}{r+s}.$$\n\nIn the next corollary, we use the fact that $(x\\dotdiv y)\\dotdiv z=x-y \\dotdiv z$.\n\\begin{corollary} \\label{cor2}\nAn $r\\times s$ $\\brho$-latin rectangle with $e_\\ell\\geq r+s-\\rho _\\ell$ for $\\ell\\in [k]$ can be completed to an $n\\times n$ $\\brho$-latin square if and only if any of the following two conditions,\n\\begin{align*}\n \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+r\\dotdiv n) &\\leq \\sum_{i\\in [r]} \\min \\Big\\{ n-s, \\mu_K(i)\\Big\\} & \\forall K\\subseteq [k],\\\\\n|I|(n-s) &\\geq \\sum_{\\ell\\in [k]} \\big( \\rho_\\ell-e_\\ell+r- n \\dotdiv \\mu_{\\bar I}(\\ell) \\big) & \\forall I\\subseteq [r],\n\\end{align*}\ntogether with any of the following two hold.\n\\begin{align*}\n \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+s\\dotdiv n) &\\leq \\sum_{j\\in [s]} \\min \\Big\\{ n-r, \\mu_K(j)\\Big\\} & \\forall K\\subseteq [k],\\\\\n|J|(n-r) &\\geq \\sum_{\\ell\\in [k]} \\big( \\rho_\\ell-e_\\ell+s- n \\dotdiv \\mu_{\\bar J}(\\ell) \\big) & \\forall J\\subseteq [s].\n\\end{align*}\n\\end{corollary}\n\\begin{proof} \nSuppose that $e_\\ell\\geq r+s-\\rho _\\ell$ for $\\ell\\in [k]$. Then \\eqref{mainryserineq} holds, $\\rho_\\ell-e_\\ell\\geq r+s-2e_\\ell$ for $\\ell\\in [k]$, and so \\eqref{colorcon3} will be simplified to the following.\n\\begin{align*}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if } i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r & \\mbox{if } j\\in [s], \\\\\n\\mult_\\Theta (\\ell X)\\geq \\rho_\\ell-e_\\ell-n+r & \\mbox{if } \\ell \\in [k],\\\\\n\\mult_\\Theta (\\ell Y)\\geq \\rho_\\ell-e_\\ell-n+s & \\mbox{if } \\ell \\in [k].\n\\end{cases}\n\\end{align*}\nLet $\\Gamma_1, \\Gamma_2$ be the subgraphs of $\\Gamma$ induced by $X\\cup [k]$, and $Y\\cup [k]$, respectively. Let \n\\begin{align*}\n \\begin{cases}\n g_1, f_1,: V(\\Gamma_1)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n f_1(x_i)=g_1(x_i)=n-s &\\mbox {for } i\\in [r],\\\\\n g_1(\\ell)=\\rho_\\ell-e_\\ell+r\\dotdiv n &\\mbox {for } \\ell\\in [k],\\\\\n f_1(\\ell)=r-e_\\ell &\\mbox {for } \\ell\\in [k],\n \\end{cases}\n\\end{align*}\n\\begin{align*}\n \\begin{cases}\n g_2, f_2,: V(\\Gamma_2)\\rightarrow \\mathbb{N}\\cup \\{0\\},\\\\\n f_2(y_j)=g_2(y_j)=n-r &\\mbox {for } j\\in [s],\\\\\n g_2(\\ell)=\\rho_\\ell-e_\\ell+s\\dotdiv n &\\mbox {for } \\ell\\in [k],\\\\\n f_2(\\ell)=s-e_\\ell &\\mbox {for } \\ell\\in [k].\n \\end{cases}\n\\end{align*}\nIt is clear that $\\Theta\\subseteq \\Gamma$ satisfying \\eqref{colorcon3} exists if and only if $\\Gamma_1$ and $\\Gamma_2$ have an $f_1$-factor and $f_2$-factor, respectively. By Theorem \\ref{gffacthmcomb}, $\\Gamma_1$ has a $(g_1,f_1)$-factor if and only if\n\\begin{align*}\n g_1(K)&\\leq \\sum_{u\\in X} \\min \\Big\\{ f_1(u), \\mult_{\\Gamma_1}(u K)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n g_1(I)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ f_1(\\ell), \\mult_{\\Gamma_1}(\\ell I)\\Big\\}\n &\\forall I\\subseteq X,\n\\end{align*}\nand $\\Gamma_2$ has a $(g_2,f_2)$-factor if and only if\n\\begin{align*}\n g_2(K)&\\leq \\sum_{u\\in Y} \\min \\Big\\{ f_2(u), \\mult_{\\Gamma_2}(u K)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n g_2(J)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ f_2(\\ell), \\mult_{\\Gamma_2}(\\ell J)\\Big\\}\n\\quad &\\forall J\\subseteq Y.\n\\end{align*}\nEquivalently, $\\Gamma_1$ has a $(g_1,f_1)$-factor if and only if\n\\begin{align*}\n \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+r\\dotdiv n)&\\leq \\sum_{i\\in [r]} \\min \\Big\\{ n-s, \\mu_{K}(i)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n |I|(n-s)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ r-e_\\ell, \\mu_I(\\ell)\\Big\\}\n &\\forall I\\subseteq [r].\n\\end{align*}\nBut the second condition is trivial, for $\\mu_I(\\ell)\\leq r-e_\\ell$, and so $\\sum_{\\ell\\in [k]}\\mu_I(\\ell)=\\mu_I([k])=|I|(n-s)$. Similarly, $\\Gamma_2$ has a $(g_2,f_2)$-factor if and only if\n\\begin{align*}\n \\sum_{\\ell\\in K} (\\rho_\\ell-e_\\ell+s\\dotdiv n)&\\leq \\sum_{j\\in [s]} \\min \\Big\\{ n-r, \\mu_{K}(j)\\Big\\}\n &\\forall K\\subseteq [k],\\\\\n |J|(n-r)&\\leq \\sum_{\\ell\\in [k]} \\min \\Big\\{ s-e_\\ell, \\mu_J(\\ell)\\Big\\}\n &\\forall J\\subseteq [s].\n\\end{align*}\n Again, since $\\mu_J(\\ell)\\leq s-e_\\ell$, we have $\\sum_{\\ell\\in [k]}\\mu_J(\\ell)=\\mu_J([k])=|J|(n-r)$, and so the second condition is trivial.\n\nNow we modify $f_1$ and $f_2$ defined above so that for $\\ell\\in [k]$, $f_1(\\ell)=f_2(\\ell)=z$ where $z$ is a sufficiently large number. By Theorem \\ref{gffacthmcomb}, $\\Gamma_1$ has a $(g_1,f_1)$-factor if and only if \n\\begin{align} \\label{gf1faccond1}\nf_1(I)+f_1(K)&\\geq \\sum_{\\ell\\in \\bar K} \\Big( g_1(\\ell) \\dotdiv \\dg_{\\Gamma_1-(I\\cup K)}(\\ell) \\Big) \\nonumber\\\\\n&\\quad+ \\sum_{i\\in \\bar I} \\Big( g_1(i) \\dotdiv \\dg_{\\Gamma_1-(I\\cup K)}(i) \\Big) \\quad \\forall I\\subseteq X, K\\subseteq [k].\n\\end{align}\nFor $K\\neq \\emptyset$, \\eqref{gf1faccond1} is trivial, and for $K=\\emptyset$, it simplifies to the following.\n\\begin{align*} \n|I|(n-s)&\\geq \\sum_{\\ell\\in [k]} \\Big( (\\rho_\\ell-e_\\ell+r\\dotdiv n) \\dotdiv \\mu_{\\bar I}(\\ell) \\Big)+ \\sum_{i\\in \\bar I} \\Big( n-s \\dotdiv \\mu_{[k]}(i) \\Big)\\\\\n&=\\sum_{\\ell\\in [k]} \\Big( (\\rho_\\ell-e_\\ell+r\\dotdiv n) \\dotdiv \\mu_{\\bar I}(\\ell) \\Big) \\quad \\forall I\\subseteq [r].\n\\end{align*}\nSimilarly, $\\Gamma_2$ has a $(g_2,f_2)$-factor if and only if\n\\begin{align*} \n|J|(n-r)\\geq \\sum_{\\ell\\in [k]} \\Big( (\\rho_\\ell-e_\\ell+s\\dotdiv n) \\dotdiv \\mu_{\\bar J}(\\ell) \\Big) \\quad \\forall J\\subseteq [s].\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary}\nAn $r\\times s$ $\\brho$-latin rectangle satisfying the following condition can always be completed to an $n\\times n$ $\\brho$-latin square. \n$$r+s-e_\\ell \\leq \\rho_\\ell \\leq e_\\ell+n-\\max\\{r,s\\}\\quad \\forall \\ell\\in [k].$$\n\\end{corollary}\n\\begin{proof}\nCondition \\eqref{mainryserineq} holds, and we have $\\rho_\\ell-e_\\ell-n+r\\leq 0$, $\\rho_\\ell-e_\\ell-n+s\\leq 0$, and $\\rho_\\ell-e_\\ell\\geq r+s-2e_\\ell$ for $\\ell\\in [k]$. Hence, ~\\eqref{colorcon3} will be simplified to the following.\n\\begin{align*}\n\\begin{cases}\n\\dg_\\Theta (x_i)=n-s & \\mbox{if } i\\in [r], \\\\\n\\dg_\\Theta (y_j)=n-r & \\mbox{if } j\\in [s].\n\\end{cases}\n\\end{align*}\nConsequently, $\\Theta\\subseteq \\Gamma$ always exists. \n\\end{proof}\n\n\n\n\\section*{Acknowledgement}\n We wish to thank J. L. Goldwasser for a fruitful discussion that led to this publication, and the anonymous referees for their constructive criticism. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nGeneralised complex oriented cohomology first appeared in the work of\nNovikov \\cite{Novikov} and Quillen \\cite{Quillen} who realised that formal\ngroups naturally enter in algebraic topology. Such a theory is known to be\ncompletely characterised by the isomorphism $h^{\\ast }(\\mathbb{C}P^{\\infty\n})\\cong h^{\\ast }(\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{pt})[x]$, where $x$ is the first Chern class of the\ncanonical line bundle over the infinite complex projective space $\\mathbb{C}%\nP^{\\infty }$, and the K\\\"{u}nneth formula, $h^{\\ast }(\\mathbb{C}P^{\\infty\n}\\times \\mathbb{C}P^{\\infty })\\cong h^{\\ast }(\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{pt})[x,y]$, which\nimplies that the first Chern class of the tensor product of two line bundles\nobeys a formal group law \\cite{Adams}. There are three known types of formal\ngroup laws which come from the one-dimensional connected algebraic groups,\nthe additive group, the multiplicative group, and elliptic curves,\ndescribing respectively (ordinary) cohomology, K-theory and elliptic\ncohomology.\n\nOn the other hand to each of the mentioned groups one can associate\nrational, trigonometric and elliptic solutions of the Yang-Baxter equation\nwhich are linked to the appropriate quantum groups. It was first suggested\nin \\cite{GKV} that there should be a connection between the latter and the\nmentioned generalised cohomology theories.\n\nThe study of solutions of the Yang-Baxter equation is at the heart of the\narea of quantum integrable systems. Based on earlier pioneering works of\nHans Bethe \\cite{Bethe} and Rodney Baxter \\cite{Baxter}, the Faddeev School \n\\cite{Faddeevetal} developed the \\emph{algebraic Bethe ansatz} or \\emph{%\nquantum inverse scattering method}, where starting from a solution of the\nYang-Baxter equation one constructs the quantum integrals of motion of the\nphysical system as a commutative subalgebra, now often called the \\emph{%\nBethe algebra}, within a larger non-commutative \\emph{Yang-Baxter algebra}.\nHistorically, Yang-Baxter algebras were the origin for the later definition\nof quantum groups by Drinfeld \\cite{Drinfeld} and Jimbo \\cite{Jimbo}. Using\nthe commutation relations of the Yang-Baxter algebra the Bethe ansatz\nculminates in the derivation of a coupled set of -- in our setting --\npolynomial equations, whose solutions describe the spectrum of the commuting\ntransfer matrices which generate the Bethe algebra. In general solving these\nequations analytically is regarded as an intractable problem within the\nintegrable systems community except for a few special cases.\n\nThe first instance were quantum integrability was used in the study of\nquantum cohomology of full flag varieties and quantum K-theory was in works\nof Givental and Kim and Givental and Lee; see \\cite{Givental,GK,Kim,Kim2}\nand \\cite{Givental2,Lee}. In recent work of Nekrasov and Shatashvili \\cite%\n{NekrasovShatashvili} which was further developed mathematically by\nBraverman, Maulik and Okounkov \\cite{BMO,MaulikOkounkov} it was established\nthat the Bethe ansatz equations of some well known integrable systems\nrelated to the quantum groups known as Yangians describe the quantum\ncohomology and quantum K-theory for a large class of algebraic varieties,\nthe Nakajima varieties. Particular examples are the cotangent spaces of\npartial flag varieties, see the work \\cite{Gorbetal}, the simplest case\nbeing the contangent space of the Grassmannian. This opens up an exciting\nnew perspective on the connection made in \\cite{GKV}.\n\nIn this article we shall instead investigate the above connection for the\nGrassmannians $\\limfunc{Gr}_{n,N}=\\limfunc{Gr}_{n}(\\mathbb{C}^{N})$\nthemselves rather than their cotangent spaces based on the earlier findings\nin \\cite{KorffStroppel}, \\cite{VicOsc} and \\cite{GoKo}; see also the work on\nnon-quantum $GL(N)$-equivariant cohomology in \\cite{Rimanyietal}. The\ndifficulty here is, that it is initially not clear which quantum group to\nexpect. So instead we start out with special solutions to the Yang-Baxter\nequation which are tied to certain exactly solvable or quantum integrable\nlattice models in statistical mechanics and consider their associated\nYang-Baxter algebras as our \\textquotedblleft quantum group\". Despite the\nmodels being physically motivated, they are special degenerations of the\nasymmetric six-vertex model which describes ferroelectrics such as ice,\ntheir resulting Bethe algebras -- for certain special cases -- describe\nrings which have been defined previously in the setting of algebraic\ntopology and geometry where they are of great mathematical interest.\nSpecialising the parameters of the quantum integrable model in different\nways, we are able to identify them as the \\emph{quantum equivariant\ncohomology} \\cite{Mihalcea} and the (non-equivariant) \\emph{quantum K-theory}\n\\cite{BM}\\ of the Grassmannians using the results in \\emph{loc. cit}. \n\nThese special cases prompt us to conjecture that our main result, the\ndescription of a complex oriented generalised quantum cohomology and its\nequivariant version for the Grassmannians, also covers the so far unknown\ncase of \\emph{equivariant quantum K-theory}. At the same time this\ndescription can be seen as solving the well-posed mathematical problem of\nfinding the solution to the Bethe ansatz equations: we state the coordinate\nring defined by the equations, identify a special basis in it and explicitly\ndescribe the multiplication of two basis elements in terms of a generalised\nSchubert calculus within the framework of Goresky-Kottwitz-MacPherson theory\nwhich we show to extend to the quantum case.\n\n\\subsection{Statement of results}\n\nDenote by $\\limfunc{Gr}_{n,N}=\\limfunc{Gr}_{n}(\\mathbb{C}^{N})$ the\nGrassmannian of $n$-dimensional hyperplanes in $\\mathbb{C}^{N}$ with $N\\geq\n3 $ and fix a maximal torus $\\mathbb{T}\\subset GL(N)$. We describe\ngeneralised $\\mathbb{T}$-equivariant quantum cohomology rings $qh_{n}^{\\ast\n}=qh^{\\ast }(\\limfunc{Gr}_{n,N};\\beta )$ for $n=0,1,\\ldots ,N$ using the\ntheory of exactly solvable lattice models in statistical mechanics \\cite%\n{Baxter}. While the latter appear in theoretical physics, we shall use them\nhere as abstract combinatorial objects -- they define a weighted counting of\nnon-intersecting lattice paths as described for $\\beta =0$ in \\cite{VicOsc}\n-- which can be rigorously defined in purely mathematical terms using\nYang-Baxter algebras. The weights or probabilities attached to the lattice\nmodels depend on\n\n\\begin{itemize}\n\\item a variable $\\beta $ (the anisotropy parameter of the six-vertex model)\nentering the multiplicative formal group law \\cite{Quillen}, \\cite%\n{Bukhshtaberetal} and its inverse, \n\\begin{equation}\nx\\oplus y=x+y+\\beta xy\\;\\;\\;\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\;\\;\\;x\\ominus y=\\frac{x-y}{1+\\beta y}%\n\\;, \\label{group_law}\n\\end{equation}\n\n\\item a \\textquotedblleft quantum\" parameter $q$ (the twist parameter\nrelated to quasiperiodic boundary conditions on the lattice) as well as\n\n\\item the equivariant parameters $t=(t_{1},\\ldots ,t_{N})$ (so-called\ninhomogeneities in the lattice) which are connected to the natural $\\mathbb{T%\n}$-action on $\\limfunc{Gr}_{n,N}$.\n\\end{itemize}\n\nThe case $\\beta =0$, which corresponds to the additive group law and in\nphysics terminology to the so-called \\emph{free fermion} point of the\nlattice models, has been treated previously for the homogeneous case ($%\nt_{j}=0$) in \\cite{VicOsc} and recently been extended to the equivariant\nsetting in \\cite{GoKo}.\n\nOur approach does not require any background knowledge in statistical\nmechanics, the lattice models are constructed in terms of special solutions\nto the quantum (as opposed to classical) Yang-Baxter equation, hence they\nare called \\emph{quantum integrable}, and their description is purely\nalgebraic. However, we find it noteworthy that they are degenerations of the\nasymmetric six-vertex model -- as mentioned previously -- and their\ncombinatorial description analogous to the one in \\cite{VicOsc} provides a\npowerful computational tool. For the latter to work we require the\npreviously mentioned restriction $N\\geq 3$.\n\nFrom these special solutions of the Yang-Baxter equation we construct\nYang-Baxter algebras, which in our case are bi-algebras only and not full\nHopf algebras. The so-called \\emph{row-to-row transfer matrices} of the\nlattice model generate a commutative subalgebra within the larger\nnon-commutative Yang-Baxter algebra which decomposes into the direct sum $%\n\\tbigoplus\\nolimits_{n=0}^{N}qh_{n}^{\\ast }$ of rings, which have the\nfollowing presentation.\n\nSet $\\mathcal{R}(\\mathbb{T)}=\\mathcal{R}(t_{1},\\ldots ,t_{N})$ where $%\n\\mathcal{R}$ is the ring of rational functions in $\\beta $ which are regular\nat $\\beta =0$ and $\\beta =-1$. Define $qh_{n}^{\\ast }$ %\nto be the polynomial algebra generated by $\\{e_{r}\\}_{r=1}^{n}$, $%\n\\{h_{r}\\}_{r=1}^{N-n}$ over $\\mathcal{R}(\\mathbb{T},q\\mathbb{)}=\\mathbb{Z}%\n[\\![q]\\!]\\otimes \\mathcal{R}(\\mathbb{T)}$ subject to the relations obtained\nby expanding the following functional relation in the variable $x$,%\n\\begin{equation}\nh(x)e(\\ominus x)=\\left( \\tprod_{i=1}^{n}t_{i}\\ominus x\\right) \\left(\n\\tprod_{i=1}^{N-n}x\\ominus t_{i+n}\\right) (1+\\beta h_{1})+q,\n\\label{ideal def}\n\\end{equation}%\nwhere $1$ is the unit element and, setting $h_{0}=e_{0}=1,$ $%\nh_{N+1-n}=e_{n+1}=0$,%\n\\begin{eqnarray}\nh(x) &=&\\sum_{r=0}^{N-n}\\left( h_{r}+\\beta h_{r+1}\\right)\n\\tprod_{i=1}^{N-n-r}\\left( x\\ominus t_{N+1-i}\\right) \\label{H def} \\\\\ne(x) &=&\\sum_{r=0}^{n}\\left( e_{r}+\\beta e_{r+1}\\right)\n\\tprod_{i=1}^{n-r}\\left( x\\oplus t_{i}\\right) \\;. \\label{E def}\n\\end{eqnarray}%\nFor the non-experts we recall that the Grothendieck $K$-functor assigns to\neach smooth compact manifold $\\mathcal{X}$ a ring which is built out of\ncomplex vector bundles on $\\mathcal{X}$ \\cite{AA}. It is the value of this\nfunctor and its quantum analogue $QK$ for $\\mathcal{X}=\\limfunc{Gr}_{n,N}$\nwhich we shall simply refer to as (quantum) ``K-theory'' of the\nGrassmannians throughout this article.\n\nDenote by $\\{e^{\\varepsilon _{j}}\\}_{j=1}^{N}$ the (formal) exponentials\ngenerating the character ring of $\\mathfrak{gl}(N)$.\n\n\\begin{theorem}\nWe have the following special cases:\n\n\\begin{itemize}\n\\item[(i)] $qh_{n}^{\\ast }\/\\langle \\beta \\rangle $ is isomorphic to the\nequivariant quantum cohomology $QH_{\\mathbb{T}}^{\\ast }(\\limfunc{Gr}_{n,N})$\nin the presentation given by Mihalcea \\cite[Thm 1.1]{Mihalcea}.\n\n\\item[(ii)] $qh_{n}^{\\ast }\/\\langle \\beta +1,t_{1},\\ldots ,t_{N}\\rangle $ is\nisomorphic to $QK(\\limfunc{Gr}_{n,N}) $ as studied in \\cite{BM}.\n\n\\item[(iii)] $qh_{n}^{\\ast }\/\\langle \\beta +1,t_{j}+e^{\\varepsilon\n_{N+1-j}}-1,q\\rangle $ is isomorphic to $K_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})$\nwhere $K_{\\mathbb{T}}$ denotes the equivariant K-functor.\n\\end{itemize}\n\\end{theorem}\n\nEach of the cases (i)--(iii) is interesting in its own right and we compare\nour findings against existing presentations of these rings in the\nliterature. In particular, in case (i) our results are linked to previous\n(unpublished) work by Peterson \\cite{Peterson} and the affine nil-Hecke ring\nof Kostant and Kumar \\cite{KostantKumar}: we explicitly construct a family\nof operators whose matrix elements give the structure constants of $%\nqh_{n}^{\\ast }$ and which for $\\beta =0$ can be identified with Peterson's\nbasis; see \\cite{GoKo} for details. The other cases can then be seen as a\ngeneralisation of this construction to $K$-theory.\n\nTo establish (ii) we compare our ring structure against the Pieri rules\nderived by Lenart \\cite{Lenart} for $q=0$ and the quantum Pieri and\nGiambelli formulae of Buch and Mihalcea \\cite{BM} for $q\\neq 0$. The new\nresult in our article is the coordinate ring presentation which follows from\n(\\ref{ideal def}).\n\nFinally, we show (iii) by defining a generalisation of\nGoresky-Kottwitz-MacPherson theory \\cite{GKM}: we identify McNamara's\nfactorial Grothendieck polynomials \\cite{McNamara} with localised Schubert\nclasses using the Bethe ansatz of quantum integrable models. We also derive\nexpressions for the localised opposite Schubert classes and identify the\npartition functions of our lattice models with classes for Richardson\nvarieties.\n\nBased on the above special cases we have the following:\n\n\\begin{conjecture}\n$qh_{n}^{\\ast }\/\\langle \\beta +1,t_{j}+e^{\\varepsilon _{N+1-j}}-1\\rangle $\ndescribes the value $QK_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})$ of the equivariant\nquantum $K$-functor for the Grassmannians.\n\\end{conjecture}\n\n\\begin{remark}\nWe note that we can define $qh^*_n$ also over the ring of Laurent\npolynomials in $\\beta$ instead, which would introduce a natural $\\mathbb{Z}$%\n-grading. This suggests that our framework might also be used to describe\nthe actual $\\mathbb{Z}$-graded quantum equivariant K-theory which is\nobtained from the K-functor in conjunction with the Bott Periodicity\nTheorem. However, there is currently not sufficient evidence available to\nfurther substantiate this claim, hence we state this here as a mere\nobservation and not as a conjecture.\n\\end{remark}\n\nBesides providing a complete description of $QK_{\\mathbb{T}}(\\limfunc{Gr}%\n_{n,N})$, which has so far been missing in the literature, the new aspects\nin our approach are\n\n\\begin{enumerate}\n\\item that our ring is defined for general $\\beta $ which allows us to treat\nall these special cases at once in a unified setting of a quantum\ngeneralised cohomology theory as first defined in \\cite{CG} and\n\n\\item that we reveal an underlying quantum group structure in terms of\nYang-Baxter algebras which we show to commute with the natural symmetric\ngroup action on the idempotents of these rings.\n\\end{enumerate}\n\nAs a byproduct of our investigation we also derive new combinatorial results\nsuch as a generalised Jacobi-Trudy formula and Cauchy identity for factorial\nGrothendieck polynomials.\n\n\\subsection{Outline of the article}\n\n\\begin{description}\n\\item[Section 2] We introduce the necessary combinatorial objects and\nnotations we will use throughout the article. In particular, we review\nMcNamara's definition of factorial Grothendieck polynomials which play a\ncentral role in our approach and derive several new results which we need to\ndescribe our generalised cohomology ring for the Grassmannian.\n\n\\item[Section 3] Starting from special solutions to the Yang-Baxter\nequation, so-called $L$-operators, we define the Yang-Baxter algebra in\nterms of endomorphisms over some vector space $\\mathcal{V}$ which will be\nidentified with the direct sum of the generalised cohomology rings $%\n\\tbigoplus\\nolimits_{n=0}^{N}qh_{n}^{\\ast }$. We describe the commutation\nrelations of the Yang-Baxter algebra and define the transfer matrices which\ngenerate a commutative subalgebra, the so-called Bethe algebra. The action\nof the latter on $\\mathcal{V}$ is described combinatorially using toric skew\ndiagrams. We also show that the transfer matrices obey the functional\nrelation (\\ref{ideal def}).\n\n\\item[Section 4] We derive the spectrum of the Bethe algebra by constructing\ntheir eigenvectors and computing their eigenvalues using the algebraic Bethe\nansatz. Both, eigenvectors and eigenvalues, are described in terms of the\nsolutions of a set of coupled equations, called the Bethe ansatz equations,\nwhich we show can be solved in terms of formal power series in the quantum\ndeformation parameter $q$ of $qh_{n}^{\\ast }$. We then initially define the\ngeneralised cohomology ring by identifying the eigenbasis of the transfer\nmatrices as the primitive, central orthogonal idempotents of $qh_{n}^{\\ast }$%\n. We also define a bilinear form which turns $qh_{n}^{\\ast }$ into a\nFrobenius algebra. Having identified the eigenvectors of the transfer\nmatrices as idempotents, we then fix the analogue of the Schubert basis and\ndescribe the product in this geometrically motivated basis instead. This\nallows us to state a residue formula for the structure constants in the\nSchubert basis in terms of the solutions of the Bethe ansatz equations and\nshow that they obey a recurrence formula which is derived from an\nequivariant quantum Pieri-Chevalley formula for $qh_{n}^{\\ast }$.\n\n\\item[Section 5] Employing the description of $qh_{n}^{\\ast }$ in terms of\nits idempotents leads to a formulation of the ring in terms of column\nvectors whose components can be thought of as generalised localised Schubert\nclasses where the localisation points are identified with the solutions of\nthe Bethe ansatz equations. We show that these generalised Schubert classes\nobey generalised Goresky-Kottwitz-MacPherson conditions which derive from an\naction of the symmetric group. Interestingly, the latter emerges naturally\nfrom solutions of the Yang-Baxter equation discussed in Section 3, gives\nrise to a representation of a generalised Iwahori-Hecke algebra and commutes\nwith the action of the Yang-Baxter algebra. Using this framework of GKM\ntheory we prove the special cases mentioned in the introduction, that is we\nshow that our ring $qh_{n}^{\\ast }$ can be specialised to equivariant\nquantum cohomology and quantum K-theory. This section also gives the proof\nof the presentation of $qh_{n}^{\\ast }$ as polynomial algebra modulo the\nrelations (\\ref{ideal def}).\\bigskip\n\\end{description}\n\n\\noindent \\textbf{Acknowledgment}. The authors would like to thank the Max\nPlanck Institute for Mathematics Bonn, where part of this work was carried\nout, for hospitality. They are grateful to Leonardo Mihalcea and Alexander\nVarchenko for comments on a draft version of the article. C. K. also\ngratefully acknowledges discussions with Gwyn Bellamy, Christian Voigt, Paul\nZinn-Justin and would like to thank the organisers Anita Ponsaing and Paul\nZinn-Justin for their kind invitation to the workshop \\emph{Combinatorics\nand Integrability}, Presqu'\\^{\\i}le de Giens, 23-27 June 2014, where the\nresults of this article were presented.\n\n\\section{Preliminaries}\n\nThis section introduces the combinatorial notions needed in the description\nof Schubert calculus in the rest of this paper. We also collect known as\nwell as a number of new results on factorial Grothendieck polynomials.\n\n\\subsection{Minimal coset representatives}\n\nDenote by $\\mathbb{S}_{N}$ the symmetric group in $N$-letters and choose $%\nk,n\\in \\mathbb{N}_{0}$ such that $N=n+k$. A set of minimal length coset\nrepresentatives $w$ for classes $[w]$ in $\\mathbb{S}_{N}\/\\mathbb{S}%\n_{n}\\times \\mathbb{S}_{k}$ is given by the permutations for which $%\nw(1)\\mu \\lbrack 0]_{i}\\}\\;.\n\\label{cyl_diagram}\n\\end{equation}%\nWe shall refer to $\\theta =\\lambda \/d\/\\mu $ as a \\emph{cylindric skew-diagram%\n} of degree $d=d(\\theta )$. Postnikov introduced \\cite{Postnikov} the\nterminology \\emph{toric skew-diagram }for those\\emph{\\ }$\\theta $ where the\nnumber of boxes in each row does not exceed $k$. Note that $\\lambda \/0\/\\mu\n=\\lambda \/\\mu $, that is cylindric or toric skew-diagrams contain ordinary\nskew diagrams as special cases.\n\nA cylindric skew diagram $\\theta $ which has at most one box in each column\nwill be called a \\emph{toric} \\emph{horizontal strip} and one which has at\nmost one box in each row a \\emph{toric} \\emph{vertical strip}. The \\emph{%\nlength} of such strips will be the number of boxes within the skew diagram,\nwhere we identify squares modulo integer shifts by $(n,-k)$ and choose as\nrepresentatives those squares $s=\\langle i,j\\rangle $ with $1\\leq j\\leq n$.\nIn what follows this identification is always understood implicitly if we\ntalk about a square in a toric strip.\n\n\n\\subsection{Bases in equivariant cohomology and K-theory\\label{sec:schubert}}\n\nWe are interested in describing equivariant quantum cohomology ($\\beta =0$)\nand K-theory ($\\beta =-1$) as special cases of our generalised cohomology\ntheory for $\\limfunc{Gr}_{n,N}$. The equivariant cohomology \\cite%\n{KostantKumar} and K-theory \\cite{KostantKumar2} of flag varieties -- of\nwhich Grassmannians are a special case -- was studied by Kostant and Kumar.\nThe equivariant quantum cohomology of flag varieties was computed in \\cite%\n{Givental}, \\cite{Kim}, \\cite{GK}, \\cite{Kim2} and quantum K-theory in \\cite%\n{Givental2}, \\cite{Lee}, \\cite{GL} and since then has been discussed by\nnumerous authors.\n\n\nSpecialising $\\beta =0$ we identify $\\mathcal{R}(\\mathbb{T)}$ with the\nequivariant cohomology $H_{\\mathbb{T}}^{\\ast }(\\limfunc{pt})=\\mathbb{Z}%\n(t_{1},\\ldots ,t_{N})$ of a point by mapping each $f_{\\beta }\\in \\mathcal{R}(%\n\\mathbb{T)}$ to its value at $\\beta =0$. Let $X_{\\lambda }$ and $X^{\\lambda\n} $ denote the \\emph{Schubert} and \\emph{opposite Schubert varieties} where $%\n\\lambda \\subset (k^{n})$. We also recall the definition of the \\emph{%\nRichardson variety} $X_{\\mu }^{\\lambda }=X_{\\mu }\\cap X^{\\lambda }$. All\nthree varieties are left invariant under the torus action. The corresponding\nSchubert classes $\\{[X_{\\lambda }]\\}_{\\lambda \\subset (k^{n})}$ and $%\n\\{[X^{\\lambda }]\\}_{\\lambda \\subset (k^{n})}$ form dual bases over $\\mathbb{Z%\n}[q]\\otimes \\mathbb{Z}(t_{1},\\ldots ,t_{N})$. Both bases are related by $%\nX^{\\lambda }=w_{0}\\cdot X_{\\lambda ^{\\vee }}$ where $\\lambda ^{\\vee }$ is\nobtained by reversing the binary string $b(\\lambda )$ and $w_{0}$ is the \n\\emph{long element} in $\\mathbb{S}_{N}$. One is interested in the\ncomputation of the \\emph{3-point genus 0 equivariant Gromov-Witten invariants%\n} $C_{\\lambda \\mu }^{\\nu }(t,q)$ which appear in the product%\n\\begin{equation}\n\\lbrack X_{\\lambda }][X_{\\mu }]=\\sum_{\\nu\\subset (k^n)}C_{\\lambda \\mu\n}^{\\nu}(t,q)[X_{\\nu }] \\label{GWinv}\n\\end{equation}%\nand for the Grassmannian are monomials in $q$, i.e. $C_{\\lambda \\mu }^{\\nu\n}(t,q)=q^{d}C_{\\lambda \\mu }^{\\nu ,d}(t)$. The invariants for $d=0$ also\nappear in the expansion%\n\\begin{equation}\n\\lbrack X_{\\mu }^{\\lambda }]=\\sum_{\\nu }C_{\\mu \\nu }^{\\lambda ,0}(t)[X^{\\nu\n}]\\;, \\label{Richardson}\n\\end{equation}%\nand, thus, $C_{\\mu \\nu }^{\\lambda ,0}(t)=c_{\\lambda \\mu }^{\\nu }(t)$ are the\nanalogue of Littlewood-Richardson coefficients for factorial Schur functions \n\\cite{MolevSagan}.\n\nIn the case of $K$-theory we specialise $\\beta =-1$ and set $%\nt_{j}=1-e^{\\varepsilon _{N+1-j}}$ where the (formal) exponentials $%\n\\{e^{\\varepsilon _{j}}\\}_{j=1}^{N}$ generate the character ring of $%\n\\mathfrak{gl}(N)$. Mapping each $f_{\\beta }\\in \\mathcal{R}(\\mathbb{T)}$ to\nits value at $\\beta =-1$ then gives us $K_{\\mathbb{T}}(\\limfunc{pt})=%\n\\limfunc{Rep}(\\mathbb{T})$, the representation ring of $\\mathbb{T}$ which is\ncanonically isomorphic to the group algebra of the free abelian group of\ncharacters $e^{\\varepsilon _{i}}$. The ring $K_{\\mathbb{T}}(\\limfunc{Gr}%\n_{n,N})$ is generated by the classes $[\\mathcal{O}_{\\lambda }]$ and $[%\n\\mathcal{O}^{\\lambda }]$ of the \\emph{structure sheaves} $\\mathcal{O}%\n_{\\lambda }$ and $\\mathcal{O}^{\\lambda }$ of the Schubert and opposite\nSchubert varieties within the Grothendieck group of coherent sheaves on the\nGrassmannian. Their product expansions%\n\\begin{equation}\n\\lbrack \\mathcal{O}_{\\lambda }][\\mathcal{O}_{\\mu }]=\\sum_{\\nu \\subset\n(k^{n})}c_{\\lambda \\mu }^{\\nu }(t)[\\mathcal{O}_{\\nu }],\\qquad \\lbrack \n\\mathcal{O}_{\\mu }][\\mathcal{O}^{\\lambda }]=\\sum_{\\nu \\subset (k^{n})}d_{\\mu\n\\nu }^{\\lambda }(t)[\\mathcal{O}^{\\nu }] \\label{Kstructure}\n\\end{equation}%\ndefine the K-theoretic Littlewood-Richardson coefficients $c_{\\lambda \\mu\n}^{\\nu }(t)$ where in case of the Grassmannian $d_{\\lambda \\mu }^{\\nu\n}(t)=c_{\\lambda \\mu }^{\\nu }(t)$; see e.g. \\cite{Knutson} and references\ntherein. There are known positivity statements for these structure\nconstants, see \\cite{GR} and \\cite[Sec 5]{AGM} as well as references therein.\n\nWe shall refer to the K-classes $\\{[\\mathcal{O}^{\\lambda }]\\}$ as Schubert\nbasis or simply Schubert classes. In contrast to the case $\\beta =0$ the\nclasses $[\\mathcal{O}_{\\lambda }]$ and $[\\mathcal{O}^{\\lambda }]$ do not\nform dual bases in $K_{\\mathbb{T}}(\\limfunc{Gr}_{n,N})$ but instead one has\nto introduce additional classes $[\\xi ^{\\lambda }]$ which can also be\ndefined in terms of sheaves (see \\cite[Prop 2.1]{GrKu}). For the\nnon-equivariant case $K(\\limfunc{Gr}_{n,N})=K_{\\mathbb{T}}(\\limfunc{Gr}%\n_{n,N})\/\\langle t_{1},\\ldots ,t_{N}\\rangle $ one has the relation \\cite[Sec 8%\n]{BuchKtheory} \n\\begin{equation}\n\\lbrack \\xi ^{\\lambda }]=(1-[\\mathcal{O}_{1}])[\\mathcal{O}_{\\lambda ^{\\vee\n}}] \\label{Kdualbasis}\n\\end{equation}%\nwhere $[\\mathcal{O}_{1}]$ is the K-class of the Schubert divisor. We shall\nstate the analogue of this relation for the equivariant case in a subsequent\nsection. %\n\n\n\\subsection{Discrete symmetries\\label{sec:symmetries}}\n\nThroughout this article we will make use of several involutions and a\nnatural $\\mathbb{Z}_{N}$-action defined on the set of cosets in $\\mathbb{S}%\n_{N}\/\\mathbb{S}_{n}\\times \\mathbb{S}_{k}$ where $k=N-n$ as before. These\nwill induce mappings between elements in the Schubert basis, in some cases\nfrom different rings, and since they in turn lead to non-trivial\ntransformation properties of the structure constants of $qh_{n}^{\\ast }$, we\nrefer to them as \\textquotedblleft symmetries\\textquotedblright .\n\n\\subsubsection{Poincar\\'{e} Duality}\n\nDefine an involution $\\vee :qh_{n}^{\\ast }\\rightarrow qh_{n}^{\\ast }$ by\nreversing a binary string, i.e. $b_{i}^{\\vee }=b_{N+1-i}$. We shall denote\nthe corresponding permutation and partition by $w^{\\vee }$ and $\\lambda\n^{\\vee }$, respectively. One easily verifies that the Young diagram of $%\n\\lambda ^{\\vee }$ is the complement of the Young diagram of $\\lambda $ in\nthe $n\\times k$ bounding box.\n\n\\subsubsection{Level-Rank Duality}\n\nDefine an involution $\\ast :qh_{n}^{\\ast }\\rightarrow qh_{N-n}^{\\ast }$ by\nswapping 0 and 1-letters in binary strings, i.e. $b_{i}^{\\ast }=1-b_{i}$.\nThe corresponding partition $\\lambda ^{\\ast }$ is obtained by taking first\nthe conjugate partition $\\lambda ^{\\prime }$ and then its complement in the\nbounding box or vice versa, i.e. $\\lambda ^{\\ast }=(\\lambda ^{\\prime\n})^{\\vee }=(\\lambda ^{\\vee })^{\\prime }$. So, in particular we can define\nthe composite involution $qh_{n}^{\\ast }\\rightarrow qh_{N-n}^{\\ast }$ by $%\n\\lambda \\mapsto \\lambda ^{\\prime }$ and shall denote the corresponding\nbinary string and permutation respectively by $b^{\\prime }$ and $w^{\\prime }$%\n.\n\n\n\\subsection{Set-Valued Tableaux and Grothendieck polynomials\\label%\n{sec:grothendieck}}\n\nWe recall some of the necessary combinatorial objects and the definition of\nfactorial Grothendieck polynomials. This is based on earlier work by Buch \n\\cite{BuchKtheory} and McNamara \\cite{McNamara}, but we shall also derive\nseveral new results which are not contained in the latter works.\n\nLet $n$ be some non-negative integer. We will use the notation $%\n[n]=\\{1,\\ldots ,n\\}$ and $\\mathbb{P}_{n}=\\mathbb{P}([n])$ for the power set\nof $[n]$, the set of all subsets of $[n]$. Denote by $\\theta $ a skew Young\ndiagram with at most $|\\theta |\\leq n$ boxes which we identify with a subset\nof $\\mathbb{Z}^{2}$.\n\n\\begin{definition}[Buch]\nA set-valued tableau is a map $T:\\theta \\rightarrow \\mathbb{P}_{n}$ such\nthat the following conditions hold%\n\\begin{equation}\n\\max T(i,j)\\leq \\min T(i,j+1)\\;\\;\\;\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\;\\;\\;\\max T(i,j)<\\min\nT(i+1,j)\\;.\n\\end{equation}\n\\end{definition}\n\nDenote by $|T|$ the sum over the cardinalities of all the subsets in the\nimage of $T$ and let $\\limfunc{SVT}(\\theta )$ be the set of all set-valued\ntableau of shape $\\theta $. Then we have the following definition of\nfactorial Grothendieck polynomials due to McNamara \\cite{McNamara} which is\nan extension of Buch's earlier realisation \\cite{BuchKtheory} of ordinary\n(skew) Grothendieck polynomials as sum over set-valued tableaux.\n\n\\begin{definition}\nThe factorial (skew) Grothendieck polynomial is the weighted sum%\n\\begin{equation}\nG_{\\theta }(x|t)=\\sum_{T}\\beta ^{|T|-|\\theta |}\\prod_{\\substack{ (i,j)\\in\n\\theta \\\\ r\\in T(i,j)}}x_{r}\\oplus t_{r+j-i} \\label{facG}\n\\end{equation}%\nover all set-valued tableaux $T\\in \\limfunc{SVT}(\\theta )$.\n\\end{definition}\n\nN.B. the factorial Grothendieck polynomials are in general defined for an \n\\emph{infinite} sequence $(t_j)_{j\\in\\mathbb{Z}}$ of parameters. For this\nsection only we shall assume these parameters to be nonzero for all $j$ but\nthen set $t_j=0$ unless $1\\leq j\\leq N$ and identify them with the\nequivariant parameters mentioned in the introduction.\n\nEmploying (\\ref{group_law}) define the $\\beta $-deformed factorial power \n\\begin{equation}\n(x_{j}|t)^{r}:=\\prod_{i=1}^{r}x_{j}\\oplus t_{i}\\;. \\label{facpower}\n\\end{equation}%\nThe following determinant formula is stated in \\cite[Eqn (2.12)]{IN}. Its\nproof follows along similar lines as indicated in \\emph{loc. cit.} where the\nfocus is on the symplectic case.\n\n\\begin{proposition}[Ikeda-Naruse]\n\\begin{equation}\nG_{\\theta }(x|t)=\\frac{\\det \\left[ (x_{j}|t)^{\\theta _{i}+n-i}(1+\\beta\nx_{j})^{i-1}\\right] _{1\\leq i,j\\leq n}}{\\det [x_{j}^{n-i}]_{1\\leq i,j\\leq n}}\n\\label{facGdet}\n\\end{equation}%\nwhere the denominator is the Vandermonde determinant $\\Delta\n(x)=\\prod_{iN\/2$ the\naction can be deduced employing Cor \\ref{cor:levelrankEH}.\n\nWe interpret partitions and their associated cylindric loops as subsets of $%\n\\mathbb{Z}^{2}$. Given a toric horizontal strip $\\theta =\\lambda \/d\/\\mu $ of\ndegree $d$ denote by\n\n\\begin{itemize}\n\\item $\\mathcal{R}_{\\theta }$ the set which contains all squares $s=\\langle\ni,j\\rangle \\in \\mathbb{Z}^{2}$, $1\\leq i\\leq n$ such that the square\nimmediately left to it, $s^{\\prime }=\\langle i,j-1\\rangle $, is the\nrightmost square in a row of $\\lambda \\lbrack d]$ intersecting $\\theta $;\n\n\\item $\\mathcal{\\bar{C}}_{\\theta }$ the set which contains all the bottom\nsquares $s=\\langle i,j\\rangle ,~1\\leq j\\leq k$ from each column of $\\mu\n\\lbrack 0]$ which does not intersect $\\theta $ as well as the squares $%\ns=\\langle 1,j\\rangle $ in empty columns if $\\lambda _{1}+n1}u_{j_{1}}u_{j_{2}}+\\beta ^{2}\\sum_{|j_{a}-j_{b}|\\func{mod}%\nN>1}u_{j_{1}}u_{j_{2}}u_{j_{3}}+\\cdots \\label{h1}\n\\end{equation}%\nas a formal power series in $\\beta $. Note that the sums only run over\nindices where $|j_{a}-j_{b}|\\func{mod}N>1$ which ensures that all the $u_{j}$%\n's in each monomial commute. Obviously, only finitely many terms act\nnon-trivially for finite $N$ and the series therefore terminates.\n\n\\begin{lemma}\nActing with $\\bar{H}_{1}$ on a spin basis vector $v_{\\mu }\\in \\mathcal{V}%\n_{n} $ one obtains%\n\\begin{equation}\n\\bar{H}_{1}v_{\\mu }=\\sum_{\\substack{ \\mu \\rightrightarrows ^{\\ast }\\lambda\n\\lbrack d] \\\\ d=0,1}}q^{d}\\beta ^{|\\lambda \/d\/\\mu |-1}v_{\\lambda },\n\\label{h1action}\n\\end{equation}%\nwhere the sum runs over all boxed partitions $\\lambda \\subset (k^{n})$ such\nthat either $\\lambda \/0\/\\mu =\\lambda \/\\mu $ or $\\lambda \/1\/\\mu $ are toric\ndiagrams which contain at most one box in each column and row and $\\lambda\n\\neq \\mu $.\n\\end{lemma}\n\n\\begin{proof}\nUsing the bijection between binary strings and partitions detailed in\nSection \\ref{sec:partitions} and the definition of cylindric loops in\nSection \\ref{sec:torictab}, one proves that either $u_{j}v_{\\mu\n}=q^{d}v_{\\lambda }$ where one adds a box with coordinates $(x,y)$ and $%\nj=n+y-x$ to obtain $\\lambda $ (or $\\lambda \\lbrack 1]$ if $d=1$ and $j=N$)\nor $u_{j}v_{\\mu }=0$. The assertion then easily follows from the fact that\nall $u_{j}$'s in each monomial term commute.\n\\end{proof}\n\n\\begin{proposition}\nThe transfer matrices obey the following functional operator identity%\n\\begin{equation}\nH(x|t)E(\\ominus x|t)=(1+\\beta \\bar{H}_{1})\\prod_{j=1}^{N}(t_{j}\\ominus\nx)^{\\sigma _{j}^{+}\\sigma _{j}^{-}}(x\\ominus t_{j})^{\\sigma _{j}^{-}\\sigma\n_{j}^{+}}+q\\cdot 1\\;. \\label{func_eqn1}\n\\end{equation}%\nIn particular, we have that $H(t_{j}|t)E(\\ominus t_{j}|t)=q\\cdot 1$ for all $%\nj=1,\\ldots ,N$ which amount to non-trivial identities between the\ncoefficients $\\{H_{r}\\}$ and $\\{E_{r}\\}$ defined in (\\ref{Hr}), (\\ref{Er}).\n\\end{proposition}\n\n\\begin{proof}\n\\begin{figure}[tbp]\n\\begin{equation*}\n\\includegraphics[scale=0.35]{func_eqn.eps}\n\\end{equation*}%\n\\caption{The vertex configurations corresponding to the operator $%\nL_{i+1j}^{\\prime }(\\ominus x_{i})L_{i,j}(x_{i})$.}\n\\label{fig:func_eqn}\n\\end{figure}\nA computation along similar lines as in \\cite{VicOsc}. The idea is to\nanalyse the action of $\\mathcal{L}_{j}=L_{1j}^{\\prime }(\\ominus\nx)L_{2j}(x):W(x)\\otimes V(t_{j})\\rightarrow W(x)\\otimes V(t_{j})$ where $%\nW(x)=V(\\ominus x)\\otimes V(x)=\\mathcal{R}(x)\\otimes V^{\\otimes 2}$ with\nrespect to the basis vectors%\n\\begin{eqnarray*}\nw_{0} &=&v_{0}\\otimes v_{0},\\quad w_{1}=v_{0}\\otimes v_{1}+v_{1}\\otimes\nv_{0}, \\\\\nw_{1^{\\prime }} &=&v_{0}\\otimes v_{1},\\quad w_{2}=v_{1}\\otimes v_{1}~.\n\\end{eqnarray*}%\nWe find that%\n\\begin{eqnarray*}\n\\mathcal{L}_{j}w_{0}\\otimes v_{0} &=&x\\ominus t_{j}~w_{0}\\otimes v_{0} \\\\\n\\mathcal{L}_{j}w_{0}\\otimes v_{1} &=&t_{j}\\ominus x~w_{0}\\otimes\nv_{1}+(1+\\beta t_{j}\\ominus x)w_{1}\\otimes v_{0}-\\beta t_{j}\\ominus\nx~w_{1^{\\prime }}\\otimes v_{0} \\\\\n\\mathcal{L}_{j}w_{1}\\otimes v_{0} &=&w_{1}\\otimes v_{0} \\\\\n\\mathcal{L}_{j}w_{1}\\otimes v_{1} &=&w_{1}\\otimes v_{1} \\\\\n\\mathcal{L}_{j}w_{1^{\\prime }}\\otimes v_{0} &=&w_1\\otimes v_0-x\\ominus\nt_{j}~w_{0}\\otimes v_{1} \\\\\n\\mathcal{L}_{j}w_{1^{\\prime }}\\otimes v_{1} &=&0 \\\\\n\\mathcal{L}_{j}w_{2}\\otimes v_{0} &=&(1+\\beta x\\ominus t_{j})w_{1}\\otimes\nv_{1}-\\beta x\\ominus t_{j}~w_{1^{\\prime }}\\otimes v_{1} \\\\\n\\mathcal{L}_{j}w_{2}\\otimes v_{1} &=&0\n\\end{eqnarray*}%\nThis action of the $\\mathcal{L}_{j}$ in the spin basis (\\ref{spin basis})\ncan be encoded in terms of the vertex configurations shown in Figure \\ref%\n{fig:func_eqn} with labels $0,1,1^{\\prime },2$ similarly as we did deduce\nthe action of $L$ and $L^{\\prime }$ from the vertex configurations in Figure %\n\\ref{fig:5vmodels}. Thus, the operator product $H(x)E(\\ominus x)$ can be\nwritten as the partial trace%\n\\begin{equation*}\nE(\\ominus x)H(x)=\\limfunc{Tr}_{V\\otimes V}\\left( \n\\begin{smallmatrix}\n1 & 0 & 0 & 0 \\\\ \n0 & q & 0 & 0 \\\\ \n0 & 0 & q & 0 \\\\ \n0 & 0 & 0 & q^{2}%\n\\end{smallmatrix}\n\\right) \\mathcal{L}_{N}\\cdots \\mathcal{L}_{2}\\mathcal{L}_{1}\n\\end{equation*}\nand its matrix elements in the quantum space $\\mathcal{V}_{n}^{q}$ are sums\nover the possible vertex configurations of Figure \\ref{fig:func_eqn} in a\nsingle lattice row of length $N$. This lattice row is closed and forms a\ncircle of circumference $N$, since the partial trace together with the\nmatrix containing the deformation parameter $q$ imposes quasi-periodic\nboundary conditions. Due to these periodicity conditions, one finds the\nfollowing constraints:\n\n\\begin{itemize}\n\\item the last vertex in the bottom row of Figure \\ref{fig:func_eqn} cannot\noccur;\n\n\\item the 2nd and 3rd vertex from the right in the top row always have to\ncome as a pair, but since one of them has weight zero their contribution can\nbe discarded;\n\n\\item configurations involving the second vertex from the left in the bottom\nrow do not contribute as they eventually lead to a vertex configuration\nshown at the 2nd position from the right in the top row which has weight\nzero;\n\n\\item the 2nd and 3rd vertex from the right in the bottom row always have to\ncome as an adjacent pair and it are these vertices which give rise to the\nterm involving $\\beta \\bar H_1$ as they correspond to shifting a 1-letter in\na binary string to the right.\n\\end{itemize}\n\nFrom these conditions, which can be checked graphically, one then deduces\nthe asserted identity (\\ref{func_eqn}) as only a very restricted number of\nvertices in Figure \\ref{fig:func_eqn} remain.\n\\end{proof}\n\n\\begin{corollary}[equivariant quantum Pieri-Chevalley rule]\nWe have the following explicit action of $H_{1}$ in terms of the basis $%\n\\{v_{\\lambda }\\}\\subset \\mathcal{V}_{n},$%\n\\begin{equation}\n(1+\\beta H_{1})v_{\\mu }=\\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\emptyset })}\\sum \n_{\\substack{ \\mu\\rightrightarrows\\lambda[d] \\\\ d=0,1}}q^{d}\\beta ^{|\\lambda\n\/d\/\\mu |}v_{\\lambda }, \\label{H1}\n\\end{equation}%\nwhere the sum runs over all $\\lambda \\subset (k^{n})$ such that either $%\n\\lambda \/\\mu $ or $\\lambda \/1\/\\mu $ is a skew diagram which contains at most\none box in each column or row. Moreover, the identity (\\ref{func_eqn1}) can\nbe rewritten as \n\\begin{equation}\nH(x|t)E(\\ominus x|t)=\\prod_{j=1}^{n}(t_{j}\\ominus\nx)\\prod_{j=n+1}^{N}(x\\ominus t_{j})\\,(1+\\beta H_{1})+q\\cdot 1\\;.\n\\label{func_eqn}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nActing with the first term on the right hand side of (\\ref{func_eqn1}) on a\nbasis vector $v_{\\lambda }$ we obtain%\n\\begin{multline*}\n(1+\\beta \\bar{H}_{1})\\prod_{j=1}^{N}(t_{j}\\ominus x)^{\\sigma _{j}^{+}\\sigma\n_{j}^{-}}(x\\ominus t_{j})^{\\sigma _{j}^{-}\\sigma _{j}^{+}}v_{\\lambda }= \\\\\n\\left( \\tprod\\nolimits_{j\\in I_{\\lambda }}t_{j}\\ominus x\\right) \\left(\n\\tprod\\nolimits_{j\\in I_{\\lambda ^{\\ast }}}x\\ominus t_{j}\\right) ~(1+\\beta \n\\bar{H}_{1})v_{\\lambda }= \\\\\n\\left( \\tprod\\nolimits_{j=1}^{n}t_{j}\\ominus x\\right) \\left(\n\\tprod\\nolimits_{j=n+1}^{N}x\\ominus t_{j}\\right) ~\\frac{\\Pi (t_{\\lambda })}{%\n\\Pi (t_{\\emptyset })}(1+\\beta \\bar{H}_{1})v_{\\lambda }\n\\end{multline*}%\nOn the other hand using the expansions (\\ref{Hr}) and (\\ref{Er}) we see that\nthe coefficients of the leading factorial powers are%\n\\begin{eqnarray*}\nH(x|t) &=&(x|\\ominus t^{\\prime })^{k}(1+\\beta H_{1})+\\cdots \\\\\nE(x|t) &=&(x|t)^{n}(1+\\beta E_{1})+\\cdots\n\\end{eqnarray*}%\nfrom which we deduce the desired identities with help of the left hand side\nof (\\ref{func_eqn1}) and (\\ref{facpower_id}). Namely, we have%\n\\begin{eqnarray*}\n(-1)^{n}\\frac{(1+\\beta x)^{n}}{\\Pi (t_{\\emptyset })}~E(\\ominus x)\n&=&(x|\\ominus t)^{n}\\sum_{r=0}^{n}(-1)^{r}\\beta ^{r}(E_{r}+\\beta\nE_{r+1})+~\\ldots \\\\\n&=&(x|\\ominus t)^{n}\\cdot 1+~\\ldots\n\\end{eqnarray*}%\nwhere the omitted terms involve factorial powers $(x|\\ominus t)^{p}$ with $%\npk$ and a solution $y=y_{\\mu }$ of the Bethe ansatz equations (%\n\\ref{BAE}), one has the identity%\n\\begin{multline}\nG_{\\lambda }(y|\\ominus t)= \\label{Greduction} \\\\\nq\\sum_{r=0}^{\\lambda _{1}-1-k}h_{\\lambda _{1}-1-k-r}(t_{1},\\ldots\n,t_{r+1})G_{(\\lambda _{2}-1,\\ldots ,\\lambda _{n}-1,r)}(y|\\ominus\nt)\\prod_{i=1}^{r}(1+\\beta t_{i}),\n\\end{multline}%\nwhere the $h_{r}$'s denote the complete symmetric functions and the\nfactorial Grothendieck polynomial on the right hand side is defined via (\\ref%\n{Gstraight}).\n\\end{lemma}\n\n\\begin{proof}\nRecall the determinant formula (\\ref{facGdet}) for factorial Grothendieck\npolynomials. Writing out the determinant in the numerator we find%\n\\begin{multline*}\na_{\\lambda }=\\left\\vert \n\\begin{array}{ccc}\n(y_{1}|\\ominus t)^{n+\\lambda _{1}-1} & \\cdots & (y_{n}|\\ominus t)^{n+\\lambda\n_{1}-1} \\\\ \n(y_{1}|\\ominus t)^{n+\\lambda _{2}-2}(1+\\beta y_{1}) & & (y_{n}|\\ominus\nt)^{n+\\lambda _{2}-2}(1+\\beta y_{n}) \\\\ \n\\vdots & & \\vdots \\\\ \n(y_{1}|\\ominus t)^{\\lambda _{n}}(1+\\beta y_{1})^{n-1} & \\cdots & \n(y_{n}|\\ominus t)^{\\lambda _{n}}(1+\\beta y_{n})^{n-1}%\n\\end{array}%\n\\right\\vert \\\\\n=\\frac{q}{\\Pi (y)}\\left\\vert \n\\begin{array}{ccc}\n(y_{1}|\\ominus t)^{n+\\lambda _{2}-2}(1+\\beta y_{1}) & \\cdots & \n(y_{n}|\\ominus t)^{n+\\lambda _{2}-2}(1+\\beta y_{n}) \\\\ \n\\vdots & & \\vdots \\\\ \n(y_{1}|\\ominus t)^{\\lambda _{n}}(1+\\beta y_{1})^{n-1} & & (y_{n}|\\ominus\nt)^{\\lambda _{n}}(1+\\beta y_{n})^{n-1} \\\\ \ny_{1}^{\\lambda _{1}-1-k}(1+\\beta y_{1})^{n} & \\cdots & y_{n}^{\\lambda\n_{1}-1-k}(1+\\beta y_{n})^{n}%\n\\end{array}%\n\\right\\vert\n\\end{multline*}%\nHere we have made use of (\\ref{BAE}), exchanged the first row with the last\nrow in the determinant and used row linearity of the determinant to pull out\nthe common factor in front. Note that $t_{j}=0$ for $j>N$, whence the powers\nin the bottom row are not factorial. To rewrite them as factorial powers we\nuse the equality%\n\\begin{equation*}\nx^{m}=\\sum_{r=0}^{m}(x|\\ominus t)^{m-r}h_{r}(t_{1},\\ldots\n,t_{m+1-r})\\prod_{i=1}^{m-r}(1+\\beta t_{i})\n\\end{equation*}%\nwhich is easily proved via induction using the known recursion relation%\n\\begin{equation*}\nh_{r+1}(t_{1},\\ldots ,t_{m+1-r})=h_{r}(t_{1},\\ldots\n,t_{m+1-r})t_{m+1-r}+h_{r+1}(t_{1},\\ldots ,t_{m-r})\n\\end{equation*}%\nof the complete symmetric functions. We leave this step to the reader.\n\nThus, after employing the above identity and column\/row linearity of the\ndeterminant we arrive at%\n\\begin{equation*}\na_{\\lambda }=q\\sum_{r=0}^{\\lambda _{1}-1-k}a_{(\\lambda _{2}-1,\\ldots\n,\\lambda _{n}-1,\\lambda _{1}-1-k-r)}h_{r}(t_{1},\\ldots ,t_{\\lambda\n_{1}-k-r})\\prod_{i=1}^{\\lambda _{1}-1-k-r}(1+\\beta t_{i})\n\\end{equation*}%\nwhich is the asserted identity (\\ref{Greduction}) after dividing by the\nVandermonde determinant.\n\\end{proof}\n\n\\begin{theorem}\nThe on-shell Bethe vectors (\\ref{Bethev}), (\\ref{Bethev'}) and (\\ref%\n{leftBethev}), (\\ref{leftBethev'}) form respectively right and left\neigenbases of the transfer matrices $H$ and $E$ in each subspace $\\mathcal{V}%\n_{n}^{q}$ with eigenvalue equations%\n\\begin{equation}\nH(x|t)|y_{\\mu }\\rangle =\\left( \\frac{\\prod\\limits_{j=1}^{N}x\\ominus\nt_{j}+(-1)^{n}q\\prod\\limits_{i\\in I_{\\mu }}\\left( 1+\\beta x\\ominus\ny_{i}\\right) }{\\prod\\limits_{i\\in I_{\\mu }}x\\ominus y_{i}}\\right) ~|y_{\\mu\n}\\rangle \\label{specH}\n\\end{equation}%\nand%\n\\begin{equation}\nE(x|t)|z_{\\mu }\\rangle =\\left( \\frac{\\prod\\limits_{j=1}^{N}x\\oplus\nt_{j}+(-1)^{n}q\\prod\\limits_{i\\in I_{\\mu ^{\\ast }}}\\left( 1+\\beta x\\ominus\nz_{i}\\right) }{\\prod\\limits_{i\\in I_{\\mu ^{\\ast }}}x\\ominus z_{i}}\\right)\n~|z_{\\mu }\\rangle \\;. \\label{specE}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nHere we use the commutation relations of the Yang-Baxter algebra as per\nLemma \\ref{lem:yba1} and (\\ref{A0_and_D0}) from which we deduce that if (\\ref%\n{BAE}) holds the Bethe vector (\\ref{Bethev}) is an eigenvector of $H=A+qD$.\nThe computation follows along the same lines for (\\ref{Bethev'}) and the\nleft eigenvectors (\\ref{leftBethev}, \\ref{leftBethev'}).\n\nOne deduces that the eigenvalues must separate points and, hence, $\\langle\ny_{\\lambda }|y_{\\mu }\\rangle =\\langle z_{\\lambda }|z_{\\mu }\\rangle =0$ for $%\n\\lambda \\neq \\mu $. That these eigenvectors form a basis then follows from\nthe fact that there exist $\\dim \\mathcal{V}_{n}=\\binom{N}{n}$ solutions to\nthe equations (\\ref{BAE}); see Lemma \\ref{lem:BAE}.\n\\end{proof}\n\nNote that the above formulae simplify if $q=0$. Then the on-shell Bethe\nvectors with $y_{\\mu }=t_{\\mu }$ are given by%\n\\begin{equation}\n|t_{\\mu }\\rangle =\\sum_{\\lambda \\subset (k^{n})}G_{\\lambda ^{\\vee }}(t_{\\mu\n}|\\ominus t^{\\prime })\\frac{\\Pi (t_{\\mu })}{\\Pi (t_{\\lambda })}~v_{\\lambda }\n\\label{Bethev0}\n\\end{equation}%\nand form an eigenbasis of the transfer matrices with eigenvalues,%\n\\begin{eqnarray}\nH(x|t)|t_{\\mu }\\rangle &=&\\left( \\tprod\\limits_{j\\in I_{\\mu ^{\\ast\n}}}x\\ominus t_{j}\\right) |t_{\\mu }\\rangle \\label{specH0} \\\\\nE(x|t)|t_{\\mu }\\rangle &=&\\left( \\tprod\\limits_{j\\in I_{\\mu }}x\\oplus\nt_{j}\\right) |t_{\\mu }\\rangle \\label{specE0}\n\\end{eqnarray}%\nAs we will discuss below this special case describes generalised equivariant\ncohomology theory, $h_{n}^{\\ast }=qh_{n}^{\\ast }\/\\langle q\\rangle $ and we\nshow below that $h_{n}^{\\ast }\/\\langle \\beta +1\\rangle \\cong K_{\\mathbb{T}}(%\n\\limfunc{Gr}_{n,N})$.\n\n\\begin{proposition}\n\\label{prop:Gduality}The eigenvectors of $H$ and $E$ coincide under the\nsubstitution $z_{\\lambda ^{\\prime }}=\\ominus y_{\\lambda ^{\\vee }}$ and,\nthus, we have the equality%\n\\begin{equation}\nG_{\\lambda ^{\\prime }}(\\ominus y_{\\mu ^{\\ast }}|t)=G_{\\lambda }(y_{\\mu\n}|\\ominus t^{\\prime }) \\label{Gduality}\n\\end{equation}%\nfor each solution $y_{\\mu }$ of (\\ref{BAE}). In particular, for $q=0$ we\nhave $G_{\\lambda ^{\\prime }}(\\ominus t_{\\mu ^{\\ast }}|t)=G_{\\lambda }(t_{\\mu\n}|\\ominus t^{\\prime })$.\n\\end{proposition}\n\n\\begin{proof}\nUsing the identity (\\ref{levelrankmom}) when acting on the Bethe vectors we\nfind%\n\\begin{eqnarray*}\n\\Theta H(x|t)|y_{\\lambda }\\rangle &=&E(x|\\ominus t^{\\prime })\\Theta\n|y_{\\lambda }\\rangle \\\\\n\\Theta E(x|t)|z_{\\lambda }\\rangle &=&H(x|\\ominus t^{\\prime })\\Theta\n|z_{\\lambda }\\rangle\n\\end{eqnarray*}%\nand%\n\\begin{eqnarray*}\n\\Theta |y_{\\lambda }\\rangle &=&C^{\\prime }(y_{1}|\\ominus t^{\\prime })\\cdots\nC^{\\prime }(y_{n}|\\ominus t^{\\prime })|N\\rangle \\\\\n\\Theta |z_{\\lambda }\\rangle &=&B(z_{1}|\\ominus t^{\\prime })\\cdots\nB(z_{k}|\\ominus t^{\\prime })|0\\rangle\n\\end{eqnarray*}%\nThese identities together with the expansion (\\ref{Brootexpansion}) allows\nus to identify $z_{\\lambda ^{\\prime }}=\\ominus y_{\\lambda ^{\\vee }}$.\n\\end{proof}\n\n\\begin{corollary}\nThe eigenvalue equation (\\ref{specE}) of the $E$-transfer matrix simplifies\nin the Bethe roots (\\ref{ylambda}) to \n\\begin{equation}\nE(x|t)|y_{\\mu }\\rangle =\\prod\\limits_{i\\in I_{\\mu }}\\left( x\\oplus\ny_{i}\\right) ~|y_{\\mu }\\rangle \\;. \\label{SpecE}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nSetting $x=t_{j}$ the functional equation (\\ref{func_eqn}) together with (%\n\\ref{specH}) implies%\n\\begin{equation*}\nq|y_{\\mu }\\rangle =E(\\ominus t_{j})H(t_{j})|y_{\\mu }\\rangle =\\frac{q}{%\n\\prod\\limits_{i=1}^{n}\\left( t_{j}\\ominus y_{i}\\right) }E(\\ominus\nt_{j})|y_{\\mu }\\rangle\n\\end{equation*}%\nfor all $j=1,2,\\ldots ,N$. Since the $t_{j}$'s are arbitrary variables and\nthe Bethe vectors form an eigenbasis the assertion follows.\n\\end{proof}\n\nSince the Bethe vectors (\\ref{Bethev}) and (\\ref{leftBethev}) form each an\neigenbasis they give rise to a resolution of the identity $\\boldsymbol{1}%\n=\\sum_{\\alpha \\in (n,k)}|y_{\\alpha }\\rangle \\langle y_{\\alpha }|$ where $%\n|y_{\\alpha }\\rangle \\langle y_{\\alpha }|$ denotes the orthogonal projector\nonto the eigenspace spanned by $|y_{\\alpha }\\rangle $. This elementary fact\nof linear algebra translates into the following non-trivial identities for\nfactorial Grothendieck polynomials evaluated at solutions of the Bethe\nansatz equations (\\ref{BAE}).\n\n\\begin{corollary}[orthogonality \\& completeness]\nFor all $\\lambda ,\\mu \\subset (k^{n})$ we have the identities%\n\\begin{equation}\n\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi (y_{\\lambda })}{\\Pi (t_{\\alpha })}%\n\\frac{G_{\\alpha ^{\\vee }}(y_{\\lambda }|\\ominus t^{\\prime })G_{\\alpha\n}(y_{\\mu }|\\ominus t)}{e(y_{\\lambda },y_{\\lambda })}=\\delta _{\\lambda \\mu }\n\\label{Cauchy2}\n\\end{equation}%\nand%\n\\begin{equation}\n\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi (y_{\\alpha })}{\\Pi (t_{\\lambda })}%\n\\frac{G_{\\lambda ^{\\vee }}(y_{\\alpha }|\\ominus t^{\\prime })G_{\\mu\n}(y_{\\alpha }|\\ominus t)}{e(y_{\\alpha },y_{\\alpha })}=\\delta _{\\lambda \\mu\n}\\;, \\label{res of 1}\n\\end{equation}%\nwhere $\\delta _{\\lambda \\mu }$ denotes the Kronecker delta with $\\delta\n_{\\lambda \\mu }=1$ if $\\lambda =\\mu $ and $0$ otherwise.\n\\end{corollary}\n\n\\subsection{Generalised matrix algebras and Frobenius structures}\n\nFollowing the suggested construction in \\cite[Section 7]{Korffproc} we now\nintroduce a ring structure on each $\\mathcal{V}_{n}^{q}=\\mathbb{Z}%\n[\\![q]\\!]\\otimes \\mathcal{V}_{n}$ by interpreting the on-shell Bethe vectors\n(\\ref{Bethev}) as central orthogonal idempotents of a semisimple algebra:\nfor each $n=0,1,\\ldots ,N$ define $qh_{n}^{\\ast }=(\\mathcal{V}%\n_{n}^{q},\\circledast )$ by fixing the product $\\circledast $ as follows,%\n\\begin{equation}\nY_{\\lambda }\\circledast Y_{\\mu }=\\delta _{\\lambda \\mu }Y_{\\mu }~,\\qquad\nY_{\\lambda }=e(y_{\\lambda },y_{\\lambda })^{-1}|y_{\\lambda }\\rangle \\;,\n\\label{idempotent_def}\n\\end{equation}%\nwhere $e(y_{\\lambda },y_{\\lambda })$ is the matrix element defined in (\\ref%\n{norm}). Note that $e(y_{\\lambda },y_{\\lambda })$ is a power series in $q$\nwith nonzero constant term (\\ref{norm0}) according to (\\ref{Brootexpansion}%\n). The unit element is given by \n\\begin{equation}\nv_{\\emptyset }=\\sum_{\\lambda \\subset (k^{n})}Y_{\\lambda }\\;. \\label{unit}\n\\end{equation}%\nThis determines $qh_{n}^{\\ast }$ via its Peirce decomposition \\cite{Peirce}.\nWe turn $qh_{n}^{\\ast }$ into a Frobenius algebra by introducing in addition\nthe following symmetric bilinear form $\\mathcal{V}_{n}^{q}\\times \\mathcal{V}%\n_{n}^{q}\\rightarrow \\mathcal{R}(\\mathbb{T},q)$,%\n\\begin{equation}\n(Y_{\\lambda },Y_{\\mu })=e(y_{\\lambda },y_{\\lambda })^{-1}\\delta _{\\lambda\n\\mu }\\;. \\label{bilinear_form}\n\\end{equation}%\nBy definition this bilinear form is invariant with respect to the product (%\n\\ref{idempotent_def}) and non-degenerate, since the Bethe vectors form a\nbasis.\n\n\\subsection{A residue formula for the structure constants}\n\nWe now describe the resulting generalised matrix algebra $qh_{n}^{\\ast }$ in\nthe spin basis $\\{v_{\\lambda }\\}_{\\lambda \\subset (k^{n})}$. Introduce a\nfamily of operators $\\{\\boldsymbol{G}_{\\lambda }\\}_{\\lambda \\subset\n(k^{n})}\\subset \\limfunc{End}\\mathcal{V}_{n}^{q}$ via the following\neigenvalue equation \n\\begin{equation}\n\\boldsymbol{G}_{\\lambda }Y_{\\mu }=G_{\\lambda }(y_{\\mu }|\\ominus t)Y_{\\mu }\\;.\n\\label{BigG}\n\\end{equation}%\nThis \\emph{defines} the operators $\\boldsymbol{G}_{\\lambda }$, since the\nBethe vectors form an eigenbasis and the eigenvalues separate points. Recall\nfrom Section \\ref{sec:grothendieck} that the factorial Grothendieck\npolynomials form a basis \\cite[Thm 4.6]{McNamara}. Below we give an\nexplicit, basis independent construction of $\\boldsymbol{G}_{\\lambda }$ in\nterms of the transfer matrix $H(x)$.\n\n\\begin{corollary}\nIn the spin basis (\\ref{spin basis}) the product (\\ref{idempotent_def}) is\ngiven by \n\\begin{equation}\nv_{\\lambda }\\circledast v_{\\mu }=\\boldsymbol{G}_{\\lambda }v_{\\mu }=\\sum_{\\nu\n\\subset (k^{n})}C_{\\lambda \\mu }^{\\nu }(t,q)v_{\\nu }, \\label{combproduct}\n\\end{equation}%\nwhere the structure constants $C_{\\lambda \\mu }^{\\nu }(t,q)=\\langle \\nu |%\n\\boldsymbol{G}_{\\lambda }|\\lambda \\rangle $ are obtained in terms of the\nBethe roots (\\ref{ylambda}) via the residue formula%\n\\begin{equation}\nC_{\\lambda \\mu }^{\\nu }(t,q)=\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi\n(y_{\\alpha })}{\\Pi (t_{\\nu })}\\frac{G_{\\lambda }(y_{\\alpha }|\\ominus\nt)G_{\\mu }(y_{\\alpha }|\\ominus t)G_{\\nu ^{\\ast }}(\\ominus y_{\\alpha ^{\\ast\n}}|t)}{e(y_{\\alpha },y_{\\alpha })}\\;. \\label{Verlinde}\n\\end{equation}%\nSimilarly, the bilinear form (\\ref{bilinear_form}) can be expressed as%\n\\begin{equation}\n(v_{\\lambda },v_{\\mu })=\\sum_{\\alpha \\subset (k^{n})}\\frac{G_{\\lambda\n}(y_{\\alpha }|\\ominus t)G_{\\mu }(y_{\\alpha }|\\ominus t)}{e(y_{\\alpha\n},y_{\\alpha })}\\;. \\label{bilinear_form_spin}\n\\end{equation}\n\\end{corollary}\n\n\\begin{remark}\nOur residue formula (\\ref{Verlinde}) is a generalisation of the\nBertram-Vafa-Intriligator formula for Gromov-Witten invariants. It holds\nalso true for $q=0$, where the Bethe roots are explicitly known, $%\ny_{i}=t_{i} $,%\n\\begin{equation}\nc_{\\lambda \\mu }^{\\nu }(t)=C_{\\lambda \\mu }^{\\nu }(t,0)=\\sum_{\\alpha \\subset\n(k^{n})}\\frac{\\Pi (t_{\\alpha })}{\\Pi (t_{\\nu })}\\frac{G_{\\lambda }(t_{\\alpha\n}|\\ominus t)G_{\\mu }(t_{\\alpha }|\\ominus t)G_{\\nu ^{\\ast }}(\\ominus\nt_{\\alpha ^{\\ast }}|t)}{\\prod_{i\\in I_{\\alpha },j\\in I_{\\alpha ^{\\ast\n}}}t_{i}\\ominus t_{j}}\\;. \\label{Verlinde0}\n\\end{equation}%\nThe bilinear form (\\ref{bilinear_form_spin}) for $q=0$ reads%\n\\begin{equation}\n(v_{\\lambda },v_{\\mu })=\\sum_{\\alpha \\subset (k^{n})}\\frac{G_{\\lambda\n}(t_{\\alpha }|\\ominus t)G_{\\mu }(t_{\\alpha }|\\ominus t)}{\\prod_{i\\in\nI_{\\alpha },j\\in I_{\\alpha ^{\\ast }}}t_{i}\\ominus t_{j}}\\;.\n\\label{bilinear_form0}\n\\end{equation}\n\\end{remark}\n\n\\begin{proof}\nAccording to (\\ref{rightBethe}) and (\\ref{res of 1}) we have the inverse\nbasis transformation%\n\\begin{equation}\nv_{\\lambda }=\\sum_{\\mu \\subset (k^{n})}G_{\\lambda }(y_{\\mu }|\\ominus\nt)Y_{\\mu }\\;. \\label{Bethe2spin}\n\\end{equation}%\nwhich allows us to compute%\n\\begin{eqnarray*}\nv_{\\lambda }\\circledast v_{\\mu } &=&\\sum_{\\rho ,\\sigma }G_{\\lambda }(y_{\\rho\n}|\\ominus t)G_{\\mu }(y_{\\sigma }|\\ominus t)Y_{\\rho }\\circledast Y_{\\sigma }\n\\\\\n&=&\\sum_{\\rho }G_{\\lambda }(y_{\\rho }|\\ominus t)G_{\\mu }(y_{\\rho }|\\ominus\nt)Y_{\\rho }=\\boldsymbol{G}_{\\lambda }v_{\\mu }=\\boldsymbol{G}_{\\mu\n}v_{\\lambda }\\;.\n\\end{eqnarray*}%\nThis proves the first assertion. Continuing the computation from the second\nline employing (\\ref{rightBethe}) we arrive at (\\ref{Verlinde}).\n\nThe expression (\\ref{bilinear_form_spin}) is also an immediate consequence\nof (\\ref{Bethe2spin}). Insert the latter and use the definition (\\ref%\n{bilinear_form}) to find the asserted identity (\\ref{bilinear_form_spin}).\n\\end{proof}\n\nAs is to be expected from our previous results (\\ref{LRduality_L}) and (\\ref%\n{levelrankmom}), the rings related by exchanging the dimension $n$ with the\ncodimension $k$ of the hyperplanes in the Grassmannian are closely related.\n\n\\begin{corollary}[level-rank duality]\nThe involution $qh_{n}^{\\ast }\\rightarrow qh_{k}^{\\ast }$ given by $%\nf(t,q)v_{\\lambda }\\mapsto f(\\ominus t^{\\prime },q)v_{\\lambda ^{\\prime }}$ is\na ring isomorphism over $\\mathcal{R}\\otimes \\mathbb{Z}[\\![q]\\!]$. That is,%\n\\begin{equation}\nC_{\\mu \\nu }^{\\lambda }(t,q)=C_{\\mu ^{\\prime }\\nu ^{\\prime }}^{\\lambda\n^{\\prime }}(\\ominus t^{\\prime },q)\\;. \\label{levelrank}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nFirst we note that (\\ref{G1}) and (\\ref{Gduality}) imply the identity%\n\\begin{eqnarray*}\n\\frac{\\Pi (y_{\\lambda })}{\\Pi (t_{\\mu })} &=&\\frac{\\Pi (t_{\\emptyset })}{\\Pi\n(t_{\\mu })}~\\left( 1+\\beta G_{1}(y_{\\lambda }|\\ominus t)\\right) \\\\\n&=&\\frac{\\Pi (\\ominus t_{\\emptyset }^{\\prime })}{\\Pi (\\ominus t_{\\mu\n^{\\prime }}^{\\prime })}~\\left( 1+\\beta G_{1}(\\ominus y_{\\lambda ^{\\ast\n}}|t^{\\prime })\\right) =\\frac{\\Pi (\\ominus y_{\\lambda ^{\\ast }})}{\\Pi\n(\\ominus t_{\\mu ^{\\prime }}^{\\prime })}\\;.\n\\end{eqnarray*}%\nNote further that according to (\\ref{Brootexpansion}) the $k$-tuple $\\ominus\ny_{\\lambda ^{\\ast }}$ is obtained from solutions $y_{i}$ by replacing $%\nt=(t_{1},\\ldots ,t_{N})$ with $\\ominus t^{\\prime }=(\\ominus t_{N},\\ldots\n,\\ominus t_{1})$, i.e. the constant terms of the components of the solution $%\n\\ominus y_{\\lambda ^{\\ast }}$ are $\\ominus t_{\\lambda ^{\\prime }}^{\\prime }$\nwhich identifies the solution uniquely. Using the residue formula (\\ref%\n{Verlinde}) and (\\ref{Gduality}) we compute%\n\\begin{multline*}\nC_{\\mu \\nu }^{\\lambda }(t,q)=\\sum_{\\alpha \\subset (k^{n})}\\frac{\\Pi\n(y_{\\alpha })}{\\Pi (t_{\\nu })}\\frac{G_{\\lambda }(y_{\\alpha }|\\ominus\nt)G_{\\mu }(y_{\\alpha }|\\ominus t)G_{\\nu ^{\\vee }}(y_{\\alpha }|\\ominus\nt^{\\prime })}{e(y_{\\alpha },y_{\\alpha })}= \\\\\n\\sum_{\\alpha }\\frac{\\Pi (\\ominus y_{\\alpha ^{\\ast }})}{\\Pi (\\ominus t_{\\nu\n^{\\prime }}^{\\prime })}\\frac{G_{\\lambda ^{\\prime }}(\\ominus y_{\\alpha ^{\\ast\n}}|t^{\\prime })G_{\\mu ^{\\prime }}(\\ominus y_{\\alpha ^{\\ast }}|t^{\\prime\n})G_{\\nu ^{\\ast }}(\\ominus y_{\\alpha ^{\\ast }}|t)}{e(y_{\\alpha },y_{\\alpha })%\n}=C_{\\mu ^{\\prime }\\nu ^{\\prime }}^{\\lambda ^{\\prime }}(\\ominus t^{\\prime\n},q),\n\\end{multline*}%\nwhere in the last step we have used the definition (\\ref{norm}) to show that%\n\\begin{eqnarray*}\ne(y_{\\alpha },y_{\\alpha }) &=&\\sum_{\\lambda \\subset (k^{n})}\\frac{\\Pi\n(y_{\\alpha })}{\\Pi (t_{\\lambda })}G_{\\lambda }(y_{\\alpha }|\\ominus\nt)G_{\\lambda ^{\\vee }}(y_{\\alpha }|\\ominus t^{\\prime }) \\\\\n&=&\\sum_{\\lambda \\subset (k^{n})}\\frac{\\Pi (\\ominus y_{\\alpha ^{\\ast }})}{%\n\\Pi (\\ominus t_{\\lambda ^{\\prime }}^{\\prime })}G_{\\lambda ^{\\prime\n}}(\\ominus y_{\\alpha ^{\\ast }}|t^{\\prime })G_{\\lambda ^{\\ast }}(\\ominus\ny_{\\alpha ^{\\ast }}|t)=e(\\ominus y_{\\alpha ^{\\ast }},\\ominus y_{\\alpha\n^{\\ast }})\\;.\n\\end{eqnarray*}\n\\end{proof}\n\n\\subsection{A recurrence formula}\n\nWe now return to the result (\\ref{H1}) and show that the latter formula\ndescribes the multiplication with the class of the Schubert divisor, i.e.\nthat (\\ref{H1}) describes indeed the equivariant quantum Pieri-Chevalley\nrule for the generalised cohomology ring $qh_{n}^{\\ast }$.\n\n\\begin{corollary}\nLet $\\lambda =(1,0,\\ldots ,0)$ then%\n\\begin{equation}\n\\boldsymbol{G}_{1}=H_{1} \\label{G1=H1}\n\\end{equation}%\nand the product $v_{1}\\circledast v_{\\lambda }=H_{1}v_{\\lambda }$ in the\nspin basis is given explicitly via (\\ref{H1}).\n\\end{corollary}\n\n\\begin{proof}\nEmploying the functional equation (\\ref{func_eqn}) and (\\ref{specH}), (\\ref%\n{SpecE}) we obtain%\n\\begin{multline*}\n\\prod_{j=1}^{n}(t_{j}\\ominus x)\\prod_{j=n+1}^{N}(x\\ominus t_{j})(1+\\beta\nH_{1})Y_{\\mu }=(H(x)E(\\ominus x)-q\\cdot 1)Y_{\\mu } \\\\\n=(-1)^{n}\\frac{\\Pi (y_{\\mu })}{(1+\\beta x)^{n}}\\prod_{j=1}^{N}(x\\ominus\nt_{j})~Y_{\\mu } \\\\\n=\\frac{\\Pi (y_{\\mu })}{\\Pi (t_{\\emptyset })}\\prod_{j=1}^{n}(t_{j}\\ominus\nx)\\prod_{j=n+1}^{N}(x\\ominus t_{j})~Y_{\\mu }\n\\end{multline*}%\nThus, according to (\\ref{G1Pieri}), (\\ref{BigG}) we have%\n\\begin{equation*}\n(1+\\beta H_{1})Y_{\\mu }=\\frac{\\Pi (y_{\\mu })}{\\Pi (t_{\\emptyset })}~Y_{\\mu\n}=(1+\\beta \\boldsymbol{G}_{1})Y_{\\mu }\n\\end{equation*}%\nand the assertion follows from the fact that the Bethe vectors form a basis.\n\\end{proof}\n\nAnalogous to the case of equivariant (quantum) cohomology one derives from\nthe quantum Pieri-Chevalley rule (\\ref{H1}) the following recurrence\nrelation for the structure constants.\n\n\\begin{corollary}[Recurrence relation]\nWe have the identity%\n\\begin{equation}\n\\left( \\Pi (t_{\\nu })-\\Pi (t_{\\lambda })\\right) C_{\\lambda \\mu }^{\\nu\n}=\\sum_{\\tilde{\\lambda}\/d^{\\prime }\/\\lambda }\\beta ^{|\\tilde{\\lambda}%\n\/d\/\\lambda |}C_{\\tilde{\\lambda}\\mu }^{\\nu }-\\sum_{\\nu \/d^{\\prime \\prime }\/%\n\\tilde{\\nu}}\\beta ^{|\\nu \/d^{\\prime \\prime }\/\\tilde{\\nu}|}\\Pi (t_{\\tilde{\\nu}%\n})C_{\\lambda \\mu }^{\\tilde{\\nu}}, \\label{recurr}\n\\end{equation}%\nwhere the sums run over all partitions $\\tilde{\\lambda}\\neq \\lambda ,\\tilde{%\n\\nu}\\neq \\nu $ such that respectively $\\tilde{\\lambda}\/d^{\\prime }\/\\lambda $\nand $\\nu \/d^{\\prime \\prime }\/\\nu $ are toric skew-diagrams with $d^{\\prime\n},d^{\\prime \\prime }$ either $0$ or $1$ and where each row and column\ncontains at most one box.\n\\end{corollary}\n\n\\begin{proof}\nThe derivation follows the same idea as in ordinary (quantum) cohomology;\nsee e.g. \\cite{KnutsonTao}. Since the product $\\circledast $ by definition\nis associative we have in light of (\\ref{G1=H1}) that%\n\\begin{equation*}\n\\lbrack (1+\\beta H_{1})v_{\\lambda }]\\circledast v_{\\mu }=(1+\\beta\nH_{1})(v_{\\lambda }\\circledast v_{\\mu })\\;.\n\\end{equation*}%\nApplying the Pieri-Chevalley rule (\\ref{H1}) on both sides of the equality\nsign and comparing coefficients the assertion follows.\n\\end{proof}\n\n\\begin{example}\n\\label{ex:4}Consider once more the simplest non-trivial case $\\limfunc{Gr}%\n_{1,3}=\\mathbb{P}^{2}$. Let $\\lambda =\\mu =(2)$ and $\\nu =\\emptyset $. Then $%\n\\Pi (t_{\\nu })=1+\\beta t_{1}$, $\\Pi (t_{\\lambda })=1+\\beta t_{3}$ and $%\n\\tilde{\\lambda}=\\emptyset ,$ $\\tilde{\\nu}=(2)$ with $d^{\\prime }=d^{\\prime\n\\prime }=1$ are the only boxed partitions which give rise to allowed\ncylindric skew diagrams. Therefore, we arrive at the relation%\n\\begin{equation*}\n\\beta (t_{1}-t_{3})C_{22}^{\\emptyset }=q\\beta C_{\\emptyset 2}^{\\emptyset\n}-q\\beta (1+\\beta t_{3})C_{22}^{2}=-q\\beta (1+\\beta t_{3})C_{22}^{2},\n\\end{equation*}%\nwhere we have used that $v_{\\emptyset }$ is the unit and we therefore must\nhave $C_{\\emptyset 2}^{\\emptyset }=0$. Similarly, setting $\\nu =1$ we obtain%\n\\begin{equation*}\n\\beta (t_{2}-t_{3})C_{22}^{1}=q\\beta C_{\\emptyset 2}^{1}-\\beta (1+\\beta\nt_{1})C_{22}^{\\emptyset }\\;.\n\\end{equation*}%\nThus, we end up with the recursion%\n\\begin{equation*}\nC_{22}^{\\emptyset }=q\\frac{1+\\beta t_{3}}{t_{3}-t_{1}}~C_{22}^{2},\\qquad\nC_{22}^{1}=\\frac{1+\\beta t_{1}}{t_{3}-t_{2}}~C_{22}^{\\emptyset }\n\\end{equation*}%\nwith $C_{22}^{2}=(t_{3}\\ominus t_{2})(t_{3}\\ominus t_{1})$. Thus,%\n\\begin{equation*}\nC_{22}^{\\emptyset }=q(t_{3}\\ominus t_{2})\\frac{1+\\beta t_{3}}{1+\\beta t_{1}}%\n,\\qquad C_{22}^{1}=q\\frac{1+\\beta t_{3}}{1+\\beta t_{2}}\n\\end{equation*}%\nwhich is in agreement with our earlier computation and the product expansion\nin \\cite[Sec 5.5]{BM} upon setting $t_{i}=1-e^{\\varepsilon _{4-i}}$ and $%\n\\beta =-1$.\n\\end{example}\n\n\\section{Localised Schubert classes and GKM theory}\n\nAn important result in (ordinary) equivariant quantum cohomology and\nequivariant K-theory is that the respective rings have a purely algebraic\nrealisation by restricting Schubert classes to the fixed points under the\ntorus action. This monomorphism becomes a ring isomorphism with respect to\npointwise multiplication if one imposes the Goresky-Kottwitz-MacPherson\n(GKM) conditions \\cite[Thm 1.2.2]{GKM}; see \\cite[Thm 3.13]{KostantKumar2}\nfor the analogous statement in K-theory. We now show that this algebraic\nrealisation naturally emerges from our lattice model approach for our\ngeneralised cohomology theories $qh_{n}^{\\ast }$.\n\n\\subsection{Generalised difference operators and Iwahori-Hecke algebras}\n\nWe recall that the ring $\\mathcal{R}(\\mathbb{T})=\\mathcal{R}(t_{1},\\ldots\n,t_{N})$ is naturally endowed with an $\\mathbb{S}_{N}$-action by permuting\nthe equivariant parameters. By abuse of notation we will identify\npermutations $w\\in \\mathbb{S}_{N}$ with their operators acting on $\\mathcal{R%\n}(\\mathbb{T})$. This $\\mathbb{S}_{N}$-action can be used to define a\nrepresentation of a generalised (affine) Hecke or Iwahori algebra $\\mathbb{H}%\n_{N}(\\beta )$.\n\n\\begin{definition}\nDenote by $\\mathbb{H}_{N}(\\beta )$ the associative unital algebra with the\nfollowing generators and relations%\n\\begin{equation}\n\\pi _{i}^{2}=\\beta \\pi _{i}\\qquad \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and}\\qquad \\left\\{ \n\\begin{array}{cc}\n\\pi _{i}\\pi _{j}=\\pi _{j}\\pi _{i}, & (i-j)\\func{mod}N\\neq \\pm 1 \\\\ \n\\pi _{i}\\pi _{i+1}\\pi _{i}=\\pi _{i+1}\\pi _{i}\\pi _{i+1}, & \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{else}%\n\\end{array}%\n\\right. \\label{Iwahori_def}\n\\end{equation}%\nwhere all indices are understood modulo $N$. Denote by $\\mathbb{H}_{N}^{%\n\\limfunc{fin}}(\\beta )$ the subalgebra generated by $\\{\\pi _{1},\\ldots ,\\pi\n_{N-1}\\}$.\n\\end{definition}\n\nThe subring $\\mathcal{R}[t_{1},\\ldots ,t_{N}]\\subset \\mathcal{R}(\\mathbb{T})$\nand $\\mathcal{R}(\\mathbb{T})$ itself are both $\\mathbb{H}_{N}^{\\limfunc{fin}%\n}(\\beta )$-modules with respect to the following action in terms of isobaric\ndivided difference operators%\n\\begin{equation}\n\\partial _{j}=(1+\\beta t_{j})~\\frac{1-s_{j}}{t_{j}-t_{j+1}}\\;,\n\\label{divided_difference}\n\\end{equation}%\nwhere $s_{j}$ is the simple transposition interchanging $t_{j}$ and $t_{j+1}$%\n. Note that setting $\\beta =0$ we obtain a representation of the nil-Coxeter\nalgebra $\\mathbb{A}_{N}=\\mathbb{H}_{N}(0)$ and when setting $\\beta =-1$ a\nrepresentation of the nil-Hecke algebra $\\mathbb{H}_{N}=\\mathbb{H}_{N}(-1)$.\n\n\\begin{proposition}[braid matrices]\nLet $p_{j}:V_{n}\\rightarrow V_{n}$ the operator which permutes vectors in\nthe $j$th and $(j+1)$th factor and acts everywhere else trivially, i.e. $%\np_{j}v_{b}=v_{s_{j}b}$. Then the matrices $\\{\\hat{r}%\n_{j}(t_{j},t_{j+1})=p_{j}r_{j+1,j}(t_{j+1}\\ominus t_{j})\\}_{j=1}^{N}$\\ act\non the standard basis $\\{v_{b}\\}_{|b|=n}$ via \n\\begin{equation}\n\\hat{r}_{j}(t_{j},t_{j+1})v_{b}=\\left\\{ \n\\begin{array}{cc}\n(1+\\beta t_{j+1}\\ominus t_{j})v_{b}+q^{-\\delta _{j,N}}t_{j+1}\\ominus\nt_{j}~v_{s_{j}b}, & b_{j}\\tw@\\int\\intkern@\\fi \n \\ifnum\\intno@>\\thr@@\\int\\intkern@\\fi \n \\int\n\\def\\multintlimits@{\\intop\\ifnum\\intno@=\\z@\\intdots@\\else\\intkern@\\fi\n \\ifnum\\intno@>\\tw@\\intop\\intkern@\\fi\n \\ifnum\\intno@>\\thr@@\\intop\\intkern@\\fi\\intop}%\n\\def\\intic@{%\n \\mathchoice{\\hskip.5em}{\\hskip.4em}{\\hskip.4em}{\\hskip.4em}}%\n\\def\\negintic@{\\mathchoice\n {\\hskip-.5em}{\\hskip-.4em}{\\hskip-.4em}{\\hskip-.4em}}%\n\\def\\ints@@{\\iflimtoken@ \n \\def\\ints@@@{\\iflimits@\\negintic@\n \\mathop{\\intic@\\multintlimits@}\\limits \n \\else\\multint@\\nolimits\\fi \n \\eat@\n \\else \n \\def\\ints@@@{\\iflimits@\\negintic@\n \\mathop{\\intic@\\multintlimits@}\\limits\\else\n \\multint@\\nolimits\\fi}\\fi\\ints@@@}%\n\\def\\intkern@{\\mathchoice{\\!\\!\\!}{\\!\\!}{\\!\\!}{\\!\\!}}%\n\\def\\plaincdots@{\\mathinner{\\cdotp\\cdotp\\cdotp}}%\n\\def\\intdots@{\\mathchoice{\\plaincdots@}%\n {{\\cdotp}\\mkern1.5mu{\\cdotp}\\mkern1.5mu{\\cdotp}}%\n {{\\cdotp}\\mkern1mu{\\cdotp}\\mkern1mu{\\cdotp}}%\n {{\\cdotp}\\mkern1mu{\\cdotp}\\mkern1mu{\\cdotp}}}%\n\\def\\RIfM@{\\relax\\protect\\ifmmode}\n\\def\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\\RIfM@\\expandafter\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi@\\else\\expandafter\\mbox\\fi}\n\\let\\nfss@text\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi\n\\def\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi@#1{\\mathchoice\n {\\textdef@\\displaystyle\\f@size{#1}}%\n {\\textdef@\\textstyle\\tf@size{\\firstchoice@false #1}}%\n {\\textdef@\\textstyle\\sf@size{\\firstchoice@false #1}}%\n {\\textdef@\\textstyle \\ssf@size{\\firstchoice@false #1}}%\n \\glb@settings}\n\n\\def\\textdef@#1#2#3{\\hbox{{%\n \\everymath{#1}%\n \\let\\f@size#2\\selectfont\n #3}}}\n\\newif\\iffirstchoice@\n\\firstchoice@true\n\\def\\Let@{\\relax\\iffalse{\\fi\\let\\\\=\\cr\\iffalse}\\fi}%\n\\def\\vspace@{\\def\\vspace##1{\\crcr\\noalign{\\vskip##1\\relax}}}%\n\\def\\multilimits@{\\bgroup\\vspace@\\Let@\n \\baselineskip\\fontdimen10 \\scriptfont\\tw@\n \\advance\\baselineskip\\fontdimen12 \\scriptfont\\tw@\n \\lineskip\\thr@@\\fontdimen8 \\scriptfont\\thr@@\n \\lineskiplimit\\lineskip\n \\vbox\\bgroup\\ialign\\bgroup\\hfil$\\m@th\\scriptstyle{##}$\\hfil\\crcr}%\n\\def\\Sb{_\\multilimits@}%\n\\def\\endSb{\\crcr\\egroup\\egroup\\egroup}%\n\\def\\Sp{^\\multilimits@}%\n\\let\\endSp\\endSb\n\\newdimen\\ex@\n\\ex@.2326ex\n\\def\\rightarrowfill@#1{$#1\\m@th\\mathord-\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\n \\mkern-6mu\\mathord\\rightarrow$}%\n\\def\\leftarrowfill@#1{$#1\\m@th\\mathord\\leftarrow\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\\mkern-6mu\\mathord-$}%\n\\def\\leftrightarrowfill@#1{$#1\\m@th\\mathord\\leftarrow\n\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\n \\mkern-6mu\\mathord\\rightarrow$}%\n\\def\\overrightarrow{\\mathpalette\\overrightarrow@}%\n\\def\\overrightarrow@#1#2{\\vbox{\\ialign{##\\crcr\\rightarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\let\\overarrow\\overrightarrow\n\\def\\overleftarrow{\\mathpalette\\overleftarrow@}%\n\\def\\overleftarrow@#1#2{\\vbox{\\ialign{##\\crcr\\leftarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\def\\overleftrightarrow{\\mathpalette\\overleftrightarrow@}%\n\\def\\overleftrightarrow@#1#2{\\vbox{\\ialign{##\\crcr\n \\leftrightarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\def\\underrightarrow{\\mathpalette\\underrightarrow@}%\n\\def\\underrightarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\\hfil#1#2\\hfil\n $\\crcr\\noalign{\\nointerlineskip}\\rightarrowfill@#1\\crcr}}}%\n\\let\\underarrow\\underrightarrow\n\\def\\underleftarrow{\\mathpalette\\underleftarrow@}%\n\\def\\underleftarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\\hfil#1#2\\hfil\n $\\crcr\\noalign{\\nointerlineskip}\\leftarrowfill@#1\\crcr}}}%\n\\def\\underleftrightarrow{\\mathpalette\\underleftrightarrow@}%\n\\def\\underleftrightarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\n \\hfil#1#2\\hfil$\\crcr\n \\noalign{\\nointerlineskip}\\leftrightarrowfill@#1\\crcr}}}%\n\n\\def\\qopnamewl@#1{\\mathop{\\operator@font#1}\\nlimits@}\n\\let\\nlimits@\\displaylimits\n\\def\\setboxz@h{\\setbox\\z@\\hbox}\n\n\n\\def\\varlim@#1#2{\\mathop{\\vtop{\\ialign{##\\crcr\n \\hfil$#1\\m@th\\operator@font lim$\\hfil\\crcr\n \\noalign{\\nointerlineskip}#2#1\\crcr\n \\noalign{\\nointerlineskip\\kern-\\ex@}\\crcr}}}}\n\n \\def\\rightarrowfill@#1{\\m@th\\setboxz@h{$#1-$}\\ht\\z@\\z@\n $#1\\copy\\z@\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\box\\z@\\mkern-2mu$}\\hfill\n \\mkern-6mu\\mathord\\rightarrow$}\n\\def\\leftarrowfill@#1{\\m@th\\setboxz@h{$#1-$}\\ht\\z@\\z@\n $#1\\mathord\\leftarrow\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\copy\\z@\\mkern-2mu$}\\hfill\n \\mkern-6mu\\box\\z@$}\n\n\n\\def\\qopnamewl@{proj\\,lim}{\\qopnamewl@{proj\\,lim}}\n\\def\\qopnamewl@{inj\\,lim}{\\qopnamewl@{inj\\,lim}}\n\\def\\mathpalette\\varlim@\\rightarrowfill@{\\mathpalette\\varlim@\\rightarrowfill@}\n\\def\\mathpalette\\varlim@\\leftarrowfill@{\\mathpalette\\varlim@\\leftarrowfill@}\n\\def\\mathpalette\\varliminf@{}{\\mathpalette\\mathpalette\\varliminf@{}@{}}\n\\def\\mathpalette\\varliminf@{}@#1{\\mathop{\\underline{\\vrule\\@depth.2\\ex@\\@width\\z@\n \\hbox{$#1\\m@th\\operator@font lim$}}}}\n\\def\\mathpalette\\varlimsup@{}{\\mathpalette\\mathpalette\\varlimsup@{}@{}}\n\\def\\mathpalette\\varlimsup@{}@#1{\\mathop{\\overline\n {\\hbox{$#1\\m@th\\operator@font lim$}}}}\n\n\\def\\stackunder#1#2{\\mathrel{\\mathop{#2}\\limits_{#1}}}%\n\\begingroup \\catcode `|=0 \\catcode `[= 1\n\\catcode`]=2 \\catcode `\\{=12 \\catcode `\\}=12\n\\catcode`\\\\=12 \n|gdef|@alignverbatim#1\\end{align}[#1|end[align]]\n|gdef|@salignverbatim#1\\end{align*}[#1|end[align*]]\n\n|gdef|@alignatverbatim#1\\end{alignat}[#1|end[alignat]]\n|gdef|@salignatverbatim#1\\end{alignat*}[#1|end[alignat*]]\n\n|gdef|@xalignatverbatim#1\\end{xalignat}[#1|end[xalignat]]\n|gdef|@sxalignatverbatim#1\\end{xalignat*}[#1|end[xalignat*]]\n\n|gdef|@gatherverbatim#1\\end{gather}[#1|end[gather]]\n|gdef|@sgatherverbatim#1\\end{gather*}[#1|end[gather*]]\n\n|gdef|@gatherverbatim#1\\end{gather}[#1|end[gather]]\n|gdef|@sgatherverbatim#1\\end{gather*}[#1|end[gather*]]\n\n\n|gdef|@multilineverbatim#1\\end{multiline}[#1|end[multiline]]\n|gdef|@smultilineverbatim#1\\end{multiline*}[#1|end[multiline*]]\n\n|gdef|@arraxverbatim#1\\end{arrax}[#1|end[arrax]]\n|gdef|@sarraxverbatim#1\\end{arrax*}[#1|end[arrax*]]\n\n|gdef|@tabulaxverbatim#1\\end{tabulax}[#1|end[tabulax]]\n|gdef|@stabulaxverbatim#1\\end{tabulax*}[#1|end[tabulax*]]\n\n\n|endgroup\n \n\n \n\\def\\align{\\@verbatim \\frenchspacing\\@vobeyspaces \\@alignverbatim\nYou are using the \"align\" environment in a style in which it is not defined.}\n\\let\\endalign=\\endtrivlist\n \n\\@namedef{align*}{\\@verbatim\\@salignverbatim\nYou are using the \"align*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endalign*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\alignat{\\@verbatim \\frenchspacing\\@vobeyspaces \\@alignatverbatim\nYou are using the \"alignat\" environment in a style in which it is not defined.}\n\\let\\endalignat=\\endtrivlist\n \n\\@namedef{alignat*}{\\@verbatim\\@salignatverbatim\nYou are using the \"alignat*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endalignat*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\xalignat{\\@verbatim \\frenchspacing\\@vobeyspaces \\@xalignatverbatim\nYou are using the \"xalignat\" environment in a style in which it is not defined.}\n\\let\\endxalignat=\\endtrivlist\n \n\\@namedef{xalignat*}{\\@verbatim\\@sxalignatverbatim\nYou are using the \"xalignat*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endxalignat*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\gather{\\@verbatim \\frenchspacing\\@vobeyspaces \\@gatherverbatim\nYou are using the \"gather\" environment in a style in which it is not defined.}\n\\let\\endgather=\\endtrivlist\n \n\\@namedef{gather*}{\\@verbatim\\@sgatherverbatim\nYou are using the \"gather*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endgather*\\endcsname =\\endtrivlist\n\n\n\\def\\multiline{\\@verbatim \\frenchspacing\\@vobeyspaces \\@multilineverbatim\nYou are using the \"multiline\" environment in a style in which it is not defined.}\n\\let\\endmultiline=\\endtrivlist\n \n\\@namedef{multiline*}{\\@verbatim\\@smultilineverbatim\nYou are using the \"multiline*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endmultiline*\\endcsname =\\endtrivlist\n\n\n\\def\\arrax{\\@verbatim \\frenchspacing\\@vobeyspaces \\@arraxverbatim\nYou are using a type of \"array\" construct that is only allowed in AmS-LaTeX.}\n\\let\\endarrax=\\endtrivlist\n\n\\def\\tabulax{\\@verbatim \\frenchspacing\\@vobeyspaces \\@tabulaxverbatim\nYou are using a type of \"tabular\" construct that is only allowed in AmS-LaTeX.}\n\\let\\endtabulax=\\endtrivlist\n\n \n\\@namedef{arrax*}{\\@verbatim\\@sarraxverbatim\nYou are using a type of \"array*\" construct that is only allowed in AmS-LaTeX.}\n\\expandafter\\let\\csname endarrax*\\endcsname =\\endtrivlist\n\n\\@namedef{tabulax*}{\\@verbatim\\@stabulaxverbatim\nYou are using a type of \"tabular*\" construct that is only allowed in AmS-LaTeX.}\n\\expandafter\\let\\csname endtabulax*\\endcsname =\\endtrivlist\n\n\n\n \\def\\endequation{%\n \\ifmmode\\ifinner\n \\iftag@\n \\addtocounter{equation}{-1}\n $\\hfil\n \\displaywidth\\linewidth\\@taggnum\\egroup \\endtrivlist\n \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@false\n \\global\\@ignoretrue \n \\else\n $\\hfil\n \\displaywidth\\linewidth\\@eqnnum\\egroup \\endtrivlist\n \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@false\n \\global\\@ignoretrue \n \\fi\n \\else \n \\iftag@\n \\addtocounter{equation}{-1}\n \\eqno \\hbox{\\@taggnum}\n \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@false%\n $$\\global\\@ignoretrue\n \\else\n \\eqno \\hbox{\\@eqnnum\n $$\\global\\@ignoretrue\n \\fi\n \\fi\\fi\n } \n\n \\newif\\iftag@ \\@ifnextchar*{\\@tagstar}{\\@tag}@false\n \n \\def\\@ifnextchar*{\\@TCItagstar}{\\@TCItag}{\\@ifnextchar*{\\@TCItagstar}{\\@TCItag}}\n \\def\\@TCItag#1{%\n \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n \\global\\def\\@taggnum{(#1)}}\n \\def\\@TCItagstar*#1{%\n \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n \\global\\def\\@taggnum{#1}}\n\n \\@ifundefined{tag}{\n \\def\\@ifnextchar*{\\@tagstar}{\\@tag}{\\@ifnextchar*{\\@tagstar}{\\@tag}}\n \\def\\@tag#1{%\n \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n \\global\\def\\@taggnum{(#1)}}\n \\def\\@tagstar*#1{%\n \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n \\global\\def\\@taggnum{#1}}\n }{}\n\n\\def\\tfrac#1#2{{\\textstyle {#1 \\over #2}}}%\n\\def\\dfrac#1#2{{\\displaystyle {#1 \\over #2}}}%\n\\def\\binom#1#2{{#1 \\choose #2}}%\n\\def\\tbinom#1#2{{\\textstyle {#1 \\choose #2}}}%\n\\def\\dbinom#1#2{{\\displaystyle {#1 \\choose #2}}}%\n\n\\makeatother\n\\endinput\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section*{Acknowledgements}\n\\label{sec:ack}\nWe thank the anonymous referees as well as the editor for insightful reviews that greatly improved this work.\nQUBIC is funded by the following agencies. France: ANR (Agence Nationale de la Recherche) 2012 and 2014, DIM-ACAV (Domaine d'Interet Majeur-Astronomie et Conditions d'Apparition de la Vie), Labex UnivEarthS (Universit\u00e9 de Paris), CNRS\/IN2P3 (Centre National de la Recherche Scientifique\/Institut National de Physique Nucl\u00e9aire et de Physique des Particules), CNRS\/INSU (Centre National de la Recherche Scientifique\/Institut National des Sciences de l'Univers). Italy: CNR\/PNRA (Consiglio Nazionale delle Ricerche\/ Programma Nazionale Ricerche in Antartide) until 2016, INFN (Istituto Nazionale di Fisica Nucleare) since 2017. Argentina: MINCyT (Ministerio de Ciencia, Tecnolog\u00eda e Innovaci\u00f3n), CNEA (Comisi\u00f3n Nacional de Energ\u00eda At\u00f3mica), CONICET (Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas).\n\\section{Impact of atmospheric fluctuations on scales smaller than the synthesized beam cutoff angles.}\n\\label{app_impact_small_scale_atm}\n\n In section~\\ref{sec_map_making_noise_structure} we have seen that the synthesized beam naturally filters-out large-scale fluctuations from atmospheric gradients. We have also explained that to understand the effect of atmosphere fluctuations on scales smaller than the synthesized beam cutoff angles (8.8\\,degrees at 150\\,GHz and 6\\,degrees at 220\\,GHz) it is necessary to build simulations based on atmosphere measurements taken at the site. Here we want to show that the available data taken at sites similar to that of QUBIC allows us to say that the instrument will be sensitive to fluctuations in a defined range of turbulence scales.\n\n Fluctuations in the atmospheric radiation load are determined mainly by turbulent cells in the water vapor column that are distributed over scales within an interval $\\left[L_0^\\mathrm{min}, L_0^\\mathrm{max}\\right]$. The values of the minimum and maximum cutoff coherence scales, $L_0^\\mathrm{min}$ and $L_0^\\mathrm{max}$, depend on the physical properties of the atmosphere and are typical of each observations site.\n \n In a study conducted in the framework of the POLARBEAR experiment Errard et al.~\\cite{errard2015} studied the distribution of various atmosphere parameters at Atacama and showed that turbulence scales are distributed in the interval $L_0 \\in \\left[200\\,\\mathrm{m}, 800\\,\\mathrm{m}\\right]$ with a peak around 300\\,m (see the left panel of figure~11 in Errard et al.~\\cite{errard2015}). Assuming that the atmosphere at Alto Chorrillo is not very different from that at Atacama, we can use this distribution to assess quickly the impact of atmosphere fluctuations on scales smaller than the synthetic beam cutoff angles (8.8~degrees at 150\\,GHz and 6~degrees at 220\\,GHz).\n \n Let us consider a fluctuations correlation length, $L_0$. This length corresponds to angular scales $\\theta^\\mathrm{atm}_\\mathrm{corr}(z)\\sim L_0\/z$, where $z$ is the height above the telescope line-of-sight.\n\n \\begin{figure}[!t]\n \\begin{center}\n \\includegraphics[width=14cm]{Figures\/bubble_size_dist}\n \\caption{\\label{fig_small_scale_atmo}Relationship between the relative water vapor concentration and atmospheric turbulence correlations angular scales. Black lines are a family of curves $\\theta^\\mathrm{atm}_\\mathrm{corr}(z)$ corresponding to values of $L_0$ in the interval $\\left[200\\,\\mathrm{m}, 800\\,\\mathrm{m}\\right]$. The curves are marked with a transparency level that reflects the distribution in figure~11 of Errard et al.~\\cite{errard2015}. The yellow curve marks the case with $L_0^\\mathrm{max}=300\\,\\mathrm{m}$, corresponding to the most likely correlation scale. The red and blue horizontal lines mark the synthetic beam cutoff angles at 150 and 220\\,GHz. The green curve (right axis) reports the relative water vapor concentration versus height, assuming a height scale of 2000\\,m. The dashed red and blue lines are guides that show the water vapor concentration corresponding to the synthesized beam cutoff scales and the case $L_0^\\mathrm{max}$. See text for a more detailed description.}\n \\end{center}\n \\end{figure}\n \n In figure~\\ref{fig_small_scale_atmo} we show, in black, a family of curves $\\theta^\\mathrm{atm}_\\mathrm{corr}(z)$ corresponding to values of $L_0$ in the interval $\\left[200\\,\\mathrm{m}, 800\\,\\mathrm{m}\\right]$. The curves are marked with a transparency level that reflects the distribution in figure~11 of Errard et al.~\\cite{errard2015}. The yellow curve marks the case with $L_0^\\mathrm{max}=300\\,\\mathrm{m}$, corresponding to the most likely correlation scale. The red and blue horizontal lines mark the synthetic beam cutoff angles at 150 and 220\\,GHz, so that for small scale fluctuations we are interested in the regions below these lines. The green curve (right axis) reports the relative water vapor concentration versus height, assuming a height scale of 2000\\,m\\footnote{The water vapor scale height is the distance over which a the water vapor content decreases by a factor of $e$}, which can be considered representative for Alto Chorrillo (see, for example, the left panel of figure 1 in Mararieva et al.~\\cite{makarieva2013}). Finally, the dashed red and blue lines are guides that show the water vapor concentration corresponding to the synthesized beam cutoff scales and the case $L_0^\\mathrm{max}$.\n \n Let us take, for example, the most likely correlation scale, $L_0^\\mathrm{max}$. The largest angular scales connected to $L_0^\\mathrm{max}$ are the synthetic beam cutoff scales and, with these values, we see that we will be affected by fluctuations involving $\\sim$40\\% of the water vapor column at 150\\,GHz and slightly more than 20\\% at 220\\,GHz. It is worth noting that the bottom-left and top-right white areas of the plot correspond to angular scales where we do not expect significant fluctuations. For example, for $z = 2000\\,\\mathrm{m}$ we do not expect fluctuations on angular scales up to 6\\,degrees.\n \n From this simple analysis we can conclude that the effect from turbulent cells larger than 300\\,m will be increasingly negligible, while we may expect some impact from cells in the range $\\left[200\\,\\mathrm{m}, 300\\,\\mathrm{m}\\right]$. Of course this analysis does not answer yet the question of what effect we can expect on the final science from atmosphere fluctuations on these scales. A complete assessment will be performed on the basis of dedicated atmospheric measurements taken at the QUBIC site.\n\n\n\\section{Bolometric interferometry and QUBIC}\n\\label{sec:boloint}\n\\subsection{Bolometric interferometry}\n\\label{subsec:boloint}\n\nThe idea of bolometric interferometry dates back to the 19$^\\mathrm{th}$ century when Fizeau proposed an adding interferometer to measure the expected difference in the velocity of light traveling in moving media \\citep{1851fizeau}. The setup split the signal from the Sun into two pipes, each filled with water flowing in opposite directions. The beams were then recombined and the interference fringes measured to determine the phase difference between them. In QUBIC, the pipes are replaced by an array of back-to-back feedhorns, and the beams are recombined by an arrangement of two mirrors \\citep[see O'Sullivan et al.][for details]{2020.QUBIC.PAPER8}.\n\nAn important advantage of the Fizeau arrangement is its inherent wide band performance. There is no wavelength selectivity which results in an ideal performance at any particular wavelength. This can be compared to the case for example of the Martin-Puplett interferometer \\citep{1970InfPh..10..105M} which introduces a quarter wavelength optical path difference by moving one of the corner mirrors, making it tuned to a particular wavelength.\n\nImaging radio interferometers generally work with electronic signals where multiplicative correlation is a natural operation with electronic components, both analog and digital. The correlations between signals from different antennas build up a sampling of the ``\\uvplane''. The Fourier transform of the \\uvplane\\ is an image of the sky, usually called the ``dirty image'' because no compensation has been done for the under-sampling of the \\uvplane, and no calibration has been done \\citep[see for example Thompson, Moran and Swenson][]{2017isra.book.....T}. In a bolometric interferometer, this ``dirty image'' is formed directly by the optical combiner, and is recorded by the detector array. QUBIC treats the entire band ``in one go'', whereas a traditional radio interferometer using narrow band electronic processors must split the band into many sub-bands, making large bandwidths expensive to realize in terms of processing and power requirements.\n\nThis method is equivalent to imaging the sky with an imager whose beam is the synthesized beam of the bolometric interferometer (the ``dirty beam'' of the interferometer), formed by the combination of all interference patterns of all possible pairs of horns in the aperture array~\\cite{2011APh....34..705Q}. In the case of the QUBIC Full Instrument, 400~horns form the synthesized beam shown in figure~\\ref{Fig:sbqubic} for one specific detector. This is the beam the detector located in this position in the focal plane scans the sky with. Note that the synthesized beam differs for different detectors in the focal plane (see~Mousset et al.~\\cite{2020.QUBIC.PAPER2} and O'Sullivan et al.~\\cite{ 2020.QUBIC.PAPER8} from this series of articles for details). The synthesized beam is very different from a classical imager beam as it exhibits multiple peaks (because the horn array has a finite extension), the angular distance between two peaks is driven by the ratio between the wavelength and the distance between two nearby horns while the angular resolution of each peak is driven by the ratio between the wavelength and the maximal distance between horns. This is detailed in O'Sullivan~et.~al~\\cite{2020.QUBIC.PAPER8} from this series of articles. In the case of QUBIC, with 400~horns, the angular resolution achieved is 23.5~arcminutes at 150~GHz.\nBeing the result of the summation of interference fringes, this synthesized beam significantly evolves with electromagnetic frequency, which is the basis of the spectral imaging capabilities of QUBIC (described briefly in section~\\ref{sec:specimg} and in detail in~Mousset et al.~\\cite{2020.QUBIC.PAPER2} in this series of articles).\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 1 \\hsize ]{Figures\/SB_QUBIC_150GHz.pdf}\n\\caption{QUBIC synthesized beam on the sky (left: laboratory measurement with the TD, right: simulations without optical aberrations) at 150~GHz. Note that the color scales are arbitrary units. Details on this measurement can be found in Torchinsky et al.~\\cite{2020.QUBIC.PAPER3} from this series of articles. The synthesized beam shrinks with increasing frequency as can be seen with the animated version of this image that can be found online at \\href{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}.}\n\\label{Fig:sbqubic}\n\\end{figure}\n\n\n\n\n\\subsection{Self-Calibration}\n\\label{sec:selfcal}\n\nSelf calibration in aperture synthesis evolved from the idea of ``phase-closure''~\\cite{1974A&AS...15..417H,1976A&A....50...19F,1978ApJ...223...25R} in a phased-array radiotelescope. Signals from the individual antenna elements of a phased-array are combined together and phase differences between the signals are corrected such that the combination of all the signals is equivalent to the signal captured by a large single-dish antenna pointing to the desired direction. With the advent of digital correlators, the correlation coefficients could be saved and a more sophisticated processing resulted in improved calibration. This came to be known as ``self-calibration''~\\cite{1981MNRAS.196.1067C,1984ARA&A..22...97P}.\n\nBolometric interferometry can also take advantage of the self-calibration used in traditional aperture synthesis in radio astronomy. \nHowever, as a bolometric interferometer directly observes the dirty image, one needs to `go backwards'' from the dirty image and reconstruct the \\uvplane.\nThe detailed procedure and analysis technique are described in Bigot-Sazy~et~al.~\\cite{2013A&A...550A..59B}.\nThe general idea is to observe a polarized artificial point source with the bolometric interferometer with only one pair of horns open at a time while the others are closed\\footnote{In practice, in order to keep a roughly constant loading on the detector array, we achieve this observation through measuring successively the point source with one horn closed, all the others being open, then only the second horn closed, then both closed and finally all horns open. We eventually combine these observations to achieve the same measurement as with a single pair of horns open.}, which can be achieved using the mechanical shutters (also called RF switches) developed for QUBIC that can open or close each horn (see Cavaliere~et~al.~\\cite{2020.QUBIC.PAPER7} in this series of articles for details on the QUBIC horns and RF switches). \nIn the absence of instrumental systematic effects, two ``redundant'' pairs of horns (same distance and orientation) would correspond to the exact same visibility (Fourier mode on the sky) and the bolometric interferometer would therefore measure the same quantity. A difference between the two can only come from instrumental effects\\footnote{Small-scale atmospheric fluctuations could also be responsible for a difference between measurements performed with ``redundant'' pairs of horns. However, we anticipate this to be a marginal effect because i) the artificial source has a strong emission (a fraction of Watt) and ii) self-calibration will be performed scanning the source over long periods, averaging-out the atmospheric fluctuations.}.\nBy measuring in this manner all possible visibilities, one can fit parameters of a very general instrument model comprising hundreds of parameters (a few for each horn, bolometer) as well as for different electromagnetic frequencies if the source can be tuned in frequency. The number of constraints scales as the number of horns squared while the number of unknowns is proportional to the number of horns. As a result the problem is heavily over constrained for an instrument like QUBIC. The following instrumental systematics, expected to be the major ones for a bolometric interferometer, have been considered~\\cite{2013A&A...550A..59B}:\n\\begin{itemize}\n \\item uncertainty on the exact location of the horns which would affect the shape of the synthesized beam;\n \\item uncertainty on the exact location of the bolometers in the focal plane. Similarly, it would affect the shape of the synthesized beam because it gradually changes across the focal plane (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8});\n \\item relative gains of the bolometers;\n \\item uncertainties on the primary beam (towards the sky) shape;\n \\item uncertainties on the secondary beam (towards the detectors) shape;\n \\item imperfections of the successive optical elements, each modeled by a Jones matrix containing differential gains for each polarization orientation as well as cross-polarization.\n\\end{itemize}\nWe have shown in Bigot-Sazy et al.~\\cite{2013A&A...550A..59B} that spending 1 second on self-calibration for each baseline would reduce the $E$ to $B$ power spectrum leakage by an order of magnitude for typical values for the above systematic effects. The leakage drops to two orders of magnitude with 100 seconds per baseline.\n\nSelf-calibration is a special feature of bolometric interferometry for measuring instrumental systematic effects and subsequently accounting for them in the data analysis process through a more accurate modeling of the synthesized beam than the purely theoretical one, hence improving the performance of the map-making (see section~\\ref{sims}).\n\n \n\\subsection{Spectral Imaging}\n\\label{sec:specimg}\nSpectral imaging is described in detail in Mousset~et~al.~\\cite{2020.QUBIC.PAPER2} from this series of articles, but we provide here a short explanation of this specific feature offered by bolometric interferometry.\nAs mentioned in section~\\ref{subsec:boloint} and shown in figure~\\ref{Fig:sbqubic}, the multiply peaked shape of the synthesized beams evolves with frequency. A cut of the QUBIC synthesized beam is shown in figure~\\ref{Fig:sb_freq} (left) for two different frequencies. \n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width =\\hsize ]{Figures\/sb_and_src.png}\n\\caption{{\\bf (left)} A cut of the QUBIC theoretical synthesized and primary beams at two different frequencies for a bolometer located at the center of the focal plane. The figure shows how the location of the multiple peaks significantly changes for a small frequency change. This behaviour is at the basis of the spectral imaging capabilities offered by bolometric interferometry. {\\bf (left)} signal detected by the same bolometer when scanning across an imaginary point source emitting 1 and 1.5 at 140 and 160 GHz respectively: we have both spatial and frequency information.}\n\\label{Fig:sb_freq}\n\\end{figure}\nAs the respective distance between peaks changes with frequency, the signal detected with a given bolometer will combine (through convolution with the synthesized beam) different directions on the sky for different incoming frequencies within the physical wide band. Let's imagine, as a simple toy model, that the sky consists in a single point source emitting only at two distinct frequencies 140 and 160 GHz with relative brightness 1 and 1.5. As the instrument scans over the point source, the response of the Time-Ordered-Data (TOD) will be the sum of the synthesized beam at 140 and 160 GHz scaled by 1 and 1.5 respectively. This is shown in figure~\\ref{Fig:sb_freq} (right) as a function of the angle from the point source. We can easily reconstruct the location of the source from the position of the central peak, as well as the spectrum of the source from the amplitudes of each of the peaks around the central one.\nThese signals from different frequencies will only be significantly different from each other if the corresponding side-peaks are more separated than their intrinsic width. \n\nThis shows how one can simultaneously recover spatial and frequency information within a wide physical band for a bolometric interferometer. Of course, the above toy-model corresponds to a source that is not extended in spatial nor frequency domain, but we have shown in Mousset et al.~\\cite{2020.QUBIC.PAPER2} that the exact same reasoning can be applied to diffuse signals with continuous Spectral Energy Distribution (SED).\nThis spectral imaging reconstruction is done at the map-making stage in the data-analysis pipeline with data collected within a wide bandwidth. It can be done by reconstructing maps in as many different sub-frequencies as allowed by the ratio between frequency shift and intrinsic peak width in the synthesized beam. We have shown in Mousset et al.~\\cite{2020.QUBIC.PAPER2} that the corresponding relative frequency resolution FWHM is $\\Delta_\\nu\/\\nu \\sim 1\/P$ where $P$ is the maximum number of apertures along an axis of the bolometric interferometer horn array. For a 20x20 horn-array, QUBIC therefore achieves 5 sub-bands with a frequency FWHM resolution of $\\Delta_\\nu\/\\nu\\sim 0.05$ within each of the 25\\% bandwidth frequency bands.\n\n\nIn the current context of search for primordial B-modes in the CMB originating from tensor perturbations from inflation, spectral imaging presents an opportunity to constrain foregrounds alternative to other current approaches. Previous analyses have indeed shown that B-modes measurements are largely dominated by foregrounds~\\cite{2014PhRvL.112x1101B,2015A&A...576A.104P,2015PhRvL.114j1301B,2018PhRvL.121v1301B} which can only be removed through their frequency behaviour which is distinct from that of the CMB. Classical imagers usually approach this issue by multiplying the number of frequencies at which they observe the CMB. Galactic dust is currently the most worrying foreground at frequencies above 100\\,GHz. It can be constrained from the ground through wide atmospheric windows around 150 and 220~GHz. The noise severely increases at higher frequencies because of atmospheric emissivity. As a result, constraints on foregrounds can only be achieved through comparisons between these few, largely separated frequency bands. While their large separation in frequency may appear as an advantage as it increases lever arm, it is also a limitation as it prevents data analyses to consider realistic electromagnetic spectra for dust emission that could exhibit changes of slope or dust decorrelation between frequencies~\\cite{2017A&A...599A..51P, 2018PhRvL.121v1301B}. With spectral imaging, one can measure the spectrum of the foreground components {\\bf locally} (i.e., within the bandwidth) and avoid large extrapolations between distant frequencies. This local approach to foreground mitigation is complementary to the usual approach with largely separated frequency bands. This is studied in section~\\ref{sect:foreground_challenge} of this article. \n\n\n\n\\subsection{The QUBIC Instrument}\nIn order to achieve bolometric interferometry, QUBIC relies on an optical system consisting of back-to-back horns that select the relevant baselines and an optical combiner focusing on a bolometric focal plane. The optical combiner forms interference fringes while the bolometers average their powers over timescales much larger than the period of the electromagnetic light. This is therefore the optical equivalent of a (wide-band) correlator in classical interferometry. Being a bolometric device, the whole instrument operates at cryogenic temperatures thanks to a large cryostat described in Masi et al.~\\cite{2020.QUBIC.PAPER5} from this series of articles.\n\n\\begin{figure}[p!]\n\\centering\n$\\vcenter{\\hbox{\\includegraphics[ width = 0.9 \\hsize ]{Figures\/qubic_schematic_flipped.pdf}}}$\n$\\vcenter{\\hbox{\\includegraphics[ width = 0.7 \\hsize ]{Figures\/cryostat-qubic.pdf}}}$\n\\caption{Schematic of the QUBIC instrument \\textbf{(top)} and sectional cut of the cryostat \\textbf{(bottom)} showing the same sub-systems in their real configuration.}\n\\label{Fig:qubic_scheme}\n\\end{figure}\n\n\\begin{table}[t]\n \\renewcommand{\\arraystretch}{1.}\n \\begin{center}\n \\caption{\\label{tab_qubic_params}QUBIC main parameters}\n \\begin{tabular}{p{6.4cm} m{2.5cm} m{5.5cm}}\n \\hline\n Parameter& TD & FI \\\\\n \\hline\n \\hline\n {\\bf Instrument}\\\\\n \\hline\n Frequency channels \\dotfill &150 GHz & 150 GHz \\& 220 GHz\\\\\n Frequency range 150 GHz \\dotfill &[131-169] GHz &[131-169] GHz\\\\\n Frequency range 220 GHz \\dotfill &- &[192.5-247.5] GHz\\\\\n Window Aperture [m]\\dotfill & 0.56 & 0.56 \\\\\n Number of horns\\dotfill &64 &400\\\\\n Number of detectors\\dotfill &248 &992$\\times$2\\\\\n Detector noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.05$\\times 10^{-16}$ & 4.7$\\times 10^{-17}$ \\\\\n Focal plane temp. [mK]\\dotfill &300 &300\\\\\n Synthesized beam FWHM [deg]\\dotfill &0.68~\\cite{2020.QUBIC.PAPER3} &0.39 (150 GHz), 0.27 (220 GHz)\\\\\n \\hline\n \\hline\n {\\bf Scanning Strategy (see sect.~\\ref{sims})}\\\\\n \\hline\n Elevation range [deg]\\dotfill &[30-70] &[30-70]\\\\\n Azimuth scan width [deg]\\dotfill &$\\pm 17.5$ &$\\pm 17.5$\\\\\n Azimuth scan speed [deg\/s]\\dotfill &0.4 &0.4\\\\\n Sky Coverage\\dotfill &1.5\\% &1.5\\%\n \\end{tabular}\n \\end{center}\n\\end{table}\n\n\nA schematic of the design of QUBIC is shown in figure~\\ref{Fig:qubic_scheme} and the main instrument parameters are listed in table~\\ref{tab_qubic_params}. The sky signal first goes through a 56~cm diameter window made of Ultra-High-Molecular-Weight Polyethylene followed by a series of filters cutting off frequencies higher than the desired ones. The next optical component is a stepped rotating Half-Wave-Plate which modulates incoming polarization. This sub-system is described in D'Alessandro~et~al.~\\cite{2020.QUBIC.PAPER6} from this series of articles. A single polarization is then selected thanks to a polarizing grid. Although reflecting half of the incoming photons may appear as a regrettable loss, it is in fact one of the key features of QUBIC for handling instrumental systematics, especially polarization-related ones: a single polarization is selected just after polarization modulation by a wire-grid, while our bolometers are not sensitive to polarization due to the XY symmetry of the absorbing grid (see Piat et al~\\cite{2020.QUBIC.PAPER4}). As a result, any cross-polarization occurring after the polarizing grid (horns, uncontrolled reflections inside the optical combiner) are negligible. This has been confirmed during calibration by measuring the cross-polarization of our TES (see figure~\\ref{Fig:fringe}-right where we show a cross-polarization measurement below 0.4\\% at 95\\% C.L. for a TES). The next optical device is an array of 400 back-to-back corrugated horns made of an assembly of two 400-horns arrays, composed of 175 aluminium platelets (0.3 mm thick) chemically etched to reproduce the corrugations required for the horns to achieve the required performance. An array of mechanical shutters (RF switches) separates the two back-to-back horn arrays in order to be able to close or open horns for self-calibration (see section~\\ref{sec:selfcal}). Both front and back horns are identical, each with a field of view of 13~degrees FWHM with secondary lobes below $-$25~dB. The horns and switches are described in detail in Cavaliere~et. al~\\cite{2020.QUBIC.PAPER7} from this series of articles. The back-horns directly illuminate the two-mirrors off-axis Gregorian optical combiner (described in detail in O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8}) that focuses the signal onto the two perpendicular focal planes, separated by a dichroic filter that splits the incoming waves into two wide bands centered at 150~GHz for the on-axis focal plane and 220~GHz for the off-axis one. The focal planes are each equipped with 992~NbSi Transition-Edge-Sensors (the detection chain is described in detail in Piat et al.~\\cite{2020.QUBIC.PAPER4} from this series of articles) cooled down to 300~mK using a sorption fridge. A realistic view of the cryostat can be seen in the right panel of figure~\\ref{Fig:qubic_scheme}. The cryostat weights roughly 800~kg and is around 1.6~meter high for a 1.4~meter diameter.\n\nThe anticipated scanning strategy for QUBIC is back and forth azimuth scans at constant elevation, following the azimuth of the center of the field as a function of sidereal time and updating elevation regularly in order to have scans with as many angles as possible in sky coordinates. The latitude of the QUBIC site allows us to cover a wide range of angles. The detailed scanning strategy will be determined when observing the real sky based on optimal mitigation of atmospheric fluctuations. We show in figure~\\ref{Fig:coverage} a typical sky coverage obtained with the baseline scanning parameters given in table~\\ref{tab_qubic_params}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 0.64 \\hsize ]{Figures\/coverage_moll.png}\\includegraphics[ width = 0.36 \\hsize ]{Figures\/coverage.png}\n\\caption{Baseline sky coverage in Galactic coordinates (see table~\\ref{tab_qubic_params} for corresponding scanning strategy parameters). The coverage is cut at 10\\% from the maximum and achieves a sky fraction of 1.5\\%. The grid spacing in latitude and longitude are 10 and 20 degrees respectively.}\n\\label{Fig:coverage}\n\\end{figure}\n\n\n\\subsubsection{The QUBIC Technological Demonstrator}\n\\label{sec_qubic_td}\nThe QUBIC Technological Demonstrator (hereafter QUBIC TD) uses the same cryostat, cooling system, filters and general sub-system architecture as described above but with only 64~back-to-back horns and mirrors reduced according to the illumination of the 64~horns. It also uses a single 248~TES bolometer array operating at 150~GHz. The QUBIC TD has been used as an intermediate step before the Full Instrument (FI) in order to characterize and demonstrate bolometric interferometry in the laboratory. We have confirmed the expected behaviour of the instrument and all the anticipated specific features of bolometric interferometry during this calibration campaign which is detailed in Torchinsky~et~al.~\\cite{2020.QUBIC.PAPER3}. A selection of the most relevant results from the calibration includes:\n\\begin{itemize}\n \\item the measurement of the bolometric interferometer synthesized beams multiple peaked shape in overall agreement with the theoretical prediction we made in Battistelli et al.~\\cite{2011APh....34..705Q}, shown in figure~\\ref{Fig:sb_freq}. The difference between theory and measurement in the locations and amplitudes of the peaks is expected from optical aberrations (see~\\cite{2020.QUBIC.PAPER8}) and can be calibrated from the data for the map reconstruction;\n \\item the evolution as a function of frequency of the inter-peak separation in the synthesized beam (available online at \\href{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}{https:\/\/box.in2p3.fr\/index.php\/s\/bzPYfmtjQW4wCGj}) which is at the basis of Spectral Imaging described briefly in section~\\ref{sec:specimg} and in detail in Mousset et al.~\\cite{2020.QUBIC.PAPER2}. This possibility is studied in detail in section~\\ref{sect:foreground_challenge} of this article;\n \\item the measurement of individual fringe patterns using the mechanical shutters~\\cite{2020.QUBIC.PAPER7} shown in the left panel of figure~\\ref{Fig:fringe} in this article and a more detailed description in figure~13 of Torchinsky et al.~\\cite{2020.QUBIC.PAPER3}. As discussed in section~\\ref{sec:selfcal}, this measurement is at the basis of the self-calibration allowing QUBIC to achieve a novel control of instrumental systematic effects, as shown in Bigot-Sazy et al.~\\cite{2013A&A...550A..59B};\n \\item Figure~\\ref{Fig:fringe} shows a measurement of the TD cross-polarization with an upper-limit 0.4\\% at 95\\% C.L. for a detector close to the center of the focal plane. As studied in details in D'Alessandro et al.~\\cite{2020.QUBIC.PAPER6} from this series of articles, we have found a median measured cross-polarization of 0.12\\% among our detectors and a 0.61\\% median 95\\% upper-limit.\n 77\\% of our detectors have a measured cross-polarization compatible with zero at the one-sigma level. Such low cross-polarization is an important feature for detecting a signal as small as the primordial B-modes. This is achieved in QUBIC thanks to its specific polarization design with a single polarization selected before the interferometer apertures and full-power detectors as discussed above in this section.\n \\item The intrinsic detector noise has been measured to be 4.7$\\times 10^{-17}~\\mathrm{W\/\\sqrt{Hz}}$ but due to noise aliasing with the TD readout chain, the effective detector+readout noise is 2.05$\\times 10^{-16}~\\mathrm{W\/\\sqrt{Hz}}$. This noise aliasing will be resolved for the FI using Nyquist inductance in order to reduce the noise bandwidth of the TES (see Piat et al.~\\cite{2020.QUBIC.PAPER4}).\n\\end{itemize}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 0.36 \\hsize ]{Figures\/fringes.png}\\includegraphics[ width = 0.36 \\hsize ]{Figures\/fringes_simulation.png}\\includegraphics[ width = 0.28\\hsize ]{Figures\/hwp_plot_tes95.pdf}\n\\caption{{\\bf (left and center)} Fringe pattern measured (left) and simulated (center) with the QUBIC TD (coordinates are in the on-axis focal plane and units arbitrary) using the mechanical shutters~\\cite{2020.QUBIC.PAPER7} in order to achieve a measurement equivalent to that of a single baseline (two horns open, all others closed). Units are arbitrary. {\\bf (right)} Cross-Polarization measured by rotating the Half-Wave-Plate and observing our polarized calibration source. This result is obtained with a TES bolometer close to the focal plane center and showing a high signal-to-noise ratio.}\n\\label{Fig:fringe}\n\\end{figure}\n\nThe QUBIC TD will be shipped to Argentina in mid-2021 and subsequently deployed in its observing site (see section~\\ref{Sec:site} for a description). The objective is to conduct a full characterization of bolometric interferometry on the sky throughout 2021. The performance expected from observation with the QUBIC TD is discussed in section~\\ref{Sec:TD_perf}.\n\n\\subsubsection{The QUBIC Full Instrument\\label{FI}}\nThe upgrade to the FI will happen after one year of operation of the TD and is therefore scheduled for early 2023. It will consist in:\n\\begin{itemize}\n \\item replacing the 64~back-to-back horn array by one with 400 horns which is already manufactured and being characterized at the sub-system level (see Cavaliere~et~al.~\\cite{2020.QUBIC.PAPER7} from this series of articles);\n \\item replacing the current mirrors by larger mirrors that are already manufactured and characterized (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8});\n \\item upgrading the single 248~TES array by eight of these (four at 150~GHz and four at 220~GHz) achieving two focal planes of 992~TES each. Note that detailed quasi-optical simulations have shown that the 150~GHz optimization of the back-short of the TES is also nearly optimal at 220~GHz allowing us to use identical detector designs for both frequencies~\\cite{2020JLTP..200..363P,PerbostPhD}. A new readout electronics will avoid the TD noise aliasing through the addition of Nyquist inductors~\\cite{2020.QUBIC.PAPER4}.\n The additional focal plane also requires the installation of a dichroic filter at 45 degrees in between the two focal planes (light-blue in figure~\\ref{Fig:qubic_scheme}-left, and in red in figure~\\ref{Fig:qubic_scheme}-right).\n\\end{itemize} \nThe back-to-back horns are common to both frequencies in the QUBIC design. They have been optimized to be single-moded at 150~GHz but are multi-moded in the 220~GHz band (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8} section~2.3) resulting in a significantly higher throughput at 220~GHz. \n\nDespite stronger emission from the atmosphere, as can be found in Table~\\ref{tab_qubic_td_simulation_parameters}, simulations have shown (see figure~\\ref{Fig:bmodes} in section~\\ref{sect_Bmodes}) that this will result in a similar sensitivity to B-modes at 220~GHz and at 150~GHz. \nSeveral effects contribute to this result. First, considering the intrinsic detector noise (see Piat et al.~\\cite{2020.QUBIC.PAPER4}), our noise budget is dominated by the detectors at 150~GHz and has almost equal contributions from photon-noise and detectors at 220~GHz (see table~\\ref{tab_qubic_td_simulation_parameters}). Because of this, the net sensitivity loss at 220~GHz due to higher emissivity from the atmosphere is not as large as expected from pure photon-noise. \nWe also collect more CMB photons at 220~GHz due to the horns higher throughput. \n\nAs a result, the signal to noise ratio in the TOD at 220~GHz is higher than for a single-moded channel. \nFinally, multimoded horns at 220~GHz also result in a flatter primary beam (see O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8} figure~5) than at 150~GHz where the primary beam is Gaussian. This strongly impacts the spatial noise correlation in our maps (see section~\\ref{sims}) resulting in a reduction of our power spectra error-bars at the lowest multipoles. This effect occurs for both bands but is stronger at 220~GHz due to the higher amplitude of the synthesized beam secondary peaks.\n\n\\subsubsection{The QUBIC site}\n\\label{Sec:site}\nQUBIC will be installed in its final observing site in Argentina at the end of 2021\\footnote{It was anticipated to ship the instrument mid-2020 but the global shutdown caused by the COVID19 pandemic induced uncontrolled delays. As a consequence, the date mentioned here is subject to changes depending on the resolution of the COVID19 pandemic crisis.}. The site is located at the Alto Chorillos ($24\n^\\circ 11'11.7''$ S;\n$66^\\circ 28'40.8''$ W, altitude of 4869 m a.s.l.) about 45 minutes drive from the city of San Antonio de los Cobres in the Salta Province~\\cite{2018BAAA...60..107B}. This site has been studied for mm-wave astronomy for many years as it will also host the LLAMA 12~m antenna (\\href{https:\/\/www.llamaobservatory.org\/en\/}{https:\/\/www.llamaobservatory.org\/en\/}), 800~m away from QUBIC. The synergy between QUBIC and LLAMA at the site simplifies the site preparatory works, logistics and deployment operations.\n\nThis site exhibits excellent quality sky for CMB studies: zenith optical depth at 210~GHz $\\tau_{210} <0.1$ for 50\\% of the time and $<0.2$ for 85\\% of the time as well as relatively quiet atmosphere (winds $<6$~ m\/s for 50\\% of the time). From the LLAMA site-testing data, we have determined an average atmospheric temperature of 270~K with an average emissivity 0.081 and 0.138 at 150 and 220~GHz respectively taken at our average elevation of 50 degrees. These values are assumed for the atmospheric background in the simulations presented in this article\\label{atm}. \n\n The whole instrument is oriented to any sky direction with elevation between 30 and 70~degrees (limitations due to Pulse-Tubes-Coolers) and any azimuth by an alt-azimuthal mount on the top of a well-adapted container. A fore-baffle will be placed at the window entrance with an absorptive inner surface in order to increase side lobes rejection for angles larger than 20~degrees from bore-sight direction. Finally, the instrument will be surrounded by a ground-shield in order to minimize brightness contrast between the sky and the ground. Figure~\\ref{Fig:dome} shows an artist view of the mounted instrument. The calibration source used for self-calibration (see Torchinsky et al.~\\cite{2020.QUBIC.PAPER3} for details) will be installed on a 50~m-high tower (telecommunication-like) placed 50~m North from the instrument. It will directly be seen at normal elevations (without the use of a mirror) in the far field of the instrument and in the direction opposite to the main CMB observations. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[ width = 0.9 \\hsize ]{Figures\/dome_GS_FB.JPG}\n\\caption{Sketch of the instrument, the cryostat on the alt-azimuthal mount as will be installed on the observing site.}\n\\label{Fig:dome}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this article, first of a series of eight, we have given an overview of the QUBIC instrument. QUBIC is the first CMB polarimeter using a new technology called ``Bolometric Interferometry'' that combines the background limited sensitivity of bolometric detectors (transition edge sensors) with the clean measurement of interference fringes. QUBIC has been designed to observe the large-scale B-modes of the CMB to detect the elusive tensor perturbations (primordial gravitational waves) created during inflation. A first version of the instrument, with reduced number of horns (64) and detectors (248) operating at 150~GHz, the QUBIC Technological Demonstrator (TD), will be deployed in 2021 and is intended to demonstrate on-sky the capabilities of Bolometric Interferometry. The TD will subsequently be upgraded to the Full Instrument (FI) with the nominal 400~horn array and 992~detectors in each of the two focal planes operating at 150 and 220~GHz.\n\nWe have described the general design of QUBIC and have shown that it can be considered as a classical imager that would scan the sky with a beam composed of multiple peaks. These are separated by an angular distance given by the distance between two apertures in the interferometer horn array (in wavelength units), while the resolution of the peaks is given by the maximum size of the interferometer horn array (in wavelength units). We have emphasized two main features of QUBIC. First, the possibility of performing self-calibration~\\cite{2013A&A...550A..59B}, similarly as in a classical interferometer, allows us to control instrumental systematic effects. Second, a bolometric interferometer has simultaneous spatial and spectral sensitivity, thanks to the spectral dependence of the synthesized beam within the physical band of the instrument. This is what allows us to carry out spectral imaging~\\cite{2020.QUBIC.PAPER2}.\n\nWe have used an extensive set of simulations to make forecasts of the QUBIC performance for both the primordial B-mode search and for testing foregrounds using spectral imaging. These preliminary forecasts assume the detector noise measured during the QUBIC calibration campaign~\\cite{2020.QUBIC.PAPER3} and a stable atmosphere corresponding to the QUBIC site in Argentina at 5000\\,m a.s.l. In this work we did not consider neither component separation nor the effect of fluctuations in the atmospheric load, both of which require a significant step forward in our pipeline that we are currently carrying out. However, we showed that regardless of component separation spectral imaging allows us to reconstruct the sky emission SED pixel-by-pixel and to recognize the presence of foregrounds residuals in the derived tensor-to-scalar ratio.\n\nThese simulations show that the QUBIC TD will be able to demonstrate spectral imaging with one year of sky integration on a field centered on the Galactic center. The QUBIC TD will measure the SED of bright regions in both intensity and polarization. After the upgrade to the FI, QUBIC will reach its nominal configuration allowing us to reach a sensitivity to B-modes corresponding to a 68\\% C.L. upper-limit on the effective tensor-to-scalar ratio (primordial tensors + dust) $\\sigma(r)=0.015$ with three years of sky integration. We have also shown how QUBIC FI will be able to put constraints on the dust contamination through direct measurement of the dust SED and properties in the reconstructed maps at multiple sub-frequencies within a physical band, or through a study of the evolution of the recovered tensor-to-scalar ratio in sub-bands as a function of frequency.\n\nFuture studies will explore in detail the impact of atmosphere fluctuations and how spectral imaging can lead to improved component separation using classical techniques as well as techniques specific to bolometric interferometry.\n\n\n\\subsubsection{Overview}\n\\label{sec_fg_challenge_overview}\n\n \\input{overview_foreground_challenge}\n \n\\subsubsection{QUBIC Technological Demonstrator expected performance}\n\\label{Sec:TD_perf}\n\n \\input{qubic_td_performance}\n \n\n\\subsubsection{QUBIC Full Instrument expected performance}\n \\input{qubic_fi_performance}\n\n\n\n\\subsubsection{Ability to recognize the presence of dust residuals with spectro-imaging}\n\\label{sss:dust_residuals}\n\n \\input{detect_dust_residuals}\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\nA phase of exponential expansion called ``Inflation'' in the early Universe was proposed as a solution to major problems with the standard Big-Bang model: the horizon, flatness and monopole problems~\\citep{1981PhRvD..23..347G,1982PhLB..108..389L}. The horizon problem relates to the observed homogeneity of the Universe as evidenced by the smoothness of the Cosmic Microwave Background (CMB). The flatness problem is the fact that the observed curvature of\nspace-time is so close to flat in present times, that it must have been flat to enormous precision in the early universe. The inflationary paradigm has been so successful in resolving the flatness and horizon problems, that it soon became a keystone component to the standard model of Cosmology. Observations of the CMB have provided several convincing arguments in favor of inflation: i) the series of acoustic peaks observed in the temperature power spectrum~\\cite{debernardis_2000, netterfield_2002} ruled out the alternative to inflation, topological defects, as the mechanism responsible for the dominant primordial density fluctuations ; ii) the observation of correlations between temperature and E-polarization of the CMB~\\cite{kogut_2003} as well as the observed phase opposition between peaks in the $TT$ and $EE$ power spectra~\\cite{2004Sci...306..836R, 2009ApJ...692.1247P} are evidence for adiabatic primordial perturbations, such as produced by inflation ; iii) the measurement of the CMB scalar spectral index slightly lower than one~\\cite{2020A&A...641A...6P} is a prediction from inflation. However, despite these strong arguments, all showing agreement between observations and inflation, there continues to be a lack of direct observational evidence of inflation despite the 40~years that have passed since it was first proposed.\n\nOne observable effect of Inflation is the generation of polarization B-modes in the CMB. This happens because during the exponential expansion of the Universe intense gravitational waves are generated. The gravitational waves induce re-orientation of the primordial plasma such that B-modes are visible in the radiation released at the surface of last scattering, which is the CMB. This weak signal has not been detected so far and many projects are operating, or are proposed, to measure the polarization B-modes in the CMB. These projects include SPTPol~\\cite{sayre2019measurements}, POLARBEAR~\\cite{ade2014measurement}, ACTPol~\\cite{louis2017atacama}, BICEP2~\\cite{ade2018constraints}, CLASS~\\cite{dahal2020class}, POLARBEAR 2 + Simons Array~\\cite{Polarbear+simons2016}, advanced ACT~\\cite{AdvACT2016}, upgrade of the BICEP3\/Keck array~\\cite{grayson2016bicep3}. Planned experiments include Simons Observatory~\\cite{2019JCAP...02..056A}, PIPER~\\cite{piper2016}, LSPE~\\cite{addamo2020large}, CMB-S4~\\cite{cmbs4}, LiteBIRD \\cite{ hazumi2019} and PICO~\\cite{PICO}.\n\nThe use of bolometric interferometry to measure polarization B-modes in the CMB was proposed for a series of projects leading up to QUBIC. These are the Millimeter-wave Bolometric Interferometry~\\citep[MBI,][]{2002AIPC..616..126A,2003NewAR..47.1173T}, the Einstein Polarization Interferometer for Cosmology~\\citep[EPIC,][]{2006NewAR..50..999T}, the Background RAdiation INterferometer \\citep[BRAIN,][]{2007NewAR..51..256P}, and CMBPol\\citep{2009JPhCS.155a2003T}. Members of all these collaborations joined together to develop the \\mbox{Q \\& U Bolometric} Interferometer for Cosmology \\citep[QUBIC,][]{2009arXiv0910.0391K,2009AIPC.1185..506G,2011APh....34..705Q,2012MNRAS.423.1293B,2016arXiv160904372A}.\n\nThis paper is part of a special issue on QUBIC which includes details on all its design aspects, as well as on the performance of an advanced prototype, the Technological Demonstrator (TD). The scientific overview and expected performance of the instrument are addressed here. The other papers in the special issue address the following topics: the ability of bolometric interferometry to do spectral imaging~\\citep{2020.QUBIC.PAPER2}, the calibration and performance in the laboratory of the TD~\\citep{2020.QUBIC.PAPER3}, the performance of the detector array and readout electronics~\\citep{2020.QUBIC.PAPER4}, the cryogenic system performance~\\citep{2020.QUBIC.PAPER5}, the Half Wave Plate rotator system~\\citep{2020.QUBIC.PAPER6}, the back-to-back feedhorn-switch system~\\citep{2020.QUBIC.PAPER7}, and the optical design and performance~\\citep{2020.QUBIC.PAPER8}.\n\nThis paper is organized as follows. A description of bolometric interferometry is provided in Section~\\ref{sec:boloint}. This is followed in Section~\\ref{sec:sciobj} with an overview of the scientific objectives, the current state-of-the-art of CMB polarization experiments, and the expected performance of QUBIC. Finally, conclusions are presented in Section~\\ref{sec:conclusions}.\n\n\n\n\n\n\\section{Science Objectives of QUBIC}\n\\label{sec:sciobj}\nQUBIC, being a bolometric interferometer, has distinctive features with respect to traditional imagers designed for observing the CMB and optimized for measuring primordial B-mode polarization. QUBIC is a clean spectral polarimeter well adapted to the purpose of detecting primordial B-modes because of the following main features:\n\\begin{itemize}\n \\item it scans the sky with a synthesized beam formed by the feedhorn array, that exhibits multiple peaks (see Fig.~\\ref{Fig:sbqubic} and~\\ref{Fig:sb_freq}). Each of the synthesized beam peaks is well approximated by a Gaussian (above -20~dB) with a resolution of 23.5~arcminutes at 150~GHz; \n \\item the angular separation on the sky between the synthesized beam peaks is given by the smallest distance between two peaks and is 8.8~degrees at 150~GHz. This distance scales as a function of frequency within the physical bandwidth of 25\\% in such a way that sub-frequency maps can be reconstructed using spectral imaging;\n \\item a specific optical design with a polarizing grid before any optical component (except for the Half-Wave-Plate, filters and window) combined with full-power detectors on the focal plane make QUBIC largely immune to cross-polarization.\n\\end{itemize}\nAll of these specific features were studied in detail and are incorporated in the forecasts shown in this section. \n\n\\subsection{Data Analysis and Simulations for QUBIC performance forecasts\\label{sims}}\n\\subsubsection{Map-making and noise structure\\label{sec_map_making_noise_structure}}\nThe non-trivial shape of the synthesized beam (figures~\\ref{Fig:sbqubic} and~\\ref{Fig:sb_freq}) requires a specific map-making method which was developed for QUBIC. It is based on forward modeling to solve the inverse problem of map-making (see Mousset et al.~\\cite{2020.QUBIC.PAPER2} for details). We start from a guess map of the sky and a detailed model of the instrument\\footnote{The software \nmakes heavy use of the massively parallel libraries developed by P.~Chanial pyoperators~\\cite{chanial2012pyoperators} (\\href{https:\/\/pchanial.github.io\/pyoperators\/}{https:\/\/pchanial.github.io\/pyoperators\/}) and pysimulators (\\href{https:\/\/pchanial.github.io\/pysimulators\/}{https:\/\/pchanial.github.io\/pysimulators\/}).}. The instrument model uses a detailed description of the synthesized beam. At first we can use an idealized instrument model inspired by an ideal synthesized beam as shown in figure~\\ref{Fig:sb_freq}, or a more realistic one as simulated including optical aberrations (see figures 11 and 12 in O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8}). Then, such a description is expected to be gradually refined during the observation campaigns using information from self-calibration~\\cite{2013A&A...550A..59B} in order to incorporate instrumental systematic effects (from optics, electronics) as well observational effects such as ground-pickup or atmospheric contamination. Our software is designed to be able to incorporate a large variety of such systematic effects through the use of a number of time-domain, frequency-domain, or map-domain operators. We observe the guess map at iteration $i$ with the same scanning strategy used with the real data, obtaining simulated TOD that are compared to the real TOD with a $\\chi^2$. Using a preconditioned conjugate gradient we modify the guess map in an iterative manner until convergence. The final guess map is the solution of the map-making linear problem. There are two ways of accounting for systematics effects in the map-making: i) directly implementing their action in the operators that produce the simulated TOD in the case we have a model with known parameters from self-calibration and ii) fitting the unknown instrumental systematics parameters (for instance Jones matrices elements~\\cite{2007MNRAS.376.1767O}) as extra-parameters, along with the maps pixels, as part of the map-making.\n\nThe synthesized beam used for the instrument model during map-making is actually just a set of Dirac functions with the relevant amplitude at the location of the peaks of the synthesized beam (ideal, including optical aberrations or resulting from self-calibration). In such a way, and similarly as with an imager, the map-making does not attempt to deconvolve from the resolution of the peaks, but only from the multiple peaks.\n\nIf the synthesized beam model accounts for the realistic frequency dependence, one can use multiple maps at multiple frequencies within the physical bandwidth of the instrument and therefore reconstruct such sub-frequency maps. This is spectral imaging~\\cite{2020.QUBIC.PAPER2}. \n\nAnother specificity of this map-making is that the presence of multiple peaks separated by 8.8~degrees on the sky at 150~GHz (6~degrees at 220~GHz) makes QUBIC insensitive to modes on the sky larger than this separation. This occurs because the deconvolution from the multiple peaks relies on the measured signal difference between observations pointing to different directions where the peaks capture different amounts of power. For sky signals at angular scales larger than the angular distance between the peaks, such a difference vanishes. This naturally filters-out large-scale information be it from the sky itself, or from atmospheric gradients. This of course only applies to atmospheric fluctuations at scales larger than the angular distance between peaks in the synthesized beam 8.8 and 6 degrees at 150 and 220 GHz respectively.\n\nTo understand the impact of fluctuations on smaller scales we need to run simulations based on dedicated atmosphere measurements taken on site. In appendix~\\ref{app_impact_small_scale_atm}, however, we briefly discuss this issue and show how we expect a significant impact only from turbulence cells in a restricted range of scales.\n\nA careful study of the noise structure in end-to-end simulated maps shows two significant features that merit further explanation.\n The first is sub-band correlations. When performing spectral imaging, we show that nearby sub-bands exhibit a significant level of noise anti-correlation~\\cite{2020.QUBIC.PAPER2}, as displayed in figure~\\ref{Fig:nunu_corr} for the three Stokes parameters at 150~GHz and 5~sub-bands. The anti-correlation is strong with the nearest sub-bands but reduces significantly beyond. Similar correlation matrices are found at 220~GHz.\n \\begin{figure}[!t]\n \\centering\n \\includegraphics[ width = \\hsize ]{Figures\/nunu_corr_5bands.png}\n \\caption{QUBIC sub-bands correlation matrices for I, Q and U Stokes parameters maps obtained from end-to-end simulations at 150 GHz reconstructing the TOD onto 5 sub-bands (equally spaced in log within the physical 150 GHz bandwidth of QUBIC that ranges from 131 to 169 GHz) using spectral imaging and averaged over pixels in the maps. }\n \\label{Fig:nunu_corr}\n \\end{figure}\n\n \n The second feature worth noticing is spatial correlations. Map-making with a multiply-peaked synthesized beam involves partial deconvolution because a given time sample in a detector's TOD receives power from distinct pixels in the sky with weights given by the shape of the synthesized beam. As a result, we expect significant spatial noise correlations in our maps. This is confirmed by end-to-end simulations as shown in the left panel of figure~\\ref{Fig:space_corr}. Anti-correlation peaks, are expected, at an angle corresponding to the angular separation between the peaks in the synthesized beam ($\\theta_{\\mathrm{peaks}}$=8.8~degrees at 150~GHz and 6 degrees at 220~GHz). A similar 2pt-correlation function is found at 220~GHz, but with even higher correlation amplitude because the secondary peaks are higher due to the top-hat shape of the primary beam resulting from multimode optics at 220~GHz (see figure~4 in O'Sullivan et al.~\\cite{2020.QUBIC.PAPER8}).\n \\begin{figure}[h]\n \\centering\n \\includegraphics[ width = 0.5\n \\hsize ]{Figures\/spatial_correlation.png}\\includegraphics[ width = 0.5 \\hsize]{Figures\/clnoise.png}\n \\caption{{\\bf (left)} QUBIC spatial noise 2pt-correlation function obtained from end-to-end simulations normalized by the variance in the maps $C(\\theta=0)$. The solid lines show an adjustment by a sine-wave modulated by an exponential with a Dirac function at $\\theta=0$ (the noise variance in the maps). The maximum anti-correlation is found as expected at the scale of the angular distance between two peaks of the synthesized beam (S.B.). The amplitude of the correlation is higher at 220~GHz than at 150~GHz because of the top-hat shape of the primary beam at 220~GHz. {\\bf (right)} Spatial noise correlation converted to multipole space. The straight-line at $C_\\ell=1$ shows the expected shape for white noise. The noise correlation results in a reduction of the noise for multipoles larger than $\\sim 40-50$, that is for angular scales $\\lessapprox\\theta_\\mathrm{peaks}$ (angular separation between the synthesized beam peaks). At lower multipoles (larger angular scales), we observe an increase of the noise. This is an advantage for measuring the recombination peak around $\\ell=100$ as discussed in section~\\ref{sect_Bmodes}}.\n \\label{Fig:space_corr}\n \\end{figure}\n\n In the right panel of figure~\\ref{Fig:space_corr} we display the spherical harmonics transform of the 2pt-correlation function:\n \\begin{equation}\n C_\\ell = 2\\pi\\int_{-1}^{1}C(x)P_\\ell(x)\\mathrm{d}x\n \\end{equation}\n where $x=\\cos\\theta$ and $P_\\ell$ are the Legendre polynomials.\n This is our noise angular power spectrum which corresponds to the equivalent for QUBIC of typical white noise for a classical imager. The shape of this noise in Harmonic Space exhibits an excess with respect to white noise at very large scales (small multipoles, below $\\ell=40$ at 150 GHz and $\\ell=50$ at 220 GHz) and a significant reduction at smaller angular scales (larger multipoles). The scale of this transition is determined by the angular distance between peaks in the synthesized beam\\footnote{It is however not strictly equal to $\\pi\/\\theta_\\mathrm{peaks}$ because of the shape of the 2pt-correlation function and the non-trivial correspondence between angles and multipoles.}. Angular scales $\\gtrapprox\\theta_\\mathrm{peaks}$ are not well constrained due to the presence of the multiple peaks that are effectively deconvolved during the map-making. Conversely, angular scales smaller than this angular separation see their noise significantly reduced thanks to the positive correlation of the noise at these angles. \n Because these angular scales correspond to those of the recombination peak in the B-mode spectrum, this specific noise feature for Bolometric Interferometry turns out to be a significant advantage for detecting primordial B-modes. This is discussed with more details from Monte-Carlo simulations in section~\\ref{sect_Bmodes} and visible in Figure~\\ref{Fig:bmodes}. Also, because our noise is not white, the RMS in the maps does not have direct significance. In our case, it is more meaningful to measure the noise level in the angular power-spectrum as we will detail in section~\\ref{sect_Bmodes}.\n \n\\subsubsection{The QUBIC Fast Simulator}\nThe peculiar noise structure of the QUBIC maps has been studied in detail using a number\\footnote{40 end-to-end simulations were used for each of the considered configurations.} of end-to-end simulations run on supercomputers as they have large memory requirements\\footnote{With a 156.25~Hz sampling rate, QUBIC produces $156.25 \\times 3600 \\times 24 \\times 1024 \\times 2 \\times 8 \/1e9 \\simeq 220$~GB\/day.}. We extract from these simulations the main features of the noise discussed above:\n\\begin{itemize}\n \\item noise scaling as a function of normalized coverage;\n \\item correlations between reconstructed sub-bands (see figure~\\ref{Fig:nunu_corr});\n \\item spatial (pixel-pixel) noise correlation measured on maps corrected for the noise scaling with respect to coverage;\n\\end{itemize}\nWe have built a \"Fast Simulator\" that directly produces maps with theses features: \n\\begin{enumerate}\n \\item we start by creating (in harmonic space) noise maps with the observed spatial correlation (see figure~\\ref{Fig:space_corr}),\n \\item we then make linear combinations of these noise maps in order to have I, Q and U maps for each sub-band with the appropriate correlation matrix (see figure~\\ref{Fig:nunu_corr}),\n \\item finally, we scale the noise in the maps according to the scaling with respect to coverage. \n\\end{enumerate} \nThe overall noise normalization is adjusted to match that of the end-to-end simulations with the same integration time. We have checked in detail the accuracy of the Fast Simulator by performing the same noise structure analysis on the output maps and verifying that they lead to the same noise modeling as with the end-to-end simulations.\n\nThe Fast Simulator allows for fast production of maps with large-number statistics (thousands of realizations) and has been used extensively for the forecasts presented in this article.\n\nWe have used a simplified sky coverage obtained using random pointings on the sky from one time sample to another, reproducing the same sky fraction as the anticipated sky coverage shown in figure~\\ref{Fig:coverage}. While this allows obtaining a fast and efficient coverage of the QUBIC observed sky, it prevents one from simulating actual $1\/f$ noise from atmospheric or any time-domain instrumental fluctuations as successive time samples do not correspond to nearby pointings as in a more realistic scanning strategy. As a consequence,\nthe simulations presented in this article implicitly assume a stable atmosphere with no $1\/f$ noise, but account for the average loading from the atmosphere (see section~\\ref{overview} for a discussion of the seasonal and diurnal variations). \nWe will include realistic atmospheric fluctuations in our simulations when more detailed information than the average emissivity and temperature will be available from the TD data on the sky. In addition, as discussed in the second paragraph of this subsection, the shape of the QUBIC synthesized beam implies a low sensitivity on scales much larger than the angular distance between our multiple peaks, which will significantly reduce the impact of the (mostly large scales) atmospheric fluctuations.\n\nNo instrumental systematics are considered in this article, they will be included in future studies as well as how they can be mitigated through self-calibration. All results presented in this article should therefore be considered as idealized and intended to estimate the ultimate sensitivity achievable by a perfect instrument observing a stable atmosphere.\n\n\\subsection{The quest for CMB B-modes}\n\\label{sec:bmodeoverview}\n \n\\subsubsection{Overview}\\label{overview}\nWe have performed simulations for a three-year observation of the sky using the QUBIC FI on a sky without any foregrounds, but with realistic instrumental noise~\\cite{2020.QUBIC.PAPER3,2020.QUBIC.PAPER4}) and with atmospheric background noise (assumed to be stable). The latter has been obtained from measurements performed over 3 years at the QUBIC site in Argentina from a tipper at 210~GHz used for the LLAMA radiotelescope\\footnote{ \\href{https:\/\/www.llamaobservatory.org\/}{https:\/\/www.llamaobservatory.org\/}} to be installed near QUBIC. Atmospheric background is averaged over the 9~best months of the year and corresponds to an atmospheric temperature of 270~K and emissivities 0.081 and 0.138 at 150 and 220~GHz respectively for an average observation elevation of 50 degrees. \n\nChanging the atmospheric parameters according to maximal diurnal and seasonal variations induces 15 and 25\\% change in the photon noise at 150 and 220 GHz respectively (with respect to the numbers given in Table~\\ref{tab_qubic_td_simulation_parameters}). This corresponds to a negligible change of sensitivity for the TD because our noise is dominated by that of the detectors. For the FI, the change in the total noise is 5\\% at 150 GHz and 20\\% at 220 GHz. As said before, more detailed simulations including the measured atmospheric fluctuations will be performed when these measurement are available with the instrument.\n\nWe have used the ``Fast Simulator'' described above to produce thousands of realizations of the noise in the maps incorporating the peculiar noise structure (spatial variations of the noise as a function of coverage as well as spatial and sub-band correlations). The parameters of our simulation are summarized in table~\\ref{tab_qubic_td_simulation_parameters}.\n\n\\begin{table}[t]\n \\renewcommand{\\arraystretch}{1.}\n \\begin{center}\n \\caption{\\label{tab_qubic_td_simulation_parameters}Main instrumental and simulation parameters used in our computations. The noise value for TD is measured from the TES calibration data~\\cite{2020.QUBIC.PAPER3} while that for the FI is the intrinsic TES noise and assumes reduction of the noise aliasing in the readout chain found in the TD. We have used average atmospheric parameters at 50 degrees elevation accounting for maximal seasonal and diurnal variations corresponds to negligible change for the total noise for the TD, 5\\% and 20\\% for the FI at 150 and 220 GHz respectively (see text).}\n \\begin{tabular}{p{6cm} p{3.7cm} m{3.8cm}}\n \\hline\n Parameter& Value TD & Value FI \\\\\n \\hline\n \\hline\n Detector noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.05$\\times 10^{-16}$ & 4.7$\\times 10^{-17}$ \\\\\n \n \\hline\n Atmosphere$^1$ temperature [K]\\dotfill& 270 &270\\\\\n \n Atmosphere emissivity$^1$ at 150\\,GHz\\dotfill& 0.081 &0.081\\\\\n \n Photon noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.6$\\times 10^{-17}$~(150 GHz) & 3.1$\\times 10^{-17}$~(150 GHz),\\hfill\\break 1.17$\\times 10^{-16}$(220 GHz)\\\\\n \\hline\n Total noise [$\\mathrm{W\/\\sqrt{Hz}}$]\\dotfill & 2.06$\\times 10^{-16}$(150 GHz) & 5.7$\\times 10^{-17}$~(150 GHz),\\hfill\\break 1.26$\\times 10^{-16}$~(220 GHz)\\\\\n \\hline\n Cumulated observation time [years]\\dotfill & 1 & 3 \\\\\n \\hline\n $r$ upper-limit (68\\% C.L., No FG) &- & 0.021~(150~GHz),\\hfill\\break 0.023~(220~GHz),\\hfill\\break 0.015~(Combined)\\\\\n \\hline\n \\multicolumn{3}{l}{$^1$The atmosphere is considered perfectly stable.}\\\\ \\end{tabular}\n \\end{center}\n\\end{table}\n\n\nFor each realization, we have used NaMaster~\\footnote{\\href{https:\/\/github.com\/LSSTDESC\/NaMaster}{https:\/\/github.com\/LSSTDESC\/NaMaster}}~\\cite{2019MNRAS.484.4127A} to compute pure TT, EE, TE and BB power spectra on the residual maps (therefore noise-only maps) in order to compute the expected noise on the power spectra. Exploring various values for the minimum multipole $\\ell_{min}$, the size of the multipole bins $\\Delta_\\ell$ and minimum value of the relative coverage (normalized to 1 at maximum) of the sky that defines the region of the sky we keep for analysis, $\\mathrm{Cov}_c$, we have found the best configuration to be $\\ell_\\mathrm{min}=40$, $\\Delta_\\ell=30$ and $\\mathrm{Cov}_c=0.1$ (keeping all pixels with relative coverage above 0.1) at 150 GHz. We have kept the same configuration for the 220~GHz for the sake of simplicity. \n\n\\subsubsection{QUBIC Full Instrument expected performance}\n\\label{sect_Bmodes}\nAs remarked before, measuring the RMS of a map in the case of non-white noise is meaningless, and the actual measurement of the effective depth of our maps is done in $\\ell$-space. The resulting uncertainties from the Monte-Carlo on the BB polarization power spectrum ($\\Delta D_\\ell$) are shown in the left panel of figure~\\ref{Fig:bmodes}. \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[ width = 0.49 \\hsize ]{Figures\/bmodes_zoom.png}\\includegraphics[ width = 0.49\\hsize ]{Figures\/likelihood_r.png}\n\\caption{{\\bf (left)} BB power spectrum error-bars $\\Delta D_\\ell$ on $D_\\ell = \\frac{\\ell(\\ell+1)}{2\\pi}C_\\ell$ on residual maps from the Fast-Simulator Monte-Carlo in the absence of foregrounds, atmospheric fluctuations and instrument systematic effects for an integration time on the sky of three years from the site in San Antonio de los Cobres, Argentina. As these are calculated on residual maps, they do not incorporate sample variance and only refer to the instrumental noise.\nThe reduction at low-$\\ell$ of our error-bars with respect to theoretical white-noise (dotted lines) is clearly visible and in agreement with the expected shape from the 2pt correlation function in figure~\\ref{Fig:space_corr} (dashed lines). The difference in the first bin is discussed in the text.\n{\\bf (right)} Posterior likelihood on the $r$ (assuming no foregrounds) using QUBIC FI (two bands) with three years integration on the sky (including both noise and sample variance). The latest Planck+Bicep\/Keck constraint from Tristram et al.~\\cite{Tristram_r} is shown with the blue arrow}\n\\label{Fig:bmodes}\n\\end{figure}\nWe have also displayed theoretical B-mode power spectra $D_\\ell$ (including lensing) for $r$=0, 0.01, and 0.044 (current best upper limit from Tristram et al.~\\cite{Tristram_r}). Besides the QUBIC error bars, we also plot the expected shape for white noise (dotted lines) and for the QUBIC noise (dashed lines) from figure~\\ref{Fig:space_corr}. \nIn both cases the theoretical error-bars are obtained through the well known formula:\n\\begin{equation}\n \\Delta D_\\ell^\\mathrm{th} = \\frac{\\ell(\\ell+1)}{2\\pi}\\times \\sqrt{\\frac{2}{(2\\ell+1)f_\\mathrm{sky}\\Delta\\ell}}\\times \\frac{1}{B_\\ell^2}\\times \\frac{1}{W_\\ell^2} \\times C_\\ell^\\mathrm{noise}\n\\end{equation}\nwhere $f_\\mathrm{sky}$ is the fraction of the sky used for analysis ($f_\\mathrm{sky}=0.015$ in our case), $\\Delta\\ell$ is the width of the $\\ell$-space binning ($\\Delta\\ell=30$ in our case). $B_\\ell$ is the beam transfer function and $W_\\ell$ is that of the Healpix pixellisation ($N_\\mathrm{side}=256$). $C_\\ell^\\mathrm{noise}$ is the expected shape for the noise power spectrum that can be either a constant in the case of white noise or a constant multiplying the shape from figure~\\ref{Fig:space_corr} (right) in the case of QUBIC. We performed a fit of the above noise normalization for QUBIC to our Monte-Carlo error-bars leading to 2.7~and 3.7~$\\mu\\mathrm{K.arcmin}$ at 150~and 220~GHz respectively. This is shown as dashed lines in figure~\\ref{Fig:bmodes}. However these numbers are hardly comparable with the case of a standard imager for which the noise is white: for each frequency we overplot the expected shape for white noise with the same normalization. The significant noise reduction with respect to the white noise case is particularly visible at the scales of the recombination peak near $\\ell=100$, giving QUBIC an enhanced sensitivity at those scales. At scales larger than the separation between peaks in the synthesized beam however, the error-bars increase sharply.\nThe first bin at 220~GHz exhibits significantly larger error-bars for all spectra. This is not surprising as we have kept the same $\\ell_{min}=40$ for both channels. In reality, the multiple-peaked shape of our synthesized beam is such that we have little sensitivity to multipoles corresponding to angular scales larger than the distance between the peaks (8.8~and 6~degrees at 150~and 220~GHz respectively). As a result, the optimal $\\ell_{min}$ at 150~GHz is slightly too low for 220~GHz, resulting in larger error-bars for the first bin. This will be optimized when analyzing real data.\n\nThe right panel of figure~\\ref{Fig:bmodes} shows the posterior on the tensor-to-scalar ratio which, in the absence of foregrounds, was the only free-parameter for this power-spectrum-based likelihood (simple $\\chi^2$ accounting for sample variance~\\cite{hamimeche-lewis}) with all parameters but $r$ fixed to their fiducial values\\footnote{We have used a fiducial cosmology with parameters [$h=0.675$, $\\Omega_b h^2=0.022$, $\\Omega_c h^2=0.122$, $\\Omega_k=0$, $\\tau=0.06$, $A_s=2e-9$, $n_s=0.965$]}.\nWe calculate the likelihood at 150 and 220 GHz separately as well as jointly. These simulations show that QUBIC has the statistical power (without foregrounds, atmospheric fluctuations and systematics) to constrain the B-modes down to a tensor-to-scalar ratio $r<0.015$ at 68\\% C.L. ($r<0.03$ at 95\\% C.L.) with three years integration on the sky from our site in Argentina. \n\nIn the presence of foregrounds, the numbers above are to be understood as our statistical sensitivity to effective B-modes including the contribution from primordial tensors as well as dust polarization. \nComponent separation has not been included in the current forecasts and will be investigated in details in a future publication. In our study we have focused only on dust contamination, neglecting a possible contribution from synchrotron. This hypothesis is supported by two facts: (i) QUBIC will search for cosmological B-modes in a well defined sky patch that will correspond roughly to the one shown in figure~\\ref{Fig:coverage}, and (ii) in this sky patch the systematic error on $r$ from synchrotron contamination at 150\\,GHz is below 0.005 (see figure~6 in Krachmalnicoff et al.~\\cite{krachmalnicoff2016}), so well below our target $r<0.021$. Moreover, QUBIC will have the capability to check for any residual foregrounds contamination in the obtained tensor-to-scalar ratio by exploiting its spectral imaging features as described in section~\\ref{sss:dust_residuals}. Finally, in the QUBIC data analysis we will also exploit the wealth of data available at low frequency (WMAP, Planck, C-Bass, QUIJOTE and other data that will be publicly available) to further improve the robustness of our final results.\n\n\\subsection{The foregrounds challenge}\n\\label{sect:foreground_challenge}\n\n \\input{foreground_challenge}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nHeterogeneous graphs (which subsume knowledge graphs) can model complex relationships among multiple types of real-world entities and have been widely used in studying biological and biomedical data. In a heterogeneous graph (\\emph{e.g.}, \\cite{fan2019metapath,zhang2019heterogeneous,nicholson2020constructing}), there can be multiple types of nodes (\\emph{e.g.}, some for compounds and others for targets, gene or diseases) and edges (\\emph{e.g.}, a compound treats a disease, a compound resembles another compound, or a gene interacts with another gene), making discovery of deep insights (\\emph{e.g.}, perform drug repurposing or identify drug-target interactions) become possible. Graph representation learning aims to construct low-dimensional embeddings (vectors) for all nodes in a graph, and has been a standard approach to further knowledge discovery and development on graphs. For instance, both predicting biomedical network links based on their structural and feature information and clustering for drug-target interaction and drug discovery are based on high-quality heterogeneous network representation. Recently, various heterogeneous graph representation learning methods have been proposed, \\emph{e.g.}, random-walk based methods Metapath2Vec \\cite{dong2017metapath2vec} and HIN2Vec \\cite{fu2017hin2vec}, message-passing based graph neural networks HAN \\cite{wang2019heterogeneous} and HGT \\cite{hu2020heterogeneous}, and relation learning based methods TransE \\cite{bordes2013translating}, ComplEx \\cite{trouillon2016complex}, and RotatE \\cite{sun2019rotate}. For a systematic review on heterogeneous graph embedding, readers are referred to a good survey \\cite{yang2020heterogeneous}.\n\nHowever, modern heterogeneous graphs can be of very large size in terms of the number of nodes or the number of edges. For instance, the benchmarking knowledge graph Freebase has over $12$ millions nodes across $8$ node types and about $63$ millions edges of $36$ edge types \\cite{yang2020heterogeneous}. Processing such large-scale heterogeneous graphs imposes high computational and storage demands. We identify two directions to meet the requirements: one approach is to develop efficient heterogeneous embedding algorithms. But this direction may not offset the effects of ever growing big heterogeneous graphs. The other is to sparsify\/compress big heterogeneous graphs while approximating some important network property and not significantly impacting the quality of graph embedding. In this work, we focus on the latter topic of sparsifying heterogeneous graphs.\n\nGraph sparsification deals with approximating an arbitrary graph by a sparse graph that usually has a linear number of edges in the number of vertices of the original graph while preserving some important property.\nSeveral notions of graph sparsification have been proposed, \\emph{e.g.}, graph spanners \\cite{PS89} that approximate the shortest path distance between every pair of vertices, cut sparsifiers \\cite{BK96} that approximately preserve all graph cut values, and spectral sparsifiers \\cite{ST11} that approximate the graph spectrum\/eigenvalues. Graph sparsification has been used to improve the computational cost of learning over graphs, \\emph{e.g.}, graph semi-supervised learning and spectral clustering \\cite{SWT16,ZLB21a,ZLB21b}. Although heterogeneous graphs can have much larger size since heterogeneity allows more nodes or edges to be included in one network, however, there has been no systematic study on sparsification techniques for heterogeneous graphs yet.\n\nIn this paper, we for the first time study the problem of heterogeneous graph sparsification and its algorithmic applications in graph representation learning. We propose an efficient algorithm for constructing provably smaller sparsifiers for heterogeneous graphs, which, when feeding into a heterogeneous graph embedding algorithm, can save significant computational and storage cost. However, interestingly, the quality of the learned representations using sparsifiers is comparable or even \\emph{better} than those based on the original graphs. We have provided theoretical analysis of the saving in time and space complexities of heterogeneous graph representation learning. Extensive experiments have also been performed to demonstrate that the developed method can improve the efficiency of heterogeneous graphs while not significantly affecting the accuracy of downstream machine learning tasks based on the learned embedding.\n\n\n{\\noindent \\bf Related Work.}\nSparsification over homogeneous graphs has been extensively studied, \\emph{e.g.}, \\cite{PS89,BK96,ST11} and used in machine learning \\cite{SWT16,CKL+18,ZSL+19a}. In heterogeneous graphs, however, how to sparsify various types of vertices and edges while preserving graph properties has received limited studies \\cite{jiang2021pre,yang2020heterogeneous}. Jiang \\emph{et al.} \\cite{jiang2021pre} developed self-supervised pre-training methods for heterogeneous graphs with relation-based sparsification for efficient pre-training. But their focus is on designing both the node- and schema-level pre-training tasks to preserve semantic and structural properties, instead of a comprehensive study on the sparsification method.\nTo the best of our knowledge, this work is the first systematical study on heterogeneous graph sparsification and its use in representation learning.\n\n\n\n\n\n\n\n\\section{Heterogeneous Graph Sparsifiers}\n\nIn this section, we develop a sampling-based sparsification technique for heterogeneous graphs, analyze its computational and space complexities, and then discuss other potential methods.\n\n{\\noindent \\bf Notations and Definitions.}\nA heterogeneous graph can be defined as $G(V,E,\\phi,\\pi,X,R)$ where $V$ is the vertex set, $E$ is the edge set, $\\phi$ specifies the type for each vertex, $\\pi$ assigns the type of each edge, and $X$ and $R$ include the node and edge features respectively. $\\phi$, $\\pi$, $X$, and $R$ can be omitted from the presentation if they are clear from the context. Let $E_{Out}(u)$ be the set of $u$'s out-edges and $E_{Out}^t(u)$ be the set of $u$'s out-edges of type $t$. Similarly, let $E_{In}(u)$ be the set of $u$'s in-edges and $E_{In}^t(u)$ be the set of $u$'s in-edges of type $t$. Let $|G(V,E)|=|E|$.\n\n{\\noindent \\bf Motivation.}\nGraph sparsification in general\/homogeneous graphs has received considerable attention and many sparsification algorithms have been developed. However, they can be separated into two types, \\emph{importance} based sampling methods and \\emph{heuristic} based sampling methods. In the former, graph edges are sampled according to some important measure. For example, spectral sparsifiers and cut sparsifiers can be obtained by non-uniformly sampling each edge based on its effective resistance and edge connectivities \\cite{SS11,FHH+11}, respectively. This type of approaches is often sound and has nice theoretical guarantees. However, such a design for heterogeneous graphs can be highly challenging considering the complexity in defining an importance measure across different types of edges and computing that measure in large-scale heterogeneous graphs. \n\nIn the latter, sampling is performed on all edges independently using only heuristics instead of complex importance measures. For example, $k$-neighbors sparsifiers are constructed by randomly sampling at most $k$ edges for each vertex where $k>0$ is a tuneable parameter. Though simple, heuristic based sparsifiers have performed practically good, \\emph{e.g.}, comparable to spectral sparsifiers in machine learning tasks over graphs \\cite{SWT16}. In this paper, we will work towards sparsifying heterogeneous graphs using simple yet effective heuristics.\n\n\n{\\noindent \\bf Details.}\nAs a warm-up, we consider the simple method of sampling a fixed number of edges and including in the sparsifier. Since node degrees can be quite different for different nodes, it is possible that a node of low degree has none of its edges sampled and become \\emph{isolated} in the sparsifier. Unfortunately, some heterogeneous embedding algorithms do not allow isolated nodes (through reporting errors such as \\cite{shi2018aspem,wang2019heterogeneous}) and thus this type of sparsifiers is not robust to embedding algorithms. \nFurthermore, the number of edges of different types for a node can be very different because of their semantic difference. Consider an extreme case: a vertex $u$ with $10K$ edges of type $1$ and only one edge of type $2$. The information carried by the edge of type $2$ is supposed to be greater than that in an individual edge of type $1$. When all edges are considered together, the single edge of type $2$ can be easily sampled away, resulting in information loss.\n\nWe propose to perform random sampling within each type of edges and then aggregate all sampled edges as the sparsfiier. In this way, the isolation problem is avoided and the distribution of edge types cannot affect the quality of the constructed sparsifier.\nSpecifically, we sparsify the neighborhood of each vertex according to ascending order of vertex degree. Since heterogeneous graphs are often directed, we sparsify out-edges and in-edges of a vertex separately. Consider out-edges $E_{Out}(u)$ of vertex $u$. For each edge type $t$ in $E_{Out}(u)$, if the number of edges of type $t$ is at most $k$, we include all of them to the sparsifier $H$. Here $k$ is an input parameter to control the size of the output sparsifier. Otherwise, we first calculate the number of edges in $E^t_{Out}(u)$ that are already included in the current $H$. Note that we process vertices one by one and the processing of some vertex before the current vertex may already add a few edges in $E_{Out}(u)$ to $H$. Let $x$ be the number of such edges, $x=|E_{Out}^t(u)\\cap H|$. Then we sample $k-x$ edges from $E_{Out}^t(u)-H$ uniformly at random and include the sampled edges in $H$. Since in heterogeneous graphs there can be many edges without edge weight, we keep the weight of a sampled edge unchanged, if any. The processing for in-edges is symmetric. The algorithms are summarized in Algorithms \\ref{alg:graph} and \\ref{alg:node1}.\n\n\n\\begin{algorithm}[t]\n\\small\n\\caption{Sparsify-Graph}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n\\REQUIRE A graph $G$ and a parameter $k$\n\\ENSURE A sparsified graph\n\\STATE $H=\\emptyset$;\n\\STATE Sort vertices in ascending order of degree;\n\\FOR {each node $u$ in the sorted order}\n \\STATE Sparsify-Node-Out($u$, $G$, $H$, $k$);\n \\STATE Sparsify-Node-In($u$, $G$, $H$, $k$);\n\\ENDFOR\n\\RETURN $(V,H)$;\n\\end{algorithmic}\n\\label{alg:graph}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\small\n\\caption{Sparsify-Node-Out(-In)}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n\\REQUIRE A node $u$ in a graph $G$, an edge set $H$, and a parameter $k$\n\\ENSURE Updated $H$\n\\FOR {each edge type $t$}\n \\IF {$|E_{Out}^t(u)|