diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlumy" "b/data_all_eng_slimpj/shuffled/split2/finalzzlumy" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlumy" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe tensor model was first introduced in \\cite{Ambjorn:1990ge,Sasakura:1990fs,Godfrey:1990dt} \nas an analytical description of simplicial quantum gravity in dimensions higher than \ntwo\\footnote{See, however, \\cite{Fukuma:2015xja,Fukuma:2015haa,Fukuma:2016zea} \nfor a matrix-model-like approach to three-dimensional quantum gravity.} \nby generalizing the matrix model, which successfully describes the two dimensional case.\nWhile the original tensor models are still remaining merely as formal descriptions due to some difficulties, \nthe analyses of the more successful model, the colored tensor model \n\\cite{Gurau:2009tw}, have produced various interesting analytical results concerning the simplicial quantum gravity\nin dimensions higher than two \\cite{Gurau:2011xp}. Among them, it has been shown that the dominant contributions \nof simplicial complexes generated \nfrom the colored tensor model are branched polymers \\cite{Bonzom:2011zz,Gurau:2013cbh}. \nSince the structure of branched polymers is far from the classical spacetime picture of our universe, \nit seems difficult to consider the tensor model as a sensible model of quantum gravity, which\nshould produce wide and smooth spacetimes in certain classical regimes. \n\nOn the other hand, while the models above basically concern the Euclidean case, \nit has been shown that Causal Dynamical Triangulation (CDT),\nwhich is the simplicial quantum gravity with a causal structure, \nsuccessfully produces the 3+1 dimensional world similar to our universe \\cite{Ambjorn:2004qm},\nwhile Dynamical Triangulation, which is the Euclidean version, does not\\footnote{\nWhen coupling many U$(1)$-fields, \nthe authors in \\cite{Horata:2000eg} found a promise of \na phase transition higher than first order, \nwhich, however, is in conflict with the result in \\cite{Ambjorn:1999ix}.\n}.\nThe comparison between the two versions suggests that a causal structure is essentially \nimportant for the emergence of a classical spacetime in quantum gravity. \nThis motivated one of the present authors to formulate\na rank-three tensor model as a totally constrained system in the canonical formalism,\nwhich we call Canonical Tensor Model (CTM) \\cite{Sasakura:2011sq,Sasakura:2012fb}.\nThe constraints of CTM are composed of kinematical symmetry generators and \nthose analogous to the Hamiltonian constraint in the Arnowitt-Deser-Misner (ADM) formalism \n\\cite{Arnowitt:1960es,Arnowitt:1962hi}, \nand form a first-class constraint algebra with a non-linear structure. \nIn fact, the algebraic structure of the constraints is very similar to that of \nthe ADM formalism of general relativity (GR), and \nit can be shown \\cite{Sasakura:2015pxa} that, in a formal continuum limit, the constraint algebra of CTM\nagrees with that of the ADM formalism of GR\\footnote{As well, a certain minisuperspace model of GR can be derived from\nCTM \\cite{Sasakura:2014gia}.}.\nThis is of physical importance, since the algebraic closure of the ADM constraints assures\nthe spacetime covariance of locally defined time evolutions, which is an essence of GR \\cite{Hojman:1976vp}. \n\n\nThe main purpose of this paper is to pursue this correspondence further.\nWe will analyze the classical equation of motion (EOM) of CTM in a formal continuum limit \nthrough a derivative expansion of the tensor of CTM up to the fourth order, \nand will show that it is the same as that of a coupled system of gravity and a scalar field derived from the Hamilton-Jacobi \nequation with an appropriate choice of an action. \nThe action has an exponential potential of the scalar field, and the system is \nclassically invariant under a dilatational symmetry. \nInterestingly, the action is meaningful only \nin spatial dimensions $2\\leq d \\leq 6$, and the system becomes unstable\nin $d>6$ due to the wrong sign of the scalar kinetic term. In the critical dimension $d=6$, \nde Sitter spacetime becomes a solution to the EOM, signaling the emergence of a conformal symmetry.\n\nThe present work may also have some implications to renormalization-group (RG) flow equations of field theories. \nIt has been argued \\cite{Sasakura:2015xxa,Sasakura:2014zwa,Sasakura:2014yoa} \nthat the Hamiltonian constraints of CTM generate the RG flows of statistical systems on random networks \\cite{revnetwork},\nwhich can equivalently be described by randomly connected tensor networks.\nIn addition, it has been shown \\cite{Chen:2016xjx} that classical spaces emerge on boundaries of randomly \nconnected tensor networks by appropriately choosing the tensors. Therefore, it can be expected that \nthe Hamiltonian constraints would generate RG flows of effective field theories on such emergent spaces.\nIf so, the present work would give a hint to the connection between RG flows of field theories and gravity, \nwhich is indeed the subject of the so-called holographic RG (See \\cite{Fukuma:2002sb} for a review.).\n\nWe heavily used a Mathematica package ``xTensor\" \\cite{xtensor} to perform tensorial computations in this paper.\nThe mathematica programs we used can be downloaded from one of the author's \nhomepage \\cite{sasahome}.\n\nThis paper is organized as follows. \nIn Section~\\ref{sec:review}, we review CTM. \nIn Section~\\ref{sec:repP}, we define the fields of CTM in a formal continuum limit in terms of a\nderivative expansion of the tensor of CTM up to the fourth order. There are four fields, a rank-0,2,3,4 tensor\nfield with the weight of negative half-density.\nIn Section~\\ref{sec:gauge}, we study the kinematical symmetry of CTM in the continuum limit. Up to the fourth order, \nwe find two gauge symmetries, the diffeomorphism and a spin-three symmetry.\nIn Section~\\ref{sec:fixed}, by deleting the rank-3 and rank-4 fields by the spin-three gauge symmetry and the EOM, respectively,\nwe write down the EOM of the remaining fields, the rank-0 and rank-2 fields,\nin a static background geometry. \nIn Section~\\ref{sec:backmetric}, we discuss another gauge symmetry which allows us to freely transform the background metric. \nThen, in Section~\\ref{sec:identify}, the background metric is gauge-fixed to a combination of the fields\nso as to remove the odd situation that there exists a static spin-two field, the background metric, other than\nthe rank-2 field of CTM. The EOM with the gauge-fixing condition is written down. \nIn Section~\\ref{sec:delete}, we rewrite the EOM after deleting the weights of the fields. \nIn Section~\\ref{sec:reparametrization}, we \nperform a reparameterization of the fields so that there are no spatial derivative terms of the lapse function in EOM. \nThis is the final form of the EOM of CTM, which is comparable with that of a gravitational system in field theory. \nIn Section~\\ref{sec:conttheory}, we show that the EOM of CTM can be made coincident with\nthat of a coupled system of gravity and a scalar field \nderived from the Hamilton-Jacobi equation by an appropriate choice of an action.\nWe find a critical spatial dimension of the gravitational system, given by six, \nover which the system becomes unstable due to the wrong sign of the kinetic term of the scalar field.\nIn Section~\\ref{sec:mini}, we discuss the time evolution of the scale factor. At the critical dimension, \nde Sitter spacetime is a solution to the EOM, signaling the emergence of a conformal symmetry, \nwhile the time evolution of the scale factor has a power-law behavior below the critical dimension.\nSection~\\ref{sec:summary} is devoted to the summary and future prospects. \n \n\\section{Review of CTM}\n\\label{sec:review}\nIn this section we review the canonical tensor model (CTM) \n\\cite{Sasakura:2011sq, Sasakura:2012fb}, \nexplaining its current status. \n\nWe consider a Hamiltonian system such that \nthe dynamical variables are the real symmetric rank-three tensors, \n$M_{abc}$ and $P_{abc}$ $(a,b,c=1,2,\\cdots, \\mathcal{N})$, \nwhich are canonically conjugate \nin the sense that they satisfy the following Poisson bracket:\n\\[\n\\{ \nM_{abc}, P_{def}\n\\}\n= \n\\sum_{\\sigma}\n\\delta_{a \\sigma_d}\\delta_{b \\sigma_e}\\delta_{c \\sigma_f}, \n\\ \\ \\ \n\\{ \nM_{abc}, M_{def}\n\\}\n= \n\\{ \nP_{abc}, P_{def}\n\\}\n=0, \n\\label{eq:poissopn}\n\\] \nwhere the summation is over all the permutations of $d$, $e$ and $f$, \nreflecting the real symmetric nature of the tensors. \nHere, it would be natural to introduce the O($\\mathcal{N}$) transformation\nas a kinematical symmetry of the system, \n\\[\n\\begin{split}\n&M_{abc} \\to M'_{abc} = L_{aa'}L_{bb'}L_{cc'}M_{a'b'c'}, \\\\\n&P_{abc} \\to P'_{abc} = L_{aa'}L_{bb'}L_{cc'}P_{a'b'c'},\n\\end{split}\n\\label{eq:ontransformation}\n\\]\nwhere the repeated indices are summed over and $L$ is an O($\\mathcal{N}$) matrix, \nsince quantities constructed by the tensors \nwith all indices being contracted are invariant under the O$(\\mathcal{N})$ transformation. \nThe Hamiltonian of CTM is given as follows:\n\\[\nH_{CTM}\n= n_a \\mathcal{H}_a + n_{ab} \\mathcal{J}_{ab},\n\\label{eq:ctmhamiltonian}\n\\] \nwhere $n_a$ and $n_{ab} (=-n_{ba})$ are non-dynamical Lagrange's multipliers, \nand\n\\[\n&\\mathcal{H}_a \n=\\frac{1}{2} \\left(P_{abc}P_{bde}M_{cde} - \\lambda M_{abb} \\right), \n\\label{eq:ctmhamiltonianconstraint}\n\\\\\n&\\mathcal{J}_{ab}\n=-\\mathcal{J}_{ba}\n= \\frac{1}{4} \\left( P_{acd}M_{bcd} - P_{bcd}M_{acd} \\right), \n\\label{eq:ctmmomentumconstraint} \n\\]\nin which $\\lambda$ is a cosntant. \nImitating the nomenclatures in the Arnowitt-Deser-Misner (ADM) formalism of general relativity, \n$\\mathcal{H}_a$ and $\\mathcal{J}_{ab}$ are dubbed as \nHamiltonian constraint and momentum constraint, respectively, \nand they form the following first-class constraint Poisson algebra: \n\\[\n\\begin{split}\n& \\{ \\mathcal{H} (\\xi^1), \\mathcal{H} (\\xi^2) \\} \n= \\mathcal{J} ( [ \\tilde{\\xi}^1, \\tilde{\\xi}^2 ] + 2 \\lambda\\, \\xi^1 \\wedge \\xi^2 ), \\\\\n& \\{ \\mathcal{J} (\\eta), \\mathcal{H} (\\xi) \\} \n= \\mathcal{H} (\\eta \\xi), \\\\\n&\\{ \\mathcal{J} (\\eta^1), \\mathcal{J} (\\eta^2) \\} \n= \\mathcal{J} ([ \\eta^1, \\eta^2 ]), \n\\label{eq:ctmconstraintalgebra}\n\\end{split}\n\\]\nwhere $\\mathcal{H}(\\xi) := \\xi_a \\mathcal{H}_a$, $\\mathcal{J}(\\eta) := \\eta_{ab}\\mathcal{J}_{ab}$, \nand $\\tilde{\\xi}_{ab}:= P_{abc}\\xi_c$. \nIn (\\ref{eq:ctmconstraintalgebra}), the bracket $[\\ , \\ ]$ denotes the matrix commutator, \nand $(\\xi^1\\wedge \\xi^2)_{ab}:= \\xi^1_a \\xi^2_b - \\xi^2_a \\xi^1_b$. \nOne notices that $\\mathcal{J}$ serves as the generators of SO($\\mathcal{N}$),\ninfinitesimally representing the kinematical symmetry of the system.\nThe form of the Hamiltonian constraint has been uniquely fixed \nby the following five assumptions: \nthe Hamiltonian constraint \n(I) carries only one index, \n(II) forms a closed Poisson algebra with $\\mathcal{J}$, \n(III) preserves the time reversal symmetry, $M_{abc} \\to M_{abc}$ and $P_{abc}\\to - P_{abc}$, \n(IV) consists of terms cubic at most, \nand \n(V) allows only ``connected terms,'' \ne.g., \n$P_{abc}P_{bde}M_{cde}$ is allowed but $M_{abb}P_{cde}P_{cde}$ is not allowed \\cite{Sasakura:2012fb}.\nWith the closed Poisson algebra (\\ref{eq:ctmconstraintalgebra}) of the constraints, CTM is a totally constrained system \ngoverned by the Hamiltonian (\\ref{eq:ctmhamiltonian}). \nThe interest of this paper is the classical equation of motion (EOM) of $P$ with $\\lambda=0$, which \nis given by\n\\[\n\\begin{split}\n\\frac{\\text{d}}{\\text{d}t} P_{abc} \n= \\{P_{abc}, H_{CTM}^{\\lambda=0} \\} \n= - \\frac{1}{2} \n\\sum_{\\sigma}\n\\left(\nn_d P_{de \\sigma_{a}} P_{\\sigma_{b}\\sigma_{c}e} \n+ n_{d \\sigma_{a}} P_{\\sigma_{b}\\sigma_{c}d}\n\\right). \n\\label{eq:ctmeom} \n\\end{split}\n\\]\nThe variable $M_{abc}$ will play no roles in this paper.\n\nQuite remarkably, CTM is closely related to general relativity in arbitrary dimensions \nin the following sense. \nFirstly, for $\\mathcal{N}=1$ case, the Hamiltonian (\\ref{eq:ctmhamiltonian}) agrees with that of a certain minisuperspace \nmodel of GR in arbitrary dimensions, \nif we consider \nthe modulus of the tensor, $|M_{111}|$, is proportional to the spatial volume in the minisuperspace model \\cite{Sasakura:2014gia}. \nSecondly, in a formal continuum limit with $\\mathcal{N}\\to \\infty$, \nthe Poisson algebra (\\ref{eq:ctmconstraintalgebra}) coincides with the Dirac algebra in the ADM formalism \\cite{Sasakura:2015pxa}. \nIn this paper we take this argument one step further: \nwe will analyze the EOM (\\ref{eq:ctmeom}) of CTM in a formal continuum limit\nthrough a derivative expansion of $P$ up to the fourth order, and will\nshow that it agrees with the EOM of a coupled system of gravity and a scalar field \nderived from the Hamilton-Jacobi equation with an appropriate choice of an action.\n\n\\section{Representation of the tensor in a derivative expansion}\n\\label{sec:repP}\nIn this paper, we consider CTM in a formal continuum limit.\nWe leave aside for future study the question of dynamics why CTM can be studied in the continuum manner: we \nsimply assume that there exist some regimes where the continuum description is valid. \nThe basic strategy to treat CTM in this limit is the same \nas that in the previous papers \\cite{Sasakura:2015pxa,Chen:2016xjx}. \nWe formally replace the discrete values of the indices to the $d$-dimensional spatial coordinates: \n\\[\na \\rightarrow x \\in R^d.\n\\] \nNamely, the tensor $P_{xyz}$ is a function of three $d$-dimensional coordinates $x,y,z$, symmetric under arbitrary permutations.\nWe further assume a locality: $P_{xyz}$ takes non-vanishing values, only when $x,y,z$ are \nin the neighborhood, $x\\sim y \\sim z$. Mathematically, this can be formulated by that $P_{xyz}$ is a \ndistribution described by delta functions and their \nderivatives\\footnote{In \\cite{Sasakura:2015pxa}, the mathematical formulation is presented\ndifferently as a moment expansion in coordinates. \nThough they are essentially the same from the physical point of view, the present formulation in terms of distributions is superior to \nthe former one in the sense that the covariance can be easily incorporated. }:\n $P_{xyz}\\sim \\delta^d(x-y)\\delta^d (y-z)+\\hbox{derivatives of }\\delta^d(x-y)\\delta^d (y-z)$. \nWe also assume that we can terminate the derivative expansion at a certain order. From the \nphysical point of view, this is an assumption that the scale of the physical process of our interest is \nmuch larger than the fuzziness of the locality of the space. \nIn general, it is more convenient to use test functions to describe distributions \nrather than directly dealing with $\\delta$-functional expressions. \nSo, let us consider a contraction of $P$ with a test function $f$ up to the fourth order of derivatives as follows:\n\\[\n\\begin{split}\nP f^3&:=\n\\int d^dx d^dy d^dz\\, P_{xyz} f(x)f(y)f(z) \\\\\n&=\\int d^dx \\left( \\beta f^3 + \\beta^{\\mu\\nu}f^2 f_{,\\mu\\nu}+ \\beta^{\\mu\\nu\\rho} f^2 f_{,\\mu\\nu\\rho}\n+\\beta^{\\mu\\nu,\\rho\\sigma}f f_{,\\mu\\nu}f_{,\\rho\\sigma}+{\\cal O}(\\nabla^5)\\right),\n\\end{split}\n\\label{eq:pf3}\n\\]\nwhere, for brevity, the arguments $x$ of $\\beta$'s and $f$ are suppressed in the last line, \nand the greek indices represent spatial directions, e.g., $\\mu=1,2,\\cdots,d$.\nHere, the test function $f$ is assumed to have a compact support, \nand the indices of $f$ represent the covariant derivatives associated with a background metric $g_{\\mu\\nu}$, \ni.e., $f_{,\\mu\\nu}:=\\nabla_\\mu \\nabla_{\\nu} f,\\ f_{,\\mu\\nu\\rho}:=\\nabla_\\mu \\nabla_{\\nu} \\nabla_\\rho f$.\nAs will be explained in more detail in Section~\\ref{sec:gauge}, the test function is not a scalar, but \nmust be treated as a scalar half-density. Therefore, the covariant derivatives are defined with \na weight contribution: $\\nabla_\\mu f=(\\partial_\\mu -\\frac{1}{2} \\Gamma_\\mu)f$ with \n$\\Gamma_{\\mu}:=\\Gamma_{\\mu\\nu}^\\nu$,\n$\\nabla_\\mu \\nabla_\\nu f=(\\partial_\\mu -\\frac{1}{2}\\Gamma_\\mu)\\nabla_\\nu f -\\Gamma_{\\mu\\nu}^\\rho \\nabla_\\rho f$, and so on. The tensor fields, $\\beta^{\\mu\\nu}$ and $\\beta^{\\mu\\nu\\rho}$, are symmetric, \nand the field $\\beta^{\\mu\\nu,\\rho\\sigma}$ has the pairwise symmetries,\n\\[\n\\beta^{\\mu\\nu,\\rho\\sigma}=\\beta^{\\nu\\mu,\\rho\\sigma}=\\beta^{\\mu\\nu,\\sigma\\rho}=\n\\beta^{\\rho\\sigma,\\mu\\nu}.\n\\]\nThus, up to the fourth order, the ``components\" of $P$ are represented by the four fields, \n$\\beta(x),\\beta^{\\mu\\nu}(x),\\beta^{\\mu\\nu\\rho}(x)$, and $\\beta^{\\mu\\nu,\\rho\\sigma}(x)$.\nBecause of the weight of $f$ and the invariance of $Pf^3$, \nthese fields are assumed to have the weight of negative half-density\n(the details will be given in Section~\\ref{sec:gauge}):\n\\[\n\\begin{split}\n&[f]=\\frac{1}{2}, \\\\\n&[\\beta]=[\\beta^{\\mu\\nu}]=[\\beta^{\\mu\\nu\\rho}]=[\\beta^{\\mu\\nu,\\rho\\sigma}]=-\\frac{1}{2}.\n\\end{split}\n\\label{eq:weightbeta}\n\\]\nHere, $[X]$ denotes the weight of a quantity $X$, meaning that $X$ has the same weight as $g^\\frac{[X]}{2}$\n($g:=\\hbox{Det}[g_{\\mu\\nu}]$). These weights cancel the weight of the integration measure $d^dx$ to secure the \ninvariance of $Pf^3$. \n\nHere, we will explain more details about the derivative expansion \\eq{eq:pf3}. \nFirstly, as proven in Appendix~\\ref{app:pf3}, a totally symmetric rank-three tensor \ncan be fully characterized by the values of the contraction with an arbitrary vector $\\phi$\\,: \n$P_{abc}\\phi_a \\phi_b \\phi_c \\hbox{ for }^\\forall \\phi$.\nThus, it is enough to know $Pf^3$ for arbitrary $f$ as in \\eq{eq:pf3} for the full characterization of $P$, instead of \nconsidering three different functions for the three indices. \nSecondly, throughout this paper, we will consider the derivative expansion of $P$ up to the fourth order of derivatives, \nas in \\eq{eq:pf3}. \nThe reason is that we are interested in the equations of motion (EOM) of the fields $\\beta,\\beta^{\\mu\\nu}$ up to the \nsecond order of derivatives: as will be discussed later, these fields describe a coupled system of gravity and a scalar field,\nwhich is of physical interest. To correctly describe the EOM of \n$\\beta^{\\mu\\nu}$ (and $\\beta$) up to the second derivatives, \nit is necessary to include the fourth order of derivatives in the expansion of $P$ as in \\eq{eq:pf3}.\nAs one can prove, an independent set of fields describing $P$ up to the fourth order\nare exhausted by the set shown in \\eq{eq:pf3}. \nMore details are given in Appendix~\\ref{app:fourth} and \\ref{app:derivative}.\nLastly, we have introduced a background metric $g_{\\mu\\nu}$, which can be taken arbitrary. \nAs will be explained in detail in Section~\\ref{sec:backmetric},\nthe introduction of the background metric does not change the physical contents,\nbut simply redefines the fields with a linear recombination of them.\nIn fact, we will see that there exists a gauge symmetry which allows one to freely change the background metric\nwith simultaneous change of the fields, and will ultimately\ngauge-fix the background metric to a certain combination of the fields.\n\nIn the analysis of the EOM \\eq{eq:ctmeom} of CTM, \nit is necessary to have an expression corresponding to $3 P_{abc}\\phi_b \\phi_c$. \nIn the continuum limit, one can obtain this by the functional derivative of $Pf^3$ in \\eq{eq:pf3}:\n\\[\n\\begin{split}\nP[f,f]:=&\\frac{\\delta}{\\delta f(x)} Pf^3 \\\\\n=&3 \\beta f^2 + 2 \\beta^{\\mu\\nu} f f_{,\\mu\\nu} + (\\beta^{\\mu\\nu} f^2 )_{,\\mu\\nu} +2 \\beta^{\\mu\\nu\\rho} f f_{,\\mu\\nu\\rho}\n-(\\beta^{\\mu\\nu\\rho}f^2)_{,\\mu\\nu\\rho} +\\beta^{\\mu\\nu,\\rho\\sigma} f_{,\\mu\\nu} f_{,\\rho\\sigma} \\\\\n&+2 (\\beta^{\\mu\\nu,\\rho\\sigma} f f_{,\\mu\\nu})_{,\\rho\\sigma} +{\\cal O}(\\nabla^5)\\\\\n=&(3 \\beta +\\beta^{\\mu\\nu}_{,\\mu\\nu}-\\beta^{\\mu\\nu\\rho}_{,\\mu\\nu\\rho})f^2+(4 \\beta^{\\mu\\nu}_{,\\nu}\n-6 \\beta_{,\\nu\\rho}^{\\mu\\nu\\rho}) f f_{,\\mu}+(2 \\beta^{\\mu\\nu}-6 \\beta^{\\mu\\nu\\rho}_{,\\rho}) f_{,\\mu} f_{,\\nu} \\\\\n&+(4 \\beta^{\\mu\\nu}-6 \\beta_{,\\rho}^{\\mu\\nu\\rho}+\n2\\beta^{\\mu\\nu,\\rho\\sigma}_{,\\rho\\sigma}) f f_{,\\mu\\nu} \n+(-6 \\beta^{\\mu\\nu\\rho}+4\\beta^{\\mu\\sigma,\\nu\\rho}_{,\\sigma})\nf_{,\\mu} f_{,\\nu\\rho} \\\\\n&+4\\beta^{\\mu\\nu,\\rho\\sigma}_{,\\sigma} f f_{,\\rho\\mu\\nu} \n+3\\beta^{\\mu\\nu,\\rho\\sigma}f_{,\\mu\\nu}f_{,\\rho\\sigma}\n+4\\beta^{\\mu\\nu,\\rho\\sigma}f_{,\\mu} f_{,\\nu\\rho\\sigma}\n+2\\beta^{\\mu\\nu,\\rho\\sigma}f f_{,\\mu\\nu\\rho\\sigma}+{\\cal O}(\\nabla^5).\n\\end{split}\n\\label{eq:pff}\n\\]\n\nSimilarly, one can define an expression corresponding to $3 P_{abc}\\phi^1_a \\phi_b^2$ for two different vectors $\\phi^{1,2}$. \nThis is denoted by $P[f,g]$, and is defined by an obvious generalization: \nputting $f,g$ into two $f$'s of each term on the right-hand side of \\eq{eq:pff}, and symmetrizing them.\n\n\\section{Kinematical symmetry in the continuum limit}\n\\label{sec:gauge}\nCTM has the kinematical symmetry generated by the orthogonal group generators ${\\cal J}_{ab}$. \nIn the continuum limit, since the indices represent coordinates, \n${\\cal J}_{xy}$ will become generators of local gauge transformations. \nIn the derivative expansion, the gauge transformations are parameterized by \ntensor fields, like those in \\eq{eq:pf3} for $P$.\nUp to the fourth order, we will find two gauge transformations,\nwhich are the diffeomorphism and a spin-three gauge transformation.\n\nThe orthogonal group transformation of CTM can be characterized by a linear transformation of $f_a$ which preserves\nthe norm square $f_a f_a$. In the continuum limit, this condition is translated to the invariance of \n\\[\n\\Vert f \\Vert^2\\equiv \\int d^dx\\ f(x)f(x),\n\\label{eq:norm}\n\\] \nwhere $f(x)$ is considered to be a scalar half-density, and is assumed to have a compact support.\nIt is easy to show that \\eq{eq:norm} is invariant under the following infinitesimal linear transformations,\n\\[\n\\begin{split}\n\\delta_1 f(x)&=\\frac{1}{2} \n\\left[\\nabla_\\mu( v^\\mu(x) f(x)) + v^\\mu(x) \\nabla_\\mu f(x)\\right]=\\frac{1}{2} v^\\mu_{,\\mu}(x) f(x)+ v^\\mu(x) f_{,\\mu}(x),\\\\\n\\delta_3 f(x)&=\\frac{1}{2} \\left[ \\nabla_\\mu\\nabla_\\nu\\nabla_\\rho( v^{\\mu\\nu\\rho}(x) f(x)) + v^{\\mu\\nu\\rho}(x) \\nabla_\\mu\\nabla_\\nu\n\\nabla_\\rho f(x)\\right] \\\\\n&=\\frac{1}{2} v^{\\mu\\nu\\rho}_{,\\mu\\nu\\rho}(x) f(x)+\\frac{3}{2} v^{\\mu\\nu\\rho}_{,\\mu\\nu}(x) f_{,\\rho}(x)+\\frac{3}{2} v^{\\mu\\nu\\rho}_{,\\mu}(x) f_{,\\nu\\rho}(x)\n+ v^{\\mu\\nu\\rho}(x) f_{,\\mu\\nu\\rho}(x),\n\\end{split}\n\\label{eq:delta13}\n\\]\nwhere $v^\\mu$ and $v^{\\mu\\nu\\rho}$ are a vector field and a symmetric rank-three tensor field, respectively, and \n$\\nabla_\\mu$ is the covariant derivative ($\\nabla_\\mu f=(\\partial_\\mu -\\frac{1}{2} \\Gamma_\\mu)f$ with \n$\\Gamma_\\mu\\equiv \\Gamma_{\\mu\\nu}^\\nu$, etc.).\nHere we use the same simplified notations as in Section~\\ref{sec:repP}, such as $f_{,\\mu\\nu}=\\nabla_\\mu\\nabla_\\nu f$.\nIndeed,\n\\[\n\\delta_1 \\Vert f \\Vert^2 =2 \\int d^dx\\, f(x) \\delta_1 f(x)=\\int d^dx \\, f(x) \\left[ \\nabla_\\mu( v^\\mu(x) f(x)) \n+ v^\\mu(x) \\nabla_\\mu f(x)\\right]=0,\n\\]\nbecause the integrand is a total derivative.\\footnote{Note that $\\Gamma$'s cancel out as \n$\\nabla_\\mu (v^\\mu f^2)=(\\partial_\\mu +\\Gamma_{\\nu\\mu}^\\nu-\\Gamma_\\mu) (v^\\mu f^2)=\\partial_\\mu (v^\\mu f^2)$}\nThe invariance under $\\delta_3$ can also be shown similarly by using partial integrations.\nAs can be seen in \\eq{eq:delta13}, the transformation $\\delta_1$ \nrepresents a diffeomorphism transformation, which transforms $f(x)$ as a scalar-half density, and \n$\\delta_3$ represents a spin-three transformation.\n\nSome comments are in order. Firstly, \n$v^{\\mu\\nu\\rho}$ must be assumed to be symmetric to remove redundancies.\nThe reason is basically the same as that for the symmetry of $\\beta$'s in \\eq{eq:pf3}, which is explained \nin detail in Appendix~\\ref{app:fourth}.\nThe anti-symmetric part of $f_{,\\mu\\nu\\rho}$ in \\eq{eq:delta13} can be rewritten \nin terms of the first derivative of $f$ by using the curvature tensor,\nand therefore the anti-symmetric components of $v^{\\mu\\nu\\rho}$ can be absorbed into $v_\\mu$. \nAnother comment is that one may consider a spin-two transformation with $v^{\\mu\\nu}$\nin a similar manner. However, this is also redundant. \nThe invariance of the norm \\eq{eq:norm} requires that the transformation should be\nin the form, $\\delta_2 f=\\nabla_\\mu\\nabla_\\nu( v^{\\mu\\nu} f) - v^{\\mu\\nu} \\nabla_\\mu\\nabla_\\nu f$, with a minus\nrelative sign in this case. Then, the terms with the second derivative of $f$ cancel, and the transformation \nis equivalent to a diffeomorphism transformation with $v^\\mu=v^{\\mu\\nu}_{,\\nu}$. \nFinally, it is obvious that there exist an infinite tower of spin-odd transformations\nwhich preserve \\eq{eq:norm}. However, \nthe transformations higher than spin-three \nare irrelevant in our treatment up to the fourth order of derivatives.\n \nLet us define the transformations of $\\beta$'s in \\eq{eq:pf3} under $\\delta_{1}$ and $\\delta_{3}$,\nby transferring the transformations of $f$ to $\\beta$'s.\nAs for $\\delta_1$, we obtain\n\\[\n\\begin{split}\n\\delta_1 \\left(Pf^3\\right)&=\\int d^dx \\left[ \n3\\beta f^2 \\delta_1 f \n+\\beta^{\\mu\\nu}\\left( 2 f (\\delta_1 f) f_{,\\mu\\nu} + f^2 (\\delta_1 f)_{,\\mu\\nu} \\right) \\right. \\\\\n&\\hspace{6cm}\n\\left. +\\beta^{\\mu\\nu,\\rho\\sigma}\\left( (\\delta_1f) f_{,\\mu\\nu} f_{,\\rho\\sigma}+2 f f_{,\\mu\\nu}(\\delta_1 f)_{,\\rho\\sigma}\\right)+{\\cal O}(\\nabla^5)\\right]\\\\\n&=\\int d^dx \\left[\n(\\delta_1\\beta) f^3 + (\\delta_1 \\beta^{\\mu\\nu}) f^2 f_{,\\mu\\nu} \n+ (\\delta_1 \\beta^{\\mu\\nu\\rho}) f^2 f_{,\\mu\\nu\\rho}\n+ (\\delta_1 \\beta^{\\mu\\nu,\\rho\\sigma}) f f_{,\\mu\\nu} f_{,\\rho\\sigma}+{\\cal O}(\\nabla^5)\n\\right],\n\\end{split}\n\\label{eq:derdel1}\n\\]\nwhere\n\\[\n\\begin{split}\n&\\delta_1 \\beta=- v^\\mu \\beta_{,\\mu} + \\frac{1}{2} v^\\mu_{,\\mu}\\beta +{\\cal O}(\\nabla^3 ), \\\\\n&\\delta_1 \\beta^{\\mu\\nu} =- v^\\rho \\beta^{\\mu\\nu}_{,\\rho}+\\frac{1}{2}v^\\rho_{,\\rho} \\beta^{\\mu\\nu}+ v^\\mu_{,\\rho} \\beta^{\\rho\\nu}+\nv^\\nu_{,\\rho} \\beta^{\\mu\\rho}+{\\cal O}(\\nabla^3 ), \\\\\n&\\delta_1 \\beta^{\\mu\\nu\\rho}={\\cal O}(\\nabla^2 ),\\\\\n&\\delta_1 \\beta^{\\mu\\nu,\\rho\\sigma} ={\\cal O}(\\nabla ).\n\\end{split}\n\\label{eq:diffeo}\n\\]\nTo derive the result, we have performed some partial integrations to transform the first line of \n\\eq{eq:derdel1} into the form of \\eq{eq:pf3} in the second line.\nWe have assumed $\\beta^{\\mu\\nu\\rho}=0$ initially, which will be discussed later as a gauge condition\nfor the spin-three gauge symmetry.\nThe terms with ${\\cal O}(\\nabla^3)$ in $\\beta$ and $\\beta^{\\mu\\nu}$\ncan also be ignored, because our interest is up to the second derivatives for these fields. \n$\\delta_1 \\beta^{\\mu\\nu\\rho}$ and $\\delta_1 \\beta^{\\mu\\nu,\\rho\\sigma}$ \ncan be ignored, because they are of the fifth order of derivatives in \\eq{eq:derdel1}.\n The result \\eq{eq:diffeo} shows that $\\beta$ transforms as a scalar of negative half-density, \nand $\\beta^{\\mu\\nu}$ as a two-tensor of negative half-density.\nIndeed, \nthis coincides with the weight assignments \\eq{eq:weightbeta},\nwhat is apparently expected from the invariance of \\eq{eq:pf3} under the diffeomorphism.\n\nAs for $\\delta_3$, in a similar manner, we obtain \n\\[\n\\delta_3 (Pf^3)&=\\int d^dx \\left[ 3 \\beta f^2 \\delta_3 f + {\\cal O}(\\nabla^5 ) \\right] \\nonumber \\\\\n&= \\int d^dx \\left[ (\\delta_3 \\beta) f^3 \n+(\\delta_3 \\beta^{\\mu\\nu}) f^2 f_{,\\mu\\nu}+(\\delta_3 \\beta^{\\mu\\nu\\rho}) f^2 f_{,\\mu\\nu\\rho} + \n(\\delta_3 \\beta^{\\mu\\nu,\\rho\\sigma}) f f_{,\\mu\\nu} f_{,\\rho\\sigma}+{\\cal O}(\\nabla^5 )\\right],\n\\]\nwhere \n\\[\n\\begin{split}\n&\\delta_3\\beta = {\\cal O}(\\nabla^3 ),\\\\\n&\\delta_3 \\beta^{\\mu\\nu}=\\frac{9}{2}\\beta v_{,\\rho}^{\\mu\\nu\\rho}, \\\\\n&\\delta_3 \\beta^{\\mu\\nu\\rho} =3 \\beta v^{\\mu\\nu\\rho},\\\\\n&\\delta_3 \\beta^{\\mu\\nu,\\rho\\sigma} = {\\cal O}(\\nabla ).\n\\end{split}\n\\label{eq:delta3}\n\\]\n\nThe equation of motion \\eq{eq:ctmeom} of CTM contains the second term\n$\\sum_\\sigma n_{d \\sigma_{a}} P_{\\sigma_{b}\\sigma_{c}d}$, which \nrepresents the freedom to perform the infinitesimal kinematical transformation along time evolution\nby freely choosing $n_{ab}$ dependent on time. \nWithin our approximation of the continuum limit, the transformations which are relevant\nare $\\delta_1$ and $\\delta_3$. \nThus, we can write \\eq{eq:ctmeom} in a schematic manner as \n\\[\n\\begin{split}\n&\\frac{d}{dt} \\beta =(nPP)+\\delta_1 \\beta, \\\\\n&\\frac{d}{dt} \\beta^{\\mu\\nu}=(nPP)^{\\mu\\nu}+\\frac{9}{2}\\beta v_{,\\rho}^{\\mu\\nu\\rho}+\\delta_1 \\beta^{\\mu\\nu}, \\\\ \n&\\frac{d}{dt} \\beta^{\\mu\\nu\\rho}=(nPP)^{\\mu\\nu\\rho}+3 \\beta v^{\\mu\\nu\\rho},\\\\\n&\\frac{d}{dt} \\beta^{\\mu\\nu,\\rho\\sigma}=(nPP)^{\\mu\\nu,\\rho\\sigma},\\\\\n\\end{split}\n\\label{eq:schemEOM}\n\\]\nwhere we have used \\eq{eq:diffeo} and \\eq{eq:delta3}, and $(nPP),(nPP)^{\\mu\\nu},(nPP)^{\\mu\\nu\\rho},\n(nPP)^{\\mu\\nu,\\rho\\sigma}$ denote \nthe spin-0,2,3,4 components of $\\sum_{\\sigma} n_{d} P_{\\sigma_a de}P_{e\\sigma_b\\sigma_c}$, respectively.\nSince $\\delta_1$ describes the diffeomorphism,\nthe terms with $\\delta_1$ in \\eq{eq:schemEOM} correspond to the freedom to choose the shift-vector in the time-evolution \nin the ADM formalism of general relativity.\nAs for the spin-3 transformation, \nby setting $v^{\\mu\\nu\\rho}=-(nPP)^{\\mu\\nu\\rho}\/3\\beta$ under the assumption $\\beta\\neq0$,\nwe can make a tuning $\\frac{d}{dt} \\beta^{\\mu\\nu\\rho}=0$.\nIn this manner, one can keep the gauge condition $\\beta^{\\mu\\nu\\rho}=0$, which gauges \naway the spin-3 component. As seen in \\eq{eq:schemEOM}, \nby doing this gauge fixing, the time evolution of the spin-2 component will get a contribution by an amount,\n\\[\n-\\frac{3}{2}\\beta \\left(\\frac{(nPP)^{\\mu\\nu\\rho}}{\\beta} \\right)_{,\\rho},\n\\]\nfrom the infinitesimal spin-3 transformation.\nNote that, even if $P$ has no spin-3 component, i.e. $\\beta^{\\mu\\nu\\rho}=0$, $(nPP)^{\\mu\\nu\\rho}$ does not\nvanish in general (This will be seen explicitly later.), \nand the spin-3 infinitesimal transformation must be carried out as above to keep $\\beta^{\\mu\\nu\\rho}=0$\nalong time evolution.\nIn later sections, this and similar procedures will frequently be used to remove the appearance of the spin-3 component.\nIn fact, the spin-3 component can appear not only from the right-hand side of the equation of motion \\eq{eq:ctmeom}, \nbut also from the left-hand side $\\frac{d}{dt}P$, when the background metric has time-dependence as will be discussed in \nSection~\\ref{sec:identify}.\nThis can also be removed by balancing it with the spin-3 transformation on the right-hand side\nin a similar manner as above. \n\n\\section{Equation of motion of CTM in a static background}\n\\label{sec:fixed}\nIn this section, we will study the continuum limit of the equation of motion (EOM) \\eq{eq:ctmeom} of CTM in the case that\nthe background metric $g_{\\mu\\nu}$ is static. Let us take the contractions of both sides of \\eq{eq:ctmeom} \nwith a test function $f$ satisfying $\\dot f=0$.\nThe left-hand side, $\\frac{d}{dt} (Pf^3)$, is simply given by \\eq{eq:pf3} with $\\beta$'s replaced by $\\dot \\beta$'s.\nThe right-hand side is given by \n\\[\n\\delta Pf^3:= \\int d^dx\\, n P[f,P[f,f]],\n\\label{eq:delpf3}\n\\]\nwhere we have left aside the SO($\\mathcal{N}$) rotational part of \\eq{eq:ctmeom} \nfor later discussions, \nhave performed a replacement $n_a\\rightarrow n(x)$, and an overall numerical factor has been \nabsorbed into a constant rescaling of $n(x)$. By rewriting \\eq{eq:delpf3} in the form of \\eq{eq:pf3}, namely,\n\\[\n\\delta P f^3 = \\int d^d x \\left[\n(\\delta\\beta)f^3+(\\delta\\beta^{\\mu\\nu} )f^2 f_{,\\mu\\nu}+(\\delta\\beta^{\\mu\\nu\\rho} )f^2 f_{,\\mu\\nu\\rho}+(\\delta \\beta^{\\mu\\nu,\\rho\\sigma})\nf f_{,\\mu\\nu} f_{,\\rho\\sigma} +{\\cal O}(\\nabla^5)\n\\right],\n\\label{eq:canpf3}\n\\]\none can obtain the explicit expression of the right-hand side of the EOM for the fields $\\beta$'s.\nHere, note that a spin-three component $\\delta \\beta^{\\mu\\nu\\rho}$ of $\\delta P$ may appear\nin general, even though the gauge condition $\\beta^{\\mu\\nu\\rho}=0$ is initially assumed on $P$. \n\nThe symmetric two-tensor field $\\beta^{\\mu\\nu}$ is particularly interesting from \nthe view point of gravity.\nThe lowest order set of fields containing it is given by $\\beta$ and $\\beta^{\\mu\\nu}$.\nTherefore, we want to compute $\\delta \\beta$ and $\\delta \\beta^{\\mu\\nu}$ up to the second order of derivatives,\nwhich would be the minimum for physically interesting dynamics to be expected.\nThe wanted order about the latter field requires that our computations must be correct up to \nthe fourth order in \\eq{eq:canpf3}.\nThis means that $\\delta \\beta^{\\mu\\nu\\rho}$ and $\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ must be \ncomputed up to the first and the zeroth order of derivatives, respectively.\n\nIt would seem that the fourth order terms\\footnote{There exist no third order terms.} \nin $\\delta \\beta$ must also be included for \nthe consistency of the fourth order computations. However, the order of derivatives of \nthe terms relevant in $\\delta \\beta^{\\mu\\nu},\\delta \\beta^{\\mu\\nu\\rho},\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ \nare less than four in our computations up to the fourth order. \nThis means that the fourth derivative terms in $\\delta \\beta$ can not affect \n$\\delta \\beta^{\\mu\\nu},\\delta \\beta^{\\mu\\nu\\rho},\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ even in our later computations,\nwhich more or less mixes $\\delta \\beta,\\delta \\beta^{\\mu\\nu},\\delta \\beta^{\\mu\\nu\\rho},\n\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$. \nTherefore, the fourth derivative terms in $\\delta \\beta$ can be ignored consistently, \nif one is not interested in them: \nour interest \nis up to the second order of derivatives in $\\delta \\beta$. \n\nEven with these upper bounds of our interest on the number of derivatives, the computation of \\eq{eq:delpf3} is \nvery complicated, and we used a Mathematica package ``xTensor\" for the \ntensorial computations.\nThe details of the procedure is explained in Appendix~\\ref{app:explicit}.\nWe have obtained\n\\[\n\\begin{split}\n\\delta \\beta^{\\mu\\nu,\\rho\\sigma}&= 11n \\beta\\beta^{\\mu\\nu,\\rho\\sigma}+4n \\beta \\beta^{\\mu(\\rho,\\sigma)\\nu}\n+4 n \\beta^{\\mu\\nu} \\beta^{\\rho\\sigma}+3 p\\, n \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)}+{\\cal O}(\\nabla^2) ,\\\\\n\\delta \\beta^{\\mu\\nu\\rho}&= -14 n \\beta^{(\\mu\\nu,\\rho)\\sigma} \\beta_{,\\sigma}\n-4 p n \\beta^{(\\mu\\nu}\\beta^{\\rho)\\sigma}_{,\\sigma}+4(1-p) n \\beta^{(\\mu\\nu}_{,\\sigma}\\beta^{\\rho)\\sigma}\n-2n \\beta \\beta^{(\\mu\\nu,\\rho)\\sigma}_{,\\sigma}\\\\\n&\\ \\ \\ +4 (1-p) \\beta^{(\\mu\\nu}\\beta^{\\rho)\\sigma} n_{,\\sigma}-8 \\beta \\beta^{(\\mu\\nu,\\rho)\\sigma} n_{,\\sigma}+{\\cal O}(\\nabla^3), \\\\\n\\delta \\beta^{\\mu\\nu} &=15 n \\beta \\beta^{\\mu\\nu}-2(1+p)n \\beta^{\\mu\\nu}_{,\\rho} \\beta^{\\rho\\sigma}_{,\\sigma}\n+2 (1-p) \\beta^{\\mu\\nu} \\beta^{\\rho\\sigma}_{,\\sigma} n_{,\\rho}-2 p n \\beta^{\\mu\\rho}_{,\\sigma}\\beta^{\\nu\\sigma}_{,\\rho} \\\\\n&\\ \\ \\ \n+4 (1-p) \\beta^{\\rho(\\mu} \\beta^{\\nu)\\sigma}_{,\\rho}n_{,\\sigma}\n+4(1-p)\\beta^{\\rho(\\mu}\\beta^{\\nu)\\sigma}_{,\\sigma} n_{,\\rho}+2 (1-p) \\beta^{\\rho\\sigma} \\beta^{\\mu\\nu}_{,\\rho} n_{,\\sigma}\n-2 p n \\beta^{\\mu\\rho}_{,\\rho}\\beta^{\\nu\\sigma}_{,\\sigma}\\\\\n&\\ \\ \\ -10 n \\beta_{,\\rho} \\beta^{\\mu\\nu,\\rho\\sigma}_{,\\sigma}\n -4 \\beta \\beta^{\\mu\\nu,\\rho\\sigma}_{,\\rho} n_{,\\sigma}-8n\\beta_{,\\rho} \\beta^{\\rho(\\mu,\\nu)\\sigma}_{,\\sigma}\n-8 \\beta \\beta^{\\rho(\\mu,\\nu)\\sigma}_{,\\rho} n_{,\\sigma}-4 \\beta_{,\\rho} \\beta^{\\mu\\nu,\\rho\\sigma}n_{,\\sigma}\n\\\\\n&\\ \\ \\ -8 \\beta^{\\rho(\\mu,\\nu)\\sigma}\\beta_{,\\rho} n_{,\\sigma}\n-2n \\beta_{,\\rho\\sigma} \\beta^{\\mu\\nu,\\rho\\sigma}-4n \\beta_{,\\rho\\sigma} \\beta^{\\mu\\rho,\\nu\\sigma}\n+(1-p) n \\beta^{\\mu\\nu} \\beta^{\\rho\\sigma}_{,\\rho\\sigma}\\\\\n&\\ \\ \\ +4 (1-p) n \\beta^{\\rho(\\mu}\\beta^{\\nu)\\sigma}_{,\\rho\\sigma}\n+(2-p) n \\beta^{\\rho\\sigma} \\beta^{\\mu\\nu}_{,\\rho\\sigma}+n \\beta \\beta^{\\mu\\nu,\\rho\\sigma}_{,\\rho\\sigma}\n-4n \\beta \\beta^{\\rho(\\mu,\\nu)\\sigma}_{,\\rho\\sigma}\\\\\n&\\ \\ \\ +(6-p) \\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}n_{,\\rho\\sigma}\n+(4-2p) \\beta^{\\mu\\rho}\\beta^{\\nu\\rho} n_{,\\rho\\sigma}\n+7\\beta \\beta^{\\mu\\nu,\\rho\\sigma}n_{,\\rho\\sigma}-4 \\beta \\beta^{\\mu\\rho,\\nu\\sigma}n_{,\\rho\\sigma}\\\\\n&\\ \\ \\ \n+n\\left(\\frac{4}{3} \\beta^{\\rho\\sigma}\\beta^{\\delta(\\mu}+2 \\beta \\beta^{\\rho\\sigma,\\delta(\\mu}\\right)R^{\\nu)}{}_{\\rho\\sigma\\delta}\n+{\\cal O}(\\nabla^4),\\\\\n\\delta \\beta&= 9 n \\beta^2 - 4n \\beta_{,\\mu}\\beta^{\\mu\\nu}_{,\\nu}+n \\beta^{\\mu\\nu}\\beta_{,\\mu\\nu}\n+n \\beta\\beta^{\\mu\\nu}_{,\\mu\\nu}+5 \\beta \\beta^{\\mu\\nu}n_{,\\mu\\nu}+{\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:explicitbetas}\n\\]\nwhere $p=\\frac{4}{3}$ must be taken\\footnote{The parameter $p$ becomes a free parameter\nin the case that the term $\\delta \\beta^{\\mu\\nu\\rho\\sigma} f^2 f_{,\\mu\\nu\\rho\\sigma}$ is also allowed in the expression of \n$\\delta P f^3$. \nAs explained in Appendix~\\ref{app:fourth}, this term can be set to zero by using \\eq{eq:ambiguity}\nfor the unique representation. \nBut, if we leave it, $\\delta \\beta^{\\mu\\nu\\rho\\sigma}=(2-3p\/2)n \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)}$, \nand the others will be given by \\eq{eq:explicitbetas} with free $p$.}.\nThe round brackets in the indices represent symmetrization of the indices contained in the pairs of the brackets.\nFor example,\n$\\beta^{\\mu(\\nu,\\rho)\\sigma}=\\frac{1}{2} \\left( \\beta^{\\mu\\nu,\\rho\\sigma}+\\beta^{\\mu\\rho,\\nu\\sigma}\\right)$, and $\\beta^{(\\mu\\nu}\n\\beta^{\\rho\\sigma)}$ represent the total symmetrization.\n\nAs seen in \\eq{eq:explicitbetas}, $\\delta \\beta$'s have complicated expressions with the derivatives of \nboth $\\beta$'s and $n$. \nThe existence of the derivatives of $n$ seems to pose a challenge in comparison with general relativity, \nsince the equation of motion \nof the metric tensor field in the Hamilton-Jacobi formalism of general relativity, \nwritten down in Section~\\ref{sec:conttheory}, contains no derivatives of the lapse function.\nThis absence comes from the fact that the Hamiltonian of the ADM \nformalism $H_{ADM}$ is expressed with no derivatives of the lapse function, and the Poisson brackets\nwith the fields do not produce them either, \nwhere the conjugate momenta to the fields are replaced by some functions of the fields in the Hamilton-Jacobi \nformalism.\n\nThe fundamental reason why we encounter the above difference between CTM and general relativity can intuitively be understood\nby the fact that, in CTM, a space is an emergent object characterized by the tensor $P$. \nAs explained at the beginning of Section~\\ref{sec:repP}, there exists intrinsic fuzziness\nwhich disturbs the exactness of a position specified by the coordinate $x$, where \nthe ambiguity would be in the order of $\\sim\\sqrt{\\beta^{\\mu\\nu}\/\\beta}$ for a dimensional reason.\nThis ambiguity of positions would also make ambiguous the value of a field, here the lapse function, \nas a function of $x$ by an amount in the order of $\\delta n(x)\\sim \\beta^{\\mu\\nu}n_{,\\mu\\nu}\/\\beta$. \nThe real expressions in \\eq{eq:explicitbetas} are much more involved, but this gives an \nintuitive understanding of the reason why the spatial derivatives of the lapse function can appear,\nirrespective of their absence in general relativity. Therefore, to make relations between CTM and general relativity, \nit would be natural to perform some redefinitions of the lapse function and the fields\nby adding some corrections of the spacial derivatives.\nIn fact, we will do so in later sections.\n\nAnother interesting thing to notice in \\eq{eq:explicitbetas}\nis that there appear terms with the background curvature in $\\delta \\beta^{\\mu\\nu}$.\nFor a static background considered in this section, the background curvature appears just as the coefficients \nof the quadratic terms \nof $\\beta$'s, and do not seem to play important roles. \nOn the other hand, as we will discuss in later sections, when the background metric\nbecomes dynamical as a result of the gauge-fixing to a combination of the fields, \nthe curvature terms play essential roles for the consistency of the time evolution. \n\nThe result \\eq{eq:explicitbetas} shows that there appears a spin-three component $\\delta \\beta^{\\mu\\nu\\rho}$,\neven if we assume $\\beta^{\\mu\\nu\\rho}=0$ initially. Therefore, as explained in Section~\\ref{sec:gauge},\nto maintain the gauge condition $\\beta^{\\mu\\nu\\rho}=0$, the spin-three gauge transformation $\\delta_3$ \nin \\eq{eq:delta3} has to be performed simultaneously.\nThis is to bring in the spin-three gauge transformation contained in the SO$({\\cal N})$ \nrotation part of EOM \\eq{eq:ctmeom}.\nBy setting $\\delta \\beta^{\\mu\\nu\\rho}+3 \\beta v^{\\mu\\nu\\rho}=0$,\nwe obtain the EOM for the fields as\n\\[\n\\begin{split}\n&\\dot \\beta=\\delta\\beta, \\\\\n&\\dot \\beta^{\\mu\\nu}=\\delta \\beta^{\\mu\\nu}-\\frac{3}{2} \\beta\\ \\nabla_\\rho \\left(\\frac{1}{\\beta}\\delta\\beta^{\\mu\\nu\\rho}\\right), \\\\\n&\\dot \\beta^{\\mu\\nu,\\rho\\sigma}=\\delta \\beta^{\\mu\\nu,\\rho\\sigma},\n\\end{split}\n\\label{eq:bareeom}\n\\]\nwhere the last term in the second line comes from the second line of \\eq{eq:delta3}, \nthe consequence of maintaining the gauge fixing condition $\\beta^{\\mu\\nu\\rho}=0$.\n\nA physically important consistency check of the EOM \\eq{eq:bareeom}\nis to compute the commutation of two successive \ninfinitesimal time evolutions. This corresponds to the commutation of the Hamiltonian constraints in CTM,\nand, from the first-class nature of the constraint algebra, this should be described by the kinematical \ntransformation ${\\cal J}_{ab}$. \nIn the present context of the continuum limit, the commutation of the time evolutions \nshould be expressed by the gauge transformations discussed in the \npreceding section. Since the spin-three transformation $\\delta_3$ has already been used for the gauge fixing, \none would expect that the commutation should be described by the diffeomorphism transformation $\\delta_1$.\nNote that the lapse function $n(x)$ is a field locally depending on $x$, and the situation is \nthe same as the time evolution in terms of the Hamiltonian constraint in general relativity: \nthe commutation of Hamiltonian constraint being equal to the diffeomorphism is nothing but the assurance of \nthe spacetime covariance of the locally generated time evolution. \nThis is directly connected to the central principle in general relativity, \nand it is highly interesting to check this in the present context.\n\nNow, let us explicitly describe the commutation of two successive infinitesimal time evolutions. \nSuppose we start with a configuration,\n$\\beta,\\beta^{\\mu\\nu},\\beta^{\\mu\\nu,\\rho\\sigma}$.\nAfter an infinitesimal time $\\Delta t$ with lapse $n_1$, the fields evolve to\n\\[\n\\beta_1^i=\\beta^i+\\Delta t \\, \\dot \\beta^i(n_1,\\beta,\\beta^{\\mu\\nu},\\beta^{\\mu\\nu,\\rho\\sigma}),\n\\label{eq:beta1}\n\\]\nwhere $\\beta^i$ represents $\\beta, \\beta^{\\mu\\nu}$, or $\\beta^{\\mu\\nu,\\rho\\sigma}$. \nHere, we have written explicitly the dependence of $\\dot \\beta$'s on $n$ and $\\beta$'s.\nThen, after the second step with lapse $n_2$, \nwe obtain\n\\[\n\\beta^i_{12}=\\beta^i_1+ \\Delta t \\, \\dot \\beta^i(n_2,\\beta_1,\\beta_1^{\\mu\\nu},\\beta_1^{\\mu\\nu,\\rho\\sigma}).\n\\label{eq:beta21}\n\\]\nBy inserting \\eq{eq:beta1} into \\eq{eq:beta21}, expanding in the infinitesimal parameter $\\Delta t$, and \nsubtracting the case that $n_1$ and $n_2$ are interchanged, \none obtains\n\\[\n\\begin{split}\n(\\delta_{n_1} \\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta^i&=\n\\beta^i_{12}-\\beta^i_{21}\\\\\n&=(\\Delta t)^2 \\int d^d x\\, \\dot \\beta^j(x,n_1,\\beta,\\ldots) \n\\frac{\\delta}{\\delta \\beta^j(x)} \\dot \\beta^i(n_2,\\beta,\\ldots) -(n_1 \\leftrightarrow n_2),\n\\end{split}\n\\label{eq:del12m21}\n\\]\nwhere $j$ is summed over, and we have taken the lowest non-trivial order in $\\Delta t$.\n\nWe have used ``xTensor\" to obtain the following explicit result of \\eq{eq:del12m21}: \n\\[\n(\\delta_{n_1} \\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta^i=\\delta_1 \\beta^i+{\\cal O}(\\nabla^4),\n\\label{eq:n1n2v}\n\\]\nwhere we have dropped the infinitesimal parameter $\\Delta t$, \n$\\beta^i=\\beta$ or $\\beta^{\\mu\\nu}$, and $\\delta_1$ is the diffeomorphism transformation \\eq{eq:diffeo} with\n\\[\nv^{\\mu}=12 \\beta \\beta^{\\mu\\nu}\\left( n_1 n_{2,\\nu}-n_2 n_{1,\\nu} \\right).\n\\label{eq:vval}\n\\]\nThe case with $\\beta^i=\\beta^{\\mu\\nu,\\rho\\sigma}$ is not considered, because this requires a higher order computation\nthan the fourth.\nIf we make the identification \n\\[\n\\frac{g^{\\mu\\nu}}{\\sqrt{g}}=\\beta \\beta^{\\mu\\nu},\n\\label{eq:geqbeta2}\n\\]\nthe commutation algebra \\eq{eq:n1n2v} with \\eq{eq:vval} agrees with that of the ADM formalism of general relativity\nexcept for a weight factor $1\/\\sqrt{g}$.\nThe weight factor is necessary for the consistency with the weights of $\\beta$ and $\\beta^{\\mu\\nu}$\nshown in \\eq{eq:weightbeta}. \nThe identification \\eq{eq:geqbeta2} was first discussed in \\cite{Sasakura:2015pxa} \nwith a different argument directly taking the formal continuum limit of the constraint algebra,\nand the extra weight factor has been interpreted consistently. \nIn Section~\\ref{sec:identify}, we will use this relation \\eq{eq:geqbeta2} to gauge-fix the background metric,\nand the issue of weights will be treated in Section~\\ref{sec:delete}.\n\nIt is worth mentioning that there exists a scale invariance in the EOM \\eq{eq:bareeom} with \\eq{eq:explicitbetas}.\nThe transformation is given by\n\\[\n\\begin{split}\nt&\\rightarrow Lt,\\ x^\\mu\\rightarrow L x^\\mu,\\\\\n\\beta&\\rightarrow \\frac{\\beta}{L},\\ \\beta^{\\mu\\nu}\\rightarrow L \\beta^{\\mu\\nu},\\ \n\\beta^{\\mu\\nu\\rho}\\rightarrow L^2 \\beta^{\\mu\\nu\\rho},\\ \\beta^{\\mu\\nu,\\rho\\sigma}\\rightarrow L^3 \\beta^{\\mu\\nu,\\rho\\sigma},\n\\end{split}\n\\label{eq:scaletrans}\n\\]\nwhere $L$ is a real free parameter. \nThe lapse function $n$ and the inverse metric $g^{\\mu\\nu}$ do not transform. \nThe transformation is consistent with the identification \\eq{eq:geqbeta2}.\nThis scale invariance will be respected throughout this paper in the other forms of EOM which will appear in due course.\n\nLastly, we will present a solution to the EOM for the highest component $\\beta^{\\mu\\nu,\\rho\\sigma}$. \nLet us assume the following form of a solution,\n\\[\n\\beta^{\\mu\\nu,\\rho\\sigma}=\\frac{a}{\\beta}\\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}+\\frac{b}{\\beta} \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)},\n\\label{eq:ansatzbe4}\n\\]\nwhere $a,b$ are real numbers. Note that the form is consistent with the scale transformation \\eq{eq:scaletrans}.\nTo check whether this satisfies the EOM,\nit is enough to compute the time-derivative of the right-hand side of \\eq{eq:ansatzbe4} \nup to non-derivative terms, since we consider $\\beta^{\\mu\\nu,\\rho\\sigma}$ up to the zeroth order. \nSince, from \\eq{eq:bareeom},\n\\[\n\\begin{split}\n\\dot \\beta&=9 n \\beta^2 +\\hbox{derivative terms}, \\\\\n\\dot \\beta^{\\mu\\nu}&=15 n \\beta \\beta^{\\mu\\nu}+\\hbox{derivative terms},\n\\end{split}\n\\label{eq:zeroth}\n\\]\none obtains\n\\[\n\\dot \\beta^{\\mu\\nu,\\rho\\sigma}=21 n \\left(a \\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}+b \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)} \\right)\n+\\hbox{derivative terms}\n\\]\nfrom the assumption \\eq{eq:ansatzbe4}. \nOn the other hand, by inserting \\eq{eq:ansatzbe4} into the EOM \n\\eq{eq:bareeom}, one obtains\n\\[\n\\dot \\beta^{\\mu\\nu,\\rho\\sigma}=(9a+4) n \\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}+(6a+15b+4)n\\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)}\n+\\hbox{derivative terms}.\n\\]\nBy equating the two expressions for $\\dot \\beta^{\\mu\\nu,\\rho\\sigma}$, one obtains\n\\[\na=\\frac{1}{3},\\ b=1.\n\\label{eq:valab}\n\\]\nThe existence of the consistent solution implies that one can \nignore the field $\\beta^{\\mu\\nu,\\rho\\sigma}$ assuming that \nit is given by \\eq{eq:ansatzbe4} with \\eq{eq:valab}. \nThis truncation for simplicity will be assumed in the further analysis in later sections.\n\n\\section{Gauge symmetry of the background metric}\n\\label{sec:backmetric}\nIn the former sections, we considered a static background metric, and this is certainly a consistent treatment. \nHowever, there exist two distinct rank-two symmetric tensors, $g^{\\mu\\nu}$ and $\\beta^{\\mu\\nu}$, and \nthis would be physically awkward from the view point of general relativity, which has a unique symmetric rank-two\ntensor called the metric.\nIn fact, as will be explained below, the background metric can be chosen arbitrarily without changing the physical contents\nof CTM: there exists a gauge symmetry which allows one to freely change the background metric\nwith compensation by the fields.\nIn other words, as illustrated in Figure~\\ref{fig:gauge},\na constant surface of $P$ forms a submanifold in the configuration space of $g_{\\mu\\nu}$ and $\\beta$'s,\nand it is extending in the directions that allow arbitrary infinitesimal changes of the background metric. \nSince the motion of $P$ is determined by $P$ itself as in \\eq{eq:ctmeom} (up to the kinematical gauge symmetry), \nthe motion is actually a time-dependent transition from a constant $P$ submanifold to another. \nSuch transitions can be described by various manners of one's own choice, as illustrated \nfor two examples in Figure~\\ref{fig:gauge}.\nTaking a representative point on each constant $P$ submanifold determines a trajectory of time evolution\nin the configuration space of $g_{\\mu\\nu}$ and $\\beta$'s. This \nis a gauge choice, and, in the former section, we take the gauge that the background metric is static, and the motion is solely \ndescribed by $\\beta$'s.\nThis is illustrated as the dotted arrow in the figure. On the other hand, we may take another choice that $g_{\\mu\\nu}$\nand $\\beta$'s are correlated. This is what we will take for the comparison with general relativity, in which the actual gauge fixing\ncondition will be taken as \\eq{eq:geqbeta2}. This is illustrated as a dashed arrow in the figure. \nNote that the two descriptions are physically equivalent: they are connected by a transformation \nof $g_{\\mu\\nu}$ and $\\beta$'s along a constant $P$ submanifold, while $g_{\\mu\\nu}$ and $\\beta$'s \ntake different values.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1]{gauge.png}\n\\caption{A schematic illustration of the time evolution in CTM. The horizontal and vertical axes \nrepresent the configurations of the fields and the background metric, respectively. \nThe solid curves represent the submanifolds of constant $P$.\nA time evolution is a transition from a constant $P$ \nsubmanifold to another in the configuration space.\nThe dotted arrow represents a time evolution in the gauge of a static background metric, while the \ndashed arrow represents an evolution which describes time evolution \nin general relativity by the gauge choice \\eq{eq:geqbeta2}.}\n\\label{fig:gauge}\n\\end{center}\n\\end{figure}\n\nLet us describe the submanifold of constant $P$ by considering the infinitesimal changes of $g_{\\mu\\nu}$ and \n$\\beta$'s which keep $P$. This condition is given by $\\delta (Pf^3)=0$ with \nthe test function unchanged $\\delta f=0$, while $g_{\\mu\\nu}$ and $\\beta$'s are allowed to be changed. \nBy taking the infinitesimal of \\eq{eq:pf3}, it is straightforward to derive\n\\[\n\\begin{split}\n\\delta (Pf^3)=\\int d^d x & \\left[\n\\left(\\delta \\beta -\\frac{1}{2} \\beta^{\\mu\\nu} \\delta\\Gamma_{\\mu,\\nu}\n+\\frac{1}{3} \\left( \\beta^{\\mu\\nu}\\delta\\tilde\\Gamma_{\\mu\\nu}^\\rho\\right)_{,\\rho} \\right)f^3 \\right.\\\\\n&\n+\\left(\n\\delta \\beta^{\\mu\\nu} -\\beta^{\\mu\\nu,\\rho\\sigma} \\delta\\Gamma_{\\rho,\\sigma}\n+\\left(\\beta^{\\mu\\nu,\\rho\\sigma} \\delta \\tilde \\Gamma_{\\rho\\sigma}^\\delta \\right)_{,\\delta}\n\\right)f^2 f_{,\\mu\\nu} \n\\\\\n&+\\left.\n\\left(\\delta \\beta^{\\delta \\rho\\sigma}+\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta \n\\right)f^2 f_{,\\delta\\rho\\sigma}\n+\\delta \\beta^{\\mu\\nu,\\rho\\sigma}f f_{,\\mu\\nu} f_{,\\rho\\sigma}\n\\right]+{\\cal O}(\\nabla^5),\n\\end{split}\n\\label{eq:delpf3met}\n\\]\nwhere \n\\[\n\\tilde \\Gamma_{\\mu\\nu}^\\rho:= \\Gamma_{\\mu\\nu}^\\rho+ \\delta^{\\rho}_{(\\mu}\\Gamma_{\\nu)},\n\\] \nand \n\\[\n\\delta \\Gamma_{\\mu\\nu}^\\rho=\\frac{1}{2} g^{\\rho\\sigma}\\left(\n\\nabla_\\mu \\delta g_{\\nu\\sigma}+\\nabla_\\nu \\delta g_{\\mu\\sigma}-\\nabla_\\sigma \\delta g_{\\mu\\nu}\\right).\n\\]\nHere, we have assumed the gauge condition $\\beta^{\\mu\\nu\\rho}=0$ as an initial input.\nTo derive the result, we have considered the change of the covariant derivatives under the change of $g_{\\mu\\nu}$, \nnamely,\n\\[\n\\delta f_{,\\mu\\nu}=-\\delta\\Gamma_{\\mu\\nu}^\\rho f_{,\\rho}-\\frac{1}{2} \\delta \\Gamma_\\mu f_{,\\nu}\n+\\nabla_{\\mu} \\left(- \\frac{1}{2}\\delta\\Gamma_\\nu f \\right),\n\\]\nand have performed some partial integrations to obtain \\eq{eq:delpf3met}.\nTo further transform it to the form \\eq{eq:pf3}, we have to symmetrize the third derivative of $f$ \nby using the following equation with the Riemann tensor: \n\\[\n\\begin{split}\n\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta f^2 f_{,(\\delta\\rho\\sigma)}\n=&\\frac{1}{3}\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta f^2\n\\left(f_{,\\delta\\rho\\sigma}+f_{,\\rho\\delta\\sigma}+f_{,\\rho\\sigma\\delta}\\right)\\\\\n=&\\frac{1}{3}\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta f^2 \n\\left(3f_{,\\delta\\rho\\sigma}+2 R_{\\rho\\delta\\sigma}{}^\\kappa f_{,\\kappa}\\right).\n\\end{split}\n\\]\nThe last term in the last line can be transformed to the non-derivative terms of $f$ by a partial integration, because \n$f^2 f_{,\\kappa}=\\frac{1}{3} (f^3)_{,\\kappa}$. Then, the condition $\\delta (Pf^3)=0$ implies\n\\[\n\\begin{split}\n&\\delta \\beta=\\frac{1}{2} \\beta^{\\mu\\nu} \\delta\\Gamma_{\\mu,\\nu}\n-\\frac{1}{3} \\left( \\beta^{\\mu\\nu}\\delta\\tilde\\Gamma_{\\mu\\nu}^\\rho\\right)_{,\\rho} \n-\\frac{2}{9}\\left( \\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta R_{\\rho\\delta\\sigma}{}^\\kappa\\right)_\\kappa, \\\\\n&\\delta \\beta^{\\mu\\nu}= \\beta^{\\mu\\nu,\\rho\\sigma} \\delta\\Gamma_{\\rho,\\sigma}\n-\\left(\\beta^{\\mu\\nu,\\rho\\sigma} \\delta \\tilde \\Gamma_{\\rho\\sigma}^\\delta \\right)_{,\\delta} ,\\\\\n&\\delta \\beta^{\\mu\\nu\\rho}=-\\beta^{\\sigma\\delta,(\\mu\\nu}\\delta \\tilde \\Gamma_{\\sigma\\delta}^{\\rho)} ,\\\\\n&\\delta \\beta^{\\mu\\nu,\\rho\\sigma}=0.\n\\end{split}\n\\label{eq:deformbeta}\n\\]\nWe have shown that an arbitrary infinitesimal deformation of the background metric can be absorbed by\nthe infinitesimal change of $\\beta$'s shown in \\eq{eq:deformbeta}.\nThe last term in the first line is actually irrelevant, because it is higher order than our range of interest.\nNote that there appear spin-3 components, which must be absorbed in the way discussed in Section~\\ref{sec:gauge}\nto maintain the gauge condition $\\beta^{\\mu\\nu\\rho}=0$.\n\n\\section{Identifying the background geometry with the fields} \n\\label{sec:identify}\nThe background geometry introduced in the preceding sections is arbitrary. \nIn fact, as discussed in Section~\\ref{sec:backmetric},\nan arbitrary change of the background geometry can be absorbed into the change of the fields $\\beta$'s\nwithout changing $P$. This means that there exists a gauge symmetry which changes the background geometry\nwithout changing the dynamical contents of the system.\nThe most reasonable choice of the background geometry is \\eq{eq:geqbeta2}, which \ndetermines the background geometry in terms of $\\beta$ and $\\beta^{\\mu\\nu}$, and \nmakes it a dynamical entity. \n\nIf we impose the identification \\eq{eq:geqbeta2},\nthe diffeomorphism transformation \\eq{eq:diffeo} derived previously for a static background will also be changed.\nThis is because we have to take into account the simultaneous transformation of $g^{\\mu\\nu}$ keeping the \nrelation \\eq{eq:geqbeta2}.\nIt is easy to see that the corrections are given by the minus of \\eq{eq:deformbeta}. \nTherefore, since \\eq{eq:diffeo} is in the first order of derivatives,\nthe corrections are higher than the second order of derivatives.\nThis is out of our range of interest, and the diffeomorphism transformation \nremains in the form \\eq{eq:diffeo}. \nThis is consistent with the naive expectation that\n$\\beta$ and $\\beta^{\\mu\\nu}$ should still behave as a scalar and a two-tensor with the weight of negative half-density,\neven after the identification of the background metric with the fields.\n\nIt is important to see whether the transformation \\eq{eq:diffeo} and the identification \\eq{eq:geqbeta2} \nreproduce the standard diffeomorphism transformation of $g^{\\mu\\nu}$. Let us define\n\\[\n\\tilde g^{\\mu\\nu}\\equiv \\beta\\beta^{\\mu\\nu}=\\frac{g^{\\mu\\nu}}{\\sqrt{g}},\n\\label{eq:deftildeg}\n\\]\nwhere we wrote \\eq{eq:geqbeta2} as well. From \\eq{eq:diffeo}, one obtains\n\\[\n\\delta_1 \\tilde g^{\\mu\\nu}&=(\\delta_1 \\beta)\\beta^{\\mu\\nu}+\\beta (\\delta_1 \\beta^{\\mu\\nu})\\nonumber \\\\\n&=-v^\\rho \\tilde g^{\\mu\\nu}_{,\\rho}+v^\\rho_{,\\rho} \\tilde g^{\\mu\\nu}+v^\\mu_{,\\rho}\\tilde g^{\\rho\\nu}+\nv^\\nu_{,\\rho}\\tilde g^{\\mu\\rho}+{\\cal O}(\\nabla^3).\n\\label{eq:deltagtilde}\n\\]\nThen, by using the second relation in \\eq{eq:deftildeg}, one obtains\n\\[\n\\begin{split}\n\\delta_1 g^{\\mu\\nu}&=\\sqrt{g}\\left( \\delta_1 \\tilde g^{\\mu\\nu}-\\frac{g^{\\mu\\nu}}{d+2} g_{\\rho\\sigma} \\delta_1 \\tilde g^{\\rho\\sigma}\n\\right) \\\\\n&=\\nabla^\\mu v^\\nu+\\nabla^\\nu v^\\mu+{\\cal O}(\\nabla^3 ),\n\\end{split}\n\\]\nwhere we have put \\eq{eq:deltagtilde}.\nThis indeed agrees with the transformation of the metric under the diffeomorphism in general relativity.\n\nOne consequence of the identification \\eq{eq:geqbeta2} is that the expression of $\\delta \\beta$'s \nin \\eq{eq:explicitbetas} is considerably simplified. This comes from $\\nabla_\\mu (\\beta \\beta^{\\nu\\rho})=0$, \nwhich is because the covariant derivative satisfies $\\nabla_\\mu g^{\\nu\\rho}=0$. \nBy substituting \\eq{eq:explicitbetas} with \\eq{eq:geqbeta2}, \\eq{eq:ansatzbe4} and \\eq{eq:valab}, \nwe obtain\n\\[\n\\begin{split}\n\\tilde \\delta \\beta&=9n \\beta^2+\\frac{6n}{\\beta^2} \\tilde g^{\\mu\\nu} \\beta_{,\\mu}\\beta_{,\\nu}+5 \\tilde g^{\\mu\\nu} n_{,\\mu\\nu}\n+{\\cal O}(\\nabla^4),\\\\\n\\tilde \\delta \\beta^{\\mu\\nu}&= 15 n \\tilde g^{\\mu\\nu} -\\frac{20n}{\\beta^4} \\tilde g^{\\mu\\rho}\\tilde g^{\\nu\\sigma} \\beta_{,\\rho}\n\\beta_{,\\sigma}-\\frac{8}{\\beta^3} \\tilde g ^{\\rho(\\mu} \\tilde g^{\\nu)\\sigma}\\beta_{,\\rho} n_{,\\sigma}\n+\\frac{10 n}{\\beta^3} \\tilde g^{\\rho\\mu}\\tilde g^{\\nu\\sigma} \\beta_{,\\rho\\sigma}\n+\\frac{14}{\\beta^2} \\tilde g^{\\mu\\rho}\\tilde g^{\\nu\\sigma} n_{,\\rho\\sigma}\\\\\n&\\ \\ \\ +\\tilde g^{\\mu\\nu}\\left(\\frac{8n}{\\beta^4} \\tilde g^{\\rho\\sigma} \\beta_{,\\rho}\\beta_{,\\sigma} \n-\\frac{4}{\\beta^3} \\tilde g ^{\\rho\\sigma} \\beta_{,\\rho} n_{,\\sigma}\n+\\frac{n}{\\beta^3} \\tilde g^{\\rho\\sigma} \\beta_{,\\rho\\sigma}\n+\\frac{14}{\\beta^2} \\tilde g^{\\rho\\sigma} n_{,\\rho\\sigma}\n\\right) -\\frac{2n}{\\beta^2} \\tilde g^{\\mu\\rho}\\tilde g^{\\nu\\sigma} R_{\\rho\\sigma}+{\\cal O}(\\nabla^4).\n\\end{split}\n\\label{eq:tildebeta}\n\\]\nHere, note that $\\tilde \\delta \\beta^{\\mu\\nu\\rho}$ and $\\tilde \\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ are not considered anymore:\n$\\tilde \\delta \\beta^{\\mu\\nu\\rho}$ has been gauged away to be included in $\\tilde \\delta \\beta^{\\mu\\nu}$ by\nthe spin-three gauge transformation to keep the gauge condition $\\beta^{\\mu\\nu\\rho}=0$,\nand $\\beta^{\\mu\\nu,\\rho\\sigma}$ is assumed to be the solution \\eq{eq:ansatzbe4} with \\eq{eq:valab}.\n\nAs can be seen in \\eq{eq:tildebeta}, while the right-hand sides of the equation of motion (EOM) have considerably been simplified\nin comparison with \\eq{eq:explicitbetas},\nthe left-hand side, $\\frac{d}{dt} (Pf^3)$, \nmust be modified with some additional terms which come from the evolution of the background metric to \nkeep the relation \\eq{eq:geqbeta2}:\nthe left-hand side can not simply be expressed by the time-derivatives of the fields \n$\\dot \\beta,\\dot \\beta^{\\mu\\nu}$, but must also contain some additional terms coming from the time-derivative of $g_{\\mu\\nu}$\ncontained in the covariant derivatives in \\eq{eq:pf3}.\nThe derivation of the explicit expression of the left-hand side is basically the same as that of \\eq{eq:deformbeta}\nthrough \\eq{eq:delpf3met},\nand the additional terms are just the minus of the right-hand sides of \\eq{eq:deformbeta} with the replacement \n$\\delta \\Gamma\\rightarrow \\dot \\Gamma$. In addition, to keep the gauge condition $\\beta^{\\mu\\nu\\rho}=0$, we\nhave to perform the spin-three gauge transformation to transfer \n$\\delta \\beta^{\\mu\\nu\\rho}$ in \\eq{eq:deformbeta} to \n$\\dot \\beta^{\\mu\\nu}$. Then, we obtain the EOM as\n\\[\n\\begin{split}\n&\\dot \\beta-\\frac{1}{2} \\beta^{\\mu\\nu} \\dot \\Gamma_{\\mu,\\nu}+\\frac{1}{3} \\nabla_{\\sigma}\n\\left(\\beta^{\\mu\\nu} \\dot{\\tilde\\Gamma}_{\\mu\\nu}^\\sigma \\right)=\\tilde \\delta \\beta,\\\\\n&\\dot \\beta^{\\mu\\nu}-\\beta^{\\mu\\nu,\\rho\\sigma}\\dot \\Gamma_{\\rho,\\sigma}+\n\\nabla_\\delta \\left(\\beta^{\\mu\\nu,\\rho\\sigma}\\,\\dot{\\tilde \\Gamma}_{\\rho\\sigma}^\\delta \\right)\n-\\frac{3}{2}\\beta \\nabla_\\rho \\left(\\frac{1}{\\beta} \\beta^{\\sigma\\delta,(\\mu\\nu} \\dot{\\tilde \\Gamma}_{\\sigma\\delta}^{\\rho)}\\right)\n=\\tilde \\delta \\beta^{\\mu\\nu},\n\\end{split}\n\\label{eq:eommiddle}\n\\]\nwhere \\eq{eq:geqbeta2}, \\eq{eq:ansatzbe4} and \\eq{eq:valab} are supposed, and the last term on the left-hand side of the \nlast line comes from the spin-three transformation.\nIt would be worth to remind that the time derivative of the Christoffel symbol can be written covariantly as \n\\[\n\\dot \\Gamma_{\\mu\\nu}^\\rho=\\frac{1}{2} g^{\\rho\\sigma}\\left( \\nabla_\\mu \\dot g_{\\nu\\sigma}+\\nabla_\\nu \\dot g_{\\mu\\sigma}-\n\\nabla_\\sigma \\dot g_{\\mu\\nu}\\right),\n\\label{eq:covgam}\n\\]\nand therefore \\eq{eq:eommiddle} is a covariant expression.\n\nLet us simplify \\eq{eq:eommiddle} further.\nIn the zeroth order of derivatives, the equation of motion (EOM) derived from \\eq{eq:eommiddle} is still given by \\eq{eq:zeroth}, \nsince all the corrections in \\eq{eq:eommiddle} are in the second order. Therefore, by using \\eq{eq:geqbeta2}, \nthe EOM of $g^{\\mu\\nu}$ in the zeroth order is given by\n\\[\n\\dot g^{\\mu\\nu}=\\frac{48 n \\beta}{d+2} g^{\\mu\\nu}+{\\cal O}(\\nabla^2).\n\\label{eq:dotg}\n\\]\nHere, the dimensional dependence appears due to\nthe determinant in \\eq{eq:geqbeta2}, while the EOM so far has been independent of it. Then, by\nputting \\eq{eq:dotg} into \\eq{eq:covgam}, one obtains\n\\[\n\\dot\\Gamma_{\\mu\\nu}^\\rho=-\\frac{48}{d+2} \n\\left(\\delta^\\rho_{(\\mu}\\nabla_{\\nu)}(n \\beta)-\\frac{1}{2}g_{\\mu\\nu} \\nabla^\\rho(n\\beta)\\right)+{\\cal O}(\\nabla^3).\n\\label{eq:dotgamma}\n\\]\nThe overall minus sign is from the fact $\\dot g_{\\mu\\nu}=-g_{\\mu\\rho}g_{\\nu\\sigma} \\dot g^{\\rho\\sigma}$.\nThis order of $\\dot \\Gamma$ is enough for our second order computation of \nthe correction terms on the left-hand side of \\eq{eq:eommiddle}. \nBy putting \\eq{eq:dotgamma} into \\eq{eq:eommiddle}, we finally obtain\n\\[\n\\begin{split}\n&\\dot \\beta\n=9 n \\beta^2+\\frac{\\tilde g^{\\mu\\nu}}{d+2}\\left(\\frac{2(3d-2)n}{\\beta^2} \\beta_{,\\mu} \\beta_{,\\nu}\n-\\frac{8(3d-2)}{\\beta} \\beta_{,\\mu}n_{,\\nu}\n-\\frac{4(3d-4)n}{\\beta} \\beta_{,\\mu\\nu}\n-(7d-26) n_{,\\mu\\nu} \\right) \\\\\n&\\ \\ \\ \\ \\ \\ +{\\cal O}(\\nabla^4), \\\\\n&\\dot \\beta^{\\mu\\nu}=15 n \\tilde g^{\\mu\\nu} -\\frac{2n \\tilde g^{\\mu\\rho} \\tilde g^{\\nu\\sigma}}{\\beta^2}R_{\\rho\\sigma} \\\\\n&\\ \\ \\ \\ \\ \\ +\\frac{\\tilde g^{\\rho(\\mu}\\tilde g^{\\nu)\\sigma}}{(d+2)\\beta^2}\\left( -\\frac{4(d-14)n}{\\beta^2}\\beta_{,\\rho}\\beta_{,\\sigma}\n-\\frac{24(d-2)}{\\beta} \\beta_{,\\rho}n_{,\\sigma} \n-\\frac{2(3d-2)n}{\\beta} \\beta_{,\\rho\\sigma}\n-2(d-6) n_{,\\rho\\sigma}\n \\right)\n\\\\\n&\\ \\ \\ \\ \\ \\ +\\frac{\\tilde g^{\\mu\\nu}\\tilde g^{\\rho\\sigma}}{(d+2)\\beta^2}\\left( \\frac{8(5d+8)n}{\\beta^2} \\beta_{,\\rho}\\beta_{,\\sigma} \n-\\frac{4(5d-6)}{\\beta}\\beta_{,\\rho}n_{,\\sigma}\n-\\frac{(23d+6)n}{\\beta} \\beta_{,\\rho\\sigma}\n-10(d-2) n_{,\\rho\\sigma} \n\\right) \\\\\n&\\ \\ \\ \\ \\ \\ +{\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:eombetawithg}\n\\]\nwhere \\eq{eq:geqbeta2} is supposed. This is the version of EOM with a dynamical background metric\ndetermined by \\eq{eq:geqbeta2}.\n\nA physically meaningful consistency check of EOM \\eq{eq:eombetawithg} is\ngiven by computing the commutation of two successive infinitesimal time evolutions, as \nthe algebraic structure \\eq{eq:n1n2v} with \\eq{eq:vval} has been obtained for the static background case. \nThe existence of the gauge symmetry discussed in Section \\ref{sec:backmetric}, which \nallows us to freely change the background metric, \nassures the covariance of the time evolution for the evolving background case, too. \nTherefore, we should obtain the same algebraic structure as the static background case. \nHowever, \nthe actual computation for the consistency check is much more complicated and non-trivial than the fixed background case. \nIn the second step of the successive infinitesimal time evolutions, one has to compute the time derivative of \nthe right-hand side of \n\\eq{eq:eombetawithg}.\\footnote{\\eq{eq:del12m21} corresponds to acting the time-derivative on $\\dot \\beta$'s.}\nIn the computation, the main difference from the static background case \nis that we have to take into account the time derivative of the metric as well, which\naffects not only the metric itself but also the covariant derivatives and the curvature tensor.\nTherefore, \nwhile the number of terms in \\eq{eq:eombetawithg} has substantially been reduced from \\eq{eq:explicitbetas}\nby the identification \\eq{eq:geqbeta2}, there appear a number of new terms in the \nsecond step, which someway set back the reduction.\nOne can compute these extra contributions in a similar manner as was done in Section~\\ref{sec:backmetric}.\nFor instance, as for $\\beta$, \n\\[\n\\begin{split}\n\\frac{d}{dt} \\beta_{,\\mu}&={\\dot \\beta}_{,\\mu}+\\frac{1}{2} \\dot \\Gamma_\\mu \\beta,\\\\\n\\frac{d}{dt} \\beta_{,\\mu\\nu}&={\\dot \\beta}_{,\\mu\\nu}-\\dot \\Gamma_{\\mu\\nu}^\\rho \\beta_{,\\rho}+\\frac{1}{2} \n\\dot \\Gamma_\\mu \\beta_{,\\nu}+\\frac{1}{2} \\nabla_\\mu(\\dot \\Gamma_\\nu \\beta),\n\\end{split}\n\\label{eq:dotsecond}\n\\]\nwhere the terms with $\\dot \\Gamma_\\mu$ are due to the weight of $\\beta$'s in \\eq{eq:weightbeta}. Here, \n$\\dot \\Gamma_{\\mu\\nu}^\\rho$ is explicitly given by \\eq{eq:dotgamma}. \nAs for the curvature tensor, since the curvature is in the second order by itself, it is enough to consider the non-derivative part \n\\eq{eq:dotg} of $\\dot g_{\\mu\\nu}$, and we obtain\\footnote{The computation is simplified by noticing that\nthe non-derivative part of \\eq{eq:dotg} is just a conformal transformation.} \n\\[\n\\dot R_{\\mu\\nu}=\\frac{24}{d+2}\\left((d-2)\\nabla_\\mu \\nabla_\\nu (n\\beta)+g_{\\mu\\nu} \\nabla^2(n\\beta)\\right)+{\\cal O}(\\nabla^4).\n\\label{eq:dotR}\n\\]\nBy using these expressions, one can compute the commutation of infinitesimal time evolutions, and obtain \n\\[\n\\begin{split}\n&(\\delta_{n_1}\\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta\n=-\\tilde g^{\\mu\\nu}v_\\mu \\beta_{,\\nu}+\\frac{1}{2} \\tilde g^{\\mu\\nu}v_{\\mu,\\nu}\\beta +{\\cal O}(\\nabla^4), \\\\\n&(\\delta_{n_1}\\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta^{\\mu\\nu}\n=\\frac{1}{\\beta^2}\\left(2\\tilde g^{\\rho(\\mu}\\tilde g^{\\nu)\\sigma}v_{\\rho,\\sigma}\n \\beta+\\tilde g^{\\mu\\nu}\\tilde g^{\\rho\\sigma}\\left(\\frac{1}{2} v_{\\rho,\\sigma} \\beta+v_\\rho \\beta_{,\\sigma}\\right)\\right)\n +{\\cal O}(\\nabla^4),\n \\label{eq:hhbeta}\n\\end{split}\n\\]\nwhere \n\\[\nv_\\mu=12 (n_1 n_{2,\\mu}-n_2 n_{1,\\mu}).\n\\]\nOne can easily check that the right-hand sides are the same as \\eq{eq:n1n2v} with \\eq{eq:vval}, when \\eq{eq:geqbeta2} \nis taken into account. \nThus, the right-hand sides of (\\ref{eq:hhbeta}) represent the diffeomorphism transformations, and \nthe consistency of the time evolution in the case of the evolving background with \\eq{eq:geqbeta2} \nhas also been established.\n\n\\section{Deletion of the weights}\n\\label{sec:delete}\nSo far, the field $\\beta$ and the lapse function $n$ have the weights of negative and positive half-densities, respectively.\nWhile these are the natural weights in the framework of CTM, \nscalars with such weights are not standard in general relativity.\nTherefore, we want to transform them into simple scalars with no weights. \nAt first glance, this seems to be a trivial task by doing the replacement,\n$\\beta\\rightarrow g^{-\\frac14} \\beta$ and\n$n\\rightarrow g^\\frac{1}{4} n$, in the equation of motion (EOM) \\eq{eq:eombetawithg}.\nHowever, while the former is obvious, there is a subtle issue in the latter replacement. \n\nWhen we have shown the algebraic relation between the commutation of two infinitesimal time evolutions and \nthe diffeomorphism in the preceding sections,\nit is implicitly assumed that $n_2$ does not change after the first infinitesimal time evolution with $n_1$, and vice versa.\nNamely, the algebraic relation has been shown in the situation that the lapse functions with the weight\nof half-density do not change after the infinitesimal time evolutions.\nOn the other hand, if we do the replacement $n\\rightarrow g^\\frac{1}{4} n$, and assume that the new lapse functions \nwith no weights do not change after a first infinitesimal time evolution, the situation becomes in fact different by the \nevolution of the weight $g^\\frac{1}{4}$ from the original one. This means that the commutation of two infinitesimal time evolutions \nis a sum of a diffeomorphism and an infinitesimal time evolution\nwith the following lapse function:\n\\[\nn_{12}=-\\frac{1}{4} g_{\\mu\\nu}\\dot g^{\\mu\\nu}(n_1)n_2+\\frac{1}{4} g_{\\mu\\nu}\\dot g^{\\mu\\nu}(n_2)n_1.\n\\label{eq:additional}\n\\]\nHere, we have explicitly written the lapse function dependence of $\\dot g^{\\mu\\nu}$, while it depends also on $\\beta$ \nand $g^{\\mu\\nu}$.\nOf course, the appearance of an additional time evolution\nis not a breakdown of the framework, because the algebraic closure of the diffeomorphism and \nthe infinitesimal time evolution anyway holds. But, this deformed algebraic structure is inconvenient, \nif we want to compare CTM with the ADM formalism of general relativity.\n\nTo fix this issue, let us consider the following reparameterization of the lapse function,\n\\[\nn\\rightarrow \\tilde n =n+h(\\beta,g^{\\mu\\nu},n),\n\\label{eq:repn}\n\\] \nwhere $h$ is a scalar function linear in $n$, \nand is assumed to be in the order of second derivatives.\\footnote{A direct way to compensate, \nsuch as $n\\rightarrow g^{-\\frac{1}{4}} n$, cannot be taken, because $n$ is supposed to be a scalar with no weights, and its\nweight should not be changed.}\nThe reason for $h$ to be taken in the second order is that we want to keep the result in the main order,\nnamely, the part expressed by the diffeomorphism. \nThen, the condition to compensate \\eq{eq:additional} is given by\n\\[\n\\begin{split}\n&\\int dx \\left[ \\dot\\beta(x,n_1) \\frac{\\delta}{\\delta \\beta(x)}\n+\\dot g^{\\mu\\nu}(x,n_1)\\frac{\\delta}{\\delta g^{\\mu\\nu}(x)}\\right]h(\\beta,g^{\\mu\\nu},n_2)\n-\\frac{1}{4} g_{\\mu\\nu} \\dot g^{\\mu\\nu}(n_1)n_2 -(n_1 \\leftrightarrow n_2)\\\\\n&\\hspace{13cm}={\\cal O}(\\nabla^4).\n\\end{split}\n\\label{eq:hcond}\n\\]\n\nBefore discussing the solution for $h$ to \\eq{eq:hcond}, \nlet us first discuss the explicit expressions of the EOM\nin the case with no weights.\nSo, let us leave aside the replacement $n\\rightarrow \\tilde n$ for the moment.\nAfter the rescaling by the weight factors, i.e., $\\beta\\rightarrow g^{-\\frac14} \\beta$ and\n$n\\rightarrow g^\\frac{1}{4} n$, the EOM has the form, \n\\[\n\\begin{split}\ng^\\frac{1}{4} \\frac{d}{dt} \\left(g^{-\\frac{1}{4}} \\beta\\right)&=K(\\beta,g^{\\mu\\nu},n), \\\\\ng^\\frac{1}{4}\\frac{d}{dt} \\left( g^{-\\frac{1}{4}}\\beta^{\\mu\\nu}\\right) &=K^{\\mu\\nu}(\\beta,g^{\\mu\\nu},n),\n\\end{split}\n\\label{eq:gK}\n\\]\nwhere $K$ and $K^{\\mu\\nu}$ are given by the right-hand sides of \\eq{eq:eombetawithg} with\nthe formal replacement $\\tilde g^{\\mu\\nu} \\rightarrow g^{\\mu\\nu}$.\nThe left-hand sides of \\eq{eq:gK} can be written in the way,\n\\[\n\\left(\n\\begin{array}{cc}\n1 & \\frac{1}{4} \\beta g_{\\rho\\sigma} \\\\\n-\\frac{1}{\\beta^2}g^{\\mu\\nu} & \\frac{1}{\\beta} I_{\\rho\\sigma}^{\\mu\\nu} +\\frac1{4\\beta} g^{\\mu\\nu}g_{\\rho\\sigma} \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\dot \\beta \\\\\n\\dot g^{\\rho\\sigma}\n\\end{array}\n\\right),\n\\label{eq:dotbdotg}\n\\]\nwhere $I^{\\mu\\nu}_{\\rho\\sigma}= \\delta^{(\\mu}_\\rho \\delta^{\\nu)}_\\sigma$, and \n\\eq{eq:geqbeta2} has been used.\nIt is easy to find the inverse of the matrix in \\eq{eq:dotbdotg}, and we obtain\n\\[\n\\left(\n\\begin{array}{c}\n\\dot \\beta \\\\\n\\dot g^{\\mu\\nu}\n\\end{array}\n\\right)=\n\\left(\n\\begin{array}{cc}\nc_1 & c_2 \\beta^2 g_{\\rho\\sigma} \\\\\n\\frac{c_3}{\\beta}g^{\\mu\\nu} & \\beta I_{\\rho\\sigma}^{\\mu\\nu} +c_4 \\beta g^{\\mu\\nu}g_{\\rho\\sigma} \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nK(\\beta,g^{\\mu\\nu}, n) \\\\\nK^{\\rho\\sigma}(\\beta,g^{\\mu\\nu},n)\n\\end{array}\n\\right),\n\\label{eq:eomscalar}\n\\]\nwhere\n\\[\nc_1=\\frac{d+4}{2(d+2)},\\ c_2=-\\frac{1}{2(d+2)},\\ c_3=\\frac{2}{d+2},\\ c_4=-\\frac{1}{d+2}.\n\\] \n\nNow let us discuss the replacement $n\\rightarrow \\tilde n$. \nTo solve the condition \\eq{eq:hcond} for $h$, let us assume the following form,\n\\[\nh(\\beta,g^{\\mu\\nu},n)=\\frac{g^{\\mu\\nu}}{\\beta^2}\\left(\nz_1 \\frac{n \\beta_{,\\mu}\\beta_{,\\nu}}{\\beta^2}+z_2 \\frac{\\beta_{,\\mu}n_{,\\nu}}{\\beta}\n+z_3 \\frac{n\\beta_{,\\mu\\nu}}{\\beta}+z_4 n_{,\\mu\\nu}\n\\right),\n\\label{eq:hass}\n\\]\nwhere $z_i$ are parameters. This form is chosen so that the reparameterization \\eq{eq:repn} preserves the\noriginal form of the EOM.\nBy substituting $\\dot g^{\\mu\\nu}$ in \\eq{eq:hcond} with \\eq{eq:eomscalar}, \nwe find that \\eq{eq:hcond} can be solved by\n\\[\n\\begin{split}\n&(d-6) z_3 + 2 (2 + d) z_4 = \\frac{-12 - 44 d + 17 d^2}{6(d+2)},\\\\\n&2 ( d-6) z_1 + (10 + d) z_2 + \n 4 (3 d - 10) z_3 + 8(2 -d) z_4 =\\frac{2 (-12 - 4 d + 11 d^2)}{3 (d + 2)}.\n \\end{split}\n \\label{eq:condz}\n \\]\nThe solutions form a two-parameter family, and any of them can be used for the purpose.\n\nThe final form of the EOM with no weights of the field and the lapse function \nis obtained by \ndoing the replacement $n\\rightarrow \\tilde n$ in \\eq{eq:eomscalar}.\nBecause our concern is up to the second order, the replacement \nis effective only in the zeroth order terms in \\eq{eq:eomscalar}. \nBy explicitly computing \\eq{eq:eomscalar}, we obtain\n\\[\n\\begin{split}\n\\dot \\beta&=-\\frac{3(d-6)}{d+2} \\beta^2 \\left( n+h(\\beta,g^{\\mu\\nu},n)\\right)\n+\\frac{1}{d+2} nR-\\frac{17d^2+20d+36}{(d+2)^2 \\beta^2} n \\beta_{,\\mu}\\beta^{,\\mu}\\\\\n&\\ \\ \\ \\ -\\frac{2(d^2+20d-4)}{(d+2)^2 \\beta} \\beta_{,\\mu} n^{,\\mu} \n+\\frac{11d^2-20 d+60}{2(d+2)^2 \\beta}n \\beta_{,\\mu}^{,\\mu}+\\frac{3 d^2-20 d+92}{2(d+2)^2}n_{,\\mu}^{,\\mu}\n+{\\cal O}(\\nabla^4), \\\\\n\\dot g^{\\mu\\nu}&=\\frac{48 \\beta}{d+2} g^{\\mu\\nu} \\left(n+h(\\beta,g^{\\rho\\sigma},n)\\right)-\\frac{2}{\\beta}nR^{\\mu\\nu}\n+ \\frac{2 }{(d+2)\\beta}nR g^{\\mu\\nu}\\\\\n&\\ \\ \\ \n-\\frac{4(d-14)}{(d+2) \\beta^3} n \\beta^{,\\mu} \\beta^{,\\nu}-\\frac{24(d-2)}{(d+2)\\beta^2} n^{,(\\mu}\\beta^{,\\nu)}\n-\\frac{2(3d-2)}{(d+2)\\beta^2} n \\beta^{,\\mu\\nu}\n-\\frac{2(d-6)}{(d+2)\\beta} n^{,\\mu\\nu}\\\\\n&\\ \\ \\ \n+g^{\\mu\\nu}\\left(\\frac{32(3d+2)}{(d+2)^2\\beta^3}n \\beta_{,\\rho}\\beta^{,\\rho}\n-\\frac{32(2d-1)}{(d+2)^2\\beta^2}n_{,\\rho}\\beta^{,\\rho} -\\frac{16(4d-1)}{(d+2)^2\\beta^2}n\\beta_{,\\rho}^{,\\rho}\n-\\frac{16(2d-5)}{(d+2)^2 \\beta} n_{,\\rho}^{,\\rho} \\right) \\\\\n& \\ \\ \\ +{\\cal O}(\\nabla^4).\n\\end{split}\n\\label{eq:eomfinal}\n\\]\n\nFor a consistency check of this result, one can compute the commutation of two infinitesimal time evolutions,\nas done before.\nThe basic strategy is the same. In the second step of the infinitesimal time evolution, one has to take the \ntime derivative of the right-hand sides of \\eq{eq:eomfinal}. Not only the metric itself, but we also take into account \nthe time derivative of the second covariant derivatives\\footnote{The difference from the previous case \\eq{eq:dotsecond}\nis the absence of weights, namely, $\\dot \\Gamma_\\mu$ is absent.\nBecause of this, the first covariant derivatives have no time-dependencies.} and the curvature.\nSince our concern is up to the second order, the time-derivative of the Christoffel symbol and the curvature can be evaluated by the zeroth order of $\\dot g^{\\mu\\nu}$, as given in \\eq{eq:dotgamma} and \\eq{eq:dotR}, respectively.\nThen, we obtain \n\\[\n\\begin{split}\n&(\\delta_{n_1}\\delta_{n_2} -\\delta_{n_2}\\delta_{n_1})\\beta=-v^\\mu \\beta_{,\\mu}+{\\cal O}(\\nabla^4), \\\\\n&(\\delta_{n_1}\\delta_{n_2} -\\delta_{n_2}\\delta_{n_1})g^{\\mu\\nu}=2v^{(\\mu,\\nu)}+{\\cal O}(\\nabla^4),\n\\end{split}\n\\] \nwhere \n\\[\nv^\\mu=12\\left( n_1 n_2^{,\\mu}-n_2 n_1^{,\\mu}\\right).\n\\label{eq:metricv}\n\\]\nThe right-hand sides certainly agree with the standard diffeomorphism in general relativity for a scalar and a metric.\nIt should be stressed that this result can be obtained, only when the correction $h(\\beta,g^{\\mu\\nu},n)$ with the parameters \nsatisfying \\eq{eq:condz} is included in the equation of motion as in \\eq{eq:eomfinal}.\n\nNow, let us briefly discuss the inclusion of the terms corresponding to those parameterized \nby a shift vector in the Hamiltonian of the ADM formalism of general relativity. \nThe last term of the EOM of CTM in \\eq{eq:ctmeom} represents an arbitrary infinitesimal \nSO$({\\cal N})$ transformation.\nAs discussed in Section~\\ref{sec:gauge}, it contains the diffeomorphism and the spin-three gauge transformation \nin the present context. However, since the latter is used to maintain the gauge-fixing condition $\\beta^{\\mu\\nu\\rho}=0$, \nonly the diffeomorphism can be set arbitrary.\nThe diffeomorphism transformation \\eq{eq:diffeo} and the identification \\eq{eq:geqbeta2} imply that \n$\\beta$ and $g^{\\mu\\nu}$ are transformed in the standard way of general relativity. \nThus, implementing the following replacement in \\eq{eq:eomfinal},\n\\[\n\\begin{split}\n\\dot \\beta&\\rightarrow \\dot\\beta+n^\\mu \\beta_{,\\mu}, \\\\\n\\dot g^{\\mu\\nu}&\\rightarrow \\dot g^{\\mu\\nu}-2n^{(\\mu,\\nu)},\n\\end{split}\n\\]\nwhere $n^\\mu$ is a newly introduced shift vector, \none obtains the EOM with the shift vector.\n\n\\section{Deletion of the derivatives of the lapse function}\n\\label{sec:reparametrization}\nThe equation of motion (EOM) \\eq{eq:eomfinal} contains some terms with the derivatives of $n$.\nAs discussed below \\eq{eq:explicitbetas},\nthis is an obstacle for a general relativistic interpretation of the EOM of CTM.\nIn this section, we will show that, by redefining the fields $\\beta,g^{\\mu\\nu}$ with some \nderivative corrections, one can actually delete all the terms with the derivatives of $n$ from the EOM. \n\nThe reparameterization of the fields we consider is given by adding some correction terms with second order of derivatives:\n\\[\n\\begin{split}\n\\beta&\\rightarrow \\beta+x_1\\, \\frac{\\beta_{,\\mu}\\beta^{,\\mu}}{\\beta^3}+x_2 \\, \\frac{\\beta_{,\\mu}^{,\\mu}}{\\beta^2}\n+x_7\\,\\frac{R}{\\beta}, \\\\\ng^{\\mu\\nu}&\\rightarrow g^{\\mu\\nu} +x_3\\, \\frac{\\beta^{,\\mu}\\beta^{,\\nu}}{\\beta^4}+x_4\\, \\frac{\\beta^{,\\mu\\nu}}{\\beta^3}\n+x_5\\, \\frac{g^{\\mu\\nu} \\beta^{,\\rho}\\beta_{,\\rho}}{\\beta^4}+x_6\\, \\frac{g^{\\mu\\nu}\\beta^{,\\rho}_{,\\rho}}{\\beta^3}\n+x_8\\, \\frac{R^{\\mu\\nu}}{\\beta^2}+x_9\\,\\frac{g^{\\mu\\nu} R}{\\beta^2},\n\\label{eq:replacement}\n\\end{split}\n\\]\nwhere $x_i$'s are parameters. Note that, since the reparameterization is covariant, the algebraic consistency \nbetween the commutation of time evolutions and the diffeomorphism obtained so far should be \nunaltered\\footnote{For sure, we have checked it through explicit computations.}.\n\nThere exist two kinds of effects from this reparameterization.\nThe first one is on the right-hand side of \\eq{eq:eomfinal}. Since the corrections are in the second order of derivatives, \nthe reparameterization is effective only on the zeroth order term, \nand causes some shifts of the coefficients of the non-derivative terms of $n$.\nOn the other hand, the reparameterization affects the left-hand side more importantly for our purpose.\n$\\dot \\beta$ will be replaced by\n\\[\n\\begin{split}\n\\dot \\beta \\rightarrow & \\dot \\beta+x_1 \\left( - \\frac{3\\dot \\beta \\beta_{,\\mu}\\beta^{,\\mu}}{\\beta^4}\n+\\frac{2\\dot\\beta_{,\\mu}\\beta^{,\\mu}+\\dot g^{\\mu\\nu}\\beta_{,\\mu}\\beta_{,\\nu}}{\\beta^3}\\right)\\\\\n&\\ \\ \\ +\nx_2 \\left( -\\frac{2 \\dot \\beta \\beta_{,\\mu}^{,\\mu}}{\\beta^3}+\\frac{\\dot \\beta_{,\\mu}^{,\\mu}+\\dot g^{\\mu\\nu}\\beta_{,\\mu\\nu}\n-g^{\\mu\\nu}\\dot\\Gamma_{\\mu\\nu}^\\rho \\beta_{,\\rho}}{\\beta^2} \\right)\n+x_7\\, \\left( -\\frac{R\\dot \\beta}{\\beta^2}+\\frac{\\dot g^{\\mu\\nu}R_{\\mu\\nu}+g^{\\mu\\nu}\\dot R_{\\mu\\nu}}{\\beta} \\right).\n\\end{split}\n\\label{eq:changedotbeta}\n\\] \nTo evaluate the correction terms in \\eq{eq:changedotbeta} up to the second order of derivatives, \nwe can put the zeroth order expressions of the time-derivative of the fields, \ni.e., the first equation of \\eq{eq:zeroth}, \\eq{eq:dotg}, \\eq{eq:dotgamma}, and \\eq{eq:dotR}, \ninto them. The things are similar for the correction terms in the replacement of $g^{\\mu\\nu}$ in \\eq{eq:replacement}. \nThen, because the zeroth order expressions contain $n$, there emerge a number of terms which contain\nthe derivatives of $n$. In fact, we can delete all the derivative terms of $n$ in the EOM \nby appropriately choosing the $x_i$'s.\nThe condition for the deletion is expressed by six equations, which are explicitly given in Appendix~\\ref{app:deleten}. \nSolving the equations for $x_1,\\cdots, x_6$, and putting the solutions into the EOM, \nwe obtain\n\\[\n\\begin{split}\n\\frac{1}{n}\\dot \\beta&=\n-\\frac{3 (-6 + d) \\beta^2 }{2 + d}+\\frac{(1 + (6 - 9 d) x_7) R}{2 + d}\n-\\frac{16 (-1 + d) (-1 + (-6 + 9 d) x_7) \n\\beta_{,\\mu}^{,\\mu}}{(-6 + d) (2 + d) \\beta}\\\\\n&+\\frac{2 (-8 (11 + 84 x_7) + d^3 (-1 + 360 x_7) + 4 d (43 + 480 x_7) - \n 2 d^2 (19 + 804 x_7)) \\beta_{,\\mu}\\beta^{,\\mu}}{(-6 + d)^2 (2 + d) \\beta^2}\\\\\n&+{\\cal O}(\\nabla^4),\\\\\n\\frac{1}{n} \\dot g^{\\mu\\nu}&=\\frac{48 \\beta g^{\\mu\\nu} }{2 + d}\n+\\frac{2 (1 + 24 x_7 - 3 (2 + d) x_9) g^{\\mu\\nu} R}{(2 + d) \\beta}\n-\\frac{2 (1 + 3 x_8) R^{\\mu\\nu}}{\\beta}\\\\\n&+\\frac{A_1 g^{\\mu\\nu} \\beta_{,\\rho}\\beta^{,\\rho}}{(-6 + d)^2 (2 + d) \\beta^3}\n+\\frac{16 (48 + 84 x_8 + 3 d^2 x_8 - 8 d (1 + 6 x_8)) \\beta^{,\\mu}\\beta^{,\\nu}}{(-6 + d)^2 \\beta^3}\\\\\n&-\\frac{16 (4 + 48 x_7 + 6 x_8 - 12 x_9 + 6 d^2 x_9 + \n d (-1 - 48 x_7 + 3 x_8 + 6 x_9)) g^{\\mu\\nu} \n\\beta_{,\\rho}^{,\\rho}}{(-6 + d) (2 + d) \\beta^2}\\\\\n&-\\frac{8 (-6 + d - 12 x_8 + 6 d x_8) \\beta^{,\\mu\\nu} }{(-6 + d) \\beta^2}+{\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:eomCTM}\n\\]\nwhere \n\\[\n\\begin{split}\nA_1&=16 (2 d (13 + 456 x_7 - 6 x_8 - 72 x_9) - 4 (20 + 168 x_7 + 33 x_8 - 42 x_9) + 30 d^3 x_9\\\\\n&\\hspace{3cm}- 3 d^2 (1 + 80 x_7 - 9 x_8 + 18 x_9)).\n\\end{split}\n\\label{eq:A1}\n\\]\nInterestingly, the EOM does not depend on the two-dimensional ambiguity of the solutions of $z_i$'s to \\eq{eq:condz},\nand is parameterized solely by $x_{7,8,9}$.\nIn the following section, we will identify \\eq{eq:eomCTM} with \nthe EOM of general relativity coupled with a scalar field \nbased on the Hamilton-Jacobi approach. \n\nIn the EOM \\eq{eq:eomCTM}, one can see that \nthe scale transformation \\eq{eq:scaletrans} is realized as \n\\[\n\\begin{split}\nt&\\rightarrow L t,\\ x^{\\mu}\\rightarrow L x^\\mu, \\\\\n\\beta&\\rightarrow \\frac{\\beta}{L},\n\\end{split}\n\\label{eq:scaletranseom}\n\\]\nwhile $n,g^{\\mu\\nu}$ are invariant.\n\n\\section{Hamilton-Jacobi equation of general relativity coupled with a scalar field}\n\\label{sec:conttheory}\nIn this section, starting with an action of general relativity coupled with a scalar field, \nand employing the Hamilton-Jacobi approach, \nwe identify the equations of motion (EOM) of this gravitational system with the EOM \\eq{eq:eomCTM} of CTM.\n\nIt is an easy task to guess a possible form of the action for the purpose:\n\\[\nS = \\int_{\\mathcal{M}} d^{d+1}x \\sqrt{- G} \\left( 2 R^{(d+1)} -\\frac{A}{2} G^{ij} \\partial_i \\phi \\partial_i \\phi -\\Lambda e^{2 B \\phi} \\right),\n\\label{eq:contaction}\n\\]\nwhere $G_{ij}$ denotes the $(d+1)$-dimensional metric with $i,j=0,1,2,\\cdots,d$; \n$R^{(d+1)}$ is the $(d+1)$-dimensional Ricci scalar; $\\phi$ is a real scalar field; \n$A,B,\\Lambda$ are real parameters. \nThe scalar field $\\phi$ is assumed to be related to the CTM field $\\beta$ through $\\beta=e^{B\\phi}$. \nThis action would be considered to be an effective action valid up to the second order of derivatives. \nThe classical EOM derived from \\eq{eq:contaction} respects\nthe dilatational symmetry \\eq{eq:scaletranseom}, because $S$ is transformed homogeneously \nby the transformation as $S\\rightarrow L^{d-1} S$. \n\nConsidering that the $(d+1)$-dimensional Lorentzian manifold $\\mathcal{M}$ is globally hyperbolic, \nwe use the following diffeomorphism,\n\\[\n\\varphi : \\ \\Sigma \\times \\mathbb{R} \\to \\mathcal{M}, \n\\label{eq:diff}\n\\] \nwhere $\\Sigma$ is a $d$-dimensional spatial hypersurface, \nto obtain the ADM metric as a pull-back, $\\varphi^*G$:\n\\[\nds^2 = - N^2 dt^2 + g_{\\mu \\nu} (dx^{\\mu} + N^{\\mu}dt )(dx^{\\nu} + N^{\\nu}dt),\n\\]\nwhere $N$, $N^{\\mu}$ and $g_{\\mu \\nu}$ are the lapse function, the shift vector and the $d$-dimensional metric on $\\Sigma$ with $\\mu, \\nu = 1,2, \\cdots, d$. \nHereafter we will turn off the shift vector, i.e., $N^{\\mu}=0$ for simplicity. \nThe terms associated with the non-zero shift vector can be recovered considering the time-dependent spatial diffeomorphism.\n\nBy the diffeomorphism (\\ref{eq:diff}), the action (\\ref{eq:contaction}) becomes\n\\[\nS= \\int dt\\ (K - V),\n\\]\nwhere $K$ is the kinetic term, \n\\[\nK= \\int_{\\Sigma_t} d^dx\\ \n\\left( \n\\frac{1}{2} \\mathcal{G}^{\\mu\\nu,\\rho\\sigma} \\dot g_{\\mu\\nu} \\dot g_{\\rho\\sigma} \n+\\frac{1}{2} \\mathcal{G}^{\\phi,\\phi}\\dot \\phi \\dot \\phi\n\\right)\n\\]\nwith \n\\[\n\\begin{split}\n\\mathcal{G}^{\\mu\\nu,\\rho\\sigma}&=\\frac{\\sqrt{g}}{N} \\left( \\frac{1}{2} (g^{\\mu\\rho}g^{\\nu\\sigma}+g^{\\mu\\sigma}g^{\\nu\\rho})\n-g^{\\mu\\nu}g^{\\rho\\sigma}\\right),\\\\\n\\mathcal{G}^{\\phi,\\phi}&=\\frac{A\\sqrt{g}}{N},\n\\label{eq:quadratic}\n\\end{split}\n\\]\nand $V$ is the potential term, \n\\[\nV \n= \\int_{\\Sigma_t} d^d x\\ N \\sqrt{g} \\left( \\Lambda e^{2B \\phi} -2 R+\\frac{A}{2} (\\nabla \\phi)^2 \\right),\n\\label{eq:potential}\n\\]\nin which $(\\nabla \\phi)^2 := g^{\\mu \\nu} \\nabla_{\\mu} \\phi \\nabla_{\\nu} \\phi$ with $\\nabla_{\\mu}$ being the covariant derivative \nassociated with the metric $g_{\\mu \\nu}$.\n\nTo employ the Hamilton-Jacobi formalism, \nlet us consider the following Hamilton's principal functional: \n\\[\nW =\\int_{\\Sigma_t} d^d x \\sqrt{g} \\left( \\lambda e^{B\\phi} -e^{-B\\phi} \\left( c_1 R+c_2 (\\nabla \\phi)^2 \\right)\\right)+{\\cal O}({\\nabla^4}),\n\\label{eq:W}\n\\]\nwhere $c_1,c_2,\\lambda$ are real parameters. \n$W$ is considered to be expressed as \nperturbative expansions in spatial derivatives up to the second order. \nThe potential in \\eq{eq:potential} and \n$W$ in \\eq{eq:W} must be related by the following Hamilton-Jacobi equation: \n\\[\nV + \\int_{\\Sigma_t} d^dx\\ \n\\frac{1}{2}\\left(\n\\mathcal{G}_{\\mu\\nu,\\rho\\sigma} \\frac{\\delta W}{\\delta g_{\\mu\\nu}} \\frac{\\delta W}{\\delta g_{\\rho\\sigma}}\n+\\mathcal{G}_{\\phi,\\phi} \\frac{\\delta W}{\\delta \\phi} \\frac{\\delta W}{\\delta \\phi} \n\\right) +{\\cal O}(\\nabla^4)\n= 0,\n\\label{eq:VandW}\n\\]\nwhere \n\\[\n\\begin{split}\n\\mathcal{G}_{\\mu\\nu,\\rho\\sigma}&=\\frac{N}{\\sqrt{g}}\\left(\\frac{1}{2} (g_{\\mu\\rho}g_{\\nu\\sigma}+g_{\\mu\\sigma}g_{\\nu\\rho})\n-\\frac{1}{d-1}g_{\\mu\\nu}g_{\\rho\\sigma}\\right),\\\\\n\\mathcal{G}_{\\phi,\\phi}&=\\frac{N}{A\\sqrt{g}},\n\\end{split}\n\\]\nbeing the inverse to \\eq{eq:quadratic}.\nInserting \\eq{eq:W} into \\eq{eq:VandW}, \nwe obtain\n\\[\nV = \n\\int_{\\Sigma_t} d^dx\\ \n\\sqrt{g}N \\,\n\\frac{1}{2}\\left[\n\\frac{1}{d-1} \n\\left( \\frac{\\lambda^2de^{2 B\\phi}}{4} +\\lambda H \\right)\n+\\frac{1}{A} \\left(2 B \\lambda F -B^2 \\lambda^2 e^{2 B \\phi} \\right)\n\\right] +{\\cal O}(\\nabla^4),\n\\label{eq:V}\n\\]\nwhere\n\\[\n\\begin{split}\nH&=\\frac{2-d}{2}\\left( c_1 R+c_2 (\\nabla \\phi)^2\\right)+(d-1)c_1 \\left(B \\nabla^2 \\phi+B^2 (\\nabla\\phi)^2\\right),\\\\\nF&=-B\\left(c_1 R -c_2 (\\nabla \\phi)^2\\right)-2 c_2 \\nabla^2 \\phi.\n\\end{split}\n\\label{eq:HandS}\n\\]\nComparing \\eq{eq:V} with \\eq{eq:potential}, we obtain some conditions for the parameters of $W$ as \n\\[\n\\begin{split}\n\\Lambda&=\\frac{\\lambda^2(-4 B^2 (-1 + d) + A d)}{8 A (-1 + d)},\\\\\n-2&=-\\frac{c_1 \\lambda(A (-2 + d) + 4 B^2 (-1 + d))}{4 A (-1 + d))},\\\\\n\\frac{A}{2}&=\\frac{\\lambda(A (-c_2 (-2 + d) + 2 B^2 c_1 (-1 + d)) + \n 4 B^2 c_2 (-1 + d))}{4 A (-1 + d)},\\\\\n0&=B \\lambda (A c_1 + 4 c_2).\n\\end{split}\n\\label{eq:relpara}\n\\]\nHere, the first equation comes from the comparison of the potential term, the second the curvature, and the third the scalar \nkinetic term. The last equation comes from the absence of $\\nabla^2 \\phi$ term in the potential. \n\nThe flow equations derived from $W$ is given by\\footnote{\nThe flow equations for $\\phi$ and $g_{\\mu \\nu}$ \nare originated with Hamilton's equations for $\\phi$ and $g_{\\mu \\nu}$ \nwith the replacement of conjugate momenta by $\\frac{\\delta W}{\\delta \\phi}$ and $\\frac{\\delta W}{\\delta g_{\\mu \\nu}}$, \nrespectively. \n} \n\\[\n\\begin{split}\n&\\frac{1}{N} \\dot \\phi=\\mathcal{G}_{\\phi,\\phi} \\frac{\\delta W}{\\delta \\phi} \n=\\frac{1}{A} \\left(B \\lambda e^{B\\phi}-e^{-B\\phi} F \\right) + {\\cal O}(\\nabla^4), \\\\\n&\\frac{1}{N}\\dot g_{\\mu\\nu}= \\mathcal{G}_{\\mu\\nu,\\rho\\sigma}\\frac{\\delta W}{\\delta g_{\\rho\\sigma}}\n=\\frac{\\lambda e^{B\\phi}}{2(1-d)}g_{\\mu\\nu}+e^{-B\\phi}\\left(H_{\\mu\\nu}+\\frac{1}{1-d} g_{\\mu\\nu} H\\right)\n+{\\cal O}(\\nabla^4), \n\\end{split}\n\\label{eq:eommetbare}\n\\]\nwhere\n\\[\n\\begin{split}\nH_{\\mu\\nu}=&c_1\\left( R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R+B \\left( \\nabla_\\mu\\nabla_{\\nu}\\phi-g_{\\mu\\nu}\\nabla^2 \\phi \\right)\n-B^2 \\left(\\nabla_\\mu\\phi \\nabla_\\nu\\phi-g_{\\mu\\nu} (\\nabla \\phi)^2 \\right)\\right) \\\\\n&+c_2 \\left(\\nabla_\\mu\\phi \\nabla_\\nu\\phi-\\frac{1}{2}g_{\\mu\\nu} (\\nabla \\phi)^2 \\right).\n\\end{split}\n\\]\nThere is a relation, $H=g^{\\mu\\nu}H_{\\mu\\nu}$. \n\nTo compare \\eq{eq:eommetbare} with the EOM \\eq{eq:eomCTM} from CTM, \nlet us perform \na change of the variable, $\\beta=\\exp[B \\phi]$. Taking into account that \n$\\dot g^{\\mu\\nu}=-g^{\\mu\\mu'}g^{\\nu\\nu'}\\dot g_{\\mu'\\nu'}$, \nthe EOM \\eq{eq:eommetbare} can be rewritten as \n\\[\n\\begin{split}\n\\frac{1}{c_3 n}\\dot \\beta=&\\frac{B^2 \\lambda \\beta^2}{A}+\\frac{B^2 c_1 R}{A}+\\frac{2 c_2 \\beta_{,\\mu}^{,\\mu}}{A \\beta}\n-\\frac{3 c_2 \\beta_{,\\mu}\\beta^{,\\mu}}{A \\beta^2}+ {\\cal O}(\\nabla^4) ,\\\\\n\\frac{1}{c_3 n} \\dot g^{\\mu\\nu}=&\\frac{\\lambda \\beta g^{\\mu\\nu}}{2 (-1 + d)}\n-\\frac{c_1 R^{\\mu\\nu}}{\\beta}\n-\\frac{c_1 \\beta^{,\\mu\\nu}}{\\beta^2}\n+\\frac{(2 B^2 c_1 - c_2) \\beta^{,\\mu}\\beta^{,\\nu}}{B^2 \\beta^3} \\\\\n&+g^{\\mu\\nu}\\left( \\frac{c_1 R}{2 (-1 + d) \\beta}\n+\\frac{c_2 \\beta_{,\\rho} \\beta^{,\\rho}}{2 B^2 (-1 + d) \\beta^3}\n\\right)+ {\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:eomcont}\n\\]\nwhere we have introduced possible difference of normalizations between the lapse functions \nof CTM and general relativity as $N=c_3\\, n$ with a constant $c_3$. \nWe want to find the values of the parameters which make \\eq{eq:eomcont} coincident with \\eq{eq:eomCTM}.\nThe number of parameters is smaller than that of the equations to be satisfied (i.e., an overdetermined set of equations), \nbut we can solve the coincidence condition by the following values:\n\\[\n\\begin{split}\n\\lambda c_3 &=\\frac{96(-1+d)}{2+d}, \\\\\nB^2&=\\frac{A(6-d)}{32(d-1)},\\\\\nc_1 c_3&=\\frac{8(2+d)}{-10+7d},\\\\\nc_2 c_3 &=-\\frac{2 A (2 + d)}{-10 + 7 d}, \\\\\nx_7&=\\frac{16 - 88 d + 26 d^2 + d^3}{12 (-20 + 64 d - 65 d^2 + 21 d^3)},\\\\\nx_8&=\\frac{6 - d}{-10 + 7 d}, \\\\\nx_9&=\\frac{14 - 67 d + 17 d^2}{-60 + 192 d - 195 d^2 + 63 d^3}.\n\\end{split}\n\\label{eq:detpara}\n\\]\nThe details of the derivation of the solution are given in Appendix~\\ref{app:solution}.\nThe parameter $c_3$ can be determined by the second (or equivalently the third) \nequation of \\eq{eq:relpara} by putting \\eq{eq:detpara}:\n\\[\nc_3^2=12.\n\\label{eq:valuec3}\n\\] \nThis rather strange value actually normalizes the overall factor\nin the algebraic relation \\eq{eq:metricv} of the CTM to the natural value in GR.\nIt can be checked that the third and fourth equations\nof \\eq{eq:relpara} are also satisfied by \\eq{eq:detpara} and \\eq{eq:valuec3}. \nFrom the first equation of \\eq{eq:relpara}, \\eq{eq:detpara} and \\eq{eq:valuec3}, we obtain\n\\[\n\\Lambda=\\frac{36 (d-1)(3d-2)}{(d+2)^2}.\n\\label{eq:valuelambda}\n\\]\nThe above solution is unique except for the rather obvious ambiguities of the signs of $B$ and $c_3$.\nThese signs are physically irrelevant, because the sign of $B$ can be absorbed by that of $\\phi$, and \nthat of $c_3$ just determines the overall sign of $W$ (or can be absorbed in $n$).\n\nIf we require the positivity of the potential energy from the spatial derivative term of $\\phi$, $A>0$ is required. Then, \nthe second equation of \\eq{eq:detpara} implies that the dimension must be in the range \n$2\\leq d\\leq 6$ (The $d=1$ case is excluded from the beginning in the Hamilton formalism, \nas can be seen at the beginning of this section.).\nIn this range, \\eq{eq:valuelambda} is positive, and one can normalize the value of $\\Lambda$ \nby rescaling the space-time coordinates as $(t,x^\\mu)\\rightarrow L (t,x^\\mu)$ with $L=1\/\\sqrt{\\Lambda}$ and \ndropping an overall factor of the action.\nWe can also rescale the scalar field as $\\phi \\rightarrow \\hbox{sign(}B\\hbox{)}\\phi\/\\sqrt{A}$.\nThen, the action describing CTM is uniquely determined, \nfor a globally hyperbolic $\\mathcal{M}$, \nto be \n\\[\nS_{CTM}=\\int_{\\mathcal{M}} d^{d+1} x\\,\\sqrt{-G} \\left( \n2 R-\\frac{1}{2} G^{ij} \\partial_{i} \\phi \\partial_{j} \\phi \n- e^{\\sqrt{\\frac{6-d}{8(d-1)}}\\phi}\n\\right),\n\\label{eq:CTMaction}\n\\]\nwhich is valid in $2\\leq d \\leq 6$. Thus, \nthe system has a critical dimension $d=6$, over which it becomes unstable due to the \nwrong sign of the scalar kinetic term. \nAt the critical dimension, the scalar is a massless field with no non-derivative couplings.\n\n\\section{Time evolution of the scale factor}\n\\label{sec:mini}\nThe coupled system of gravity and a scalar field described by the action \\eq{eq:CTMaction} \nhas been discussed in the context of models of dark energy \n(See \\cite{Copeland:2006wr} for a comprehensive review.).\nThe exponential potential in \\eq{eq:CTMaction} of the scalar field is known to lead to a power-law \nbehavior (or an exponential behavior in the critical case) of the scale factor. \nLet us see this in our case, analyzing \\eq{eq:eomCTM}.\n\nDiscarding the spatial derivative terms of \\eq{eq:eomCTM} and putting $n=1$, the equation of motion is given by\n\\[\n&\\dot \\beta=d_1 \\beta^2,\n\\label{eq:minibeta} \\\\\n&\\dot g^{\\mu\\nu}=d_2 \\beta g^{\\mu\\nu},\n\\label{eq:minig}\n\\]\nwhere \n\\[\nd_1=\\frac{3(6-d)}{d+2},\\ d_2=\\frac{48}{d+2}.\n\\]\nSubstituting \\eq{eq:minig} with an ansatz $g^{\\mu\\nu}=a(t)^{-2} \\delta^{\\mu\\nu}$ \nwith a scale factor $a(t)$, we obtain\n\\[\n\\frac{2\\dot a}{a}=-d_2 \\beta. \n\\label{eq:minia}\n\\]\nThen, for $d_1\\neq 0\\ (\\hbox{i.e., }d\\neq 6)$, the solution to \\eq{eq:minibeta} and \\eq{eq:minia} is obtained as\n\\[\n\\begin{split}\n&\\beta=\\frac{1}{d_1(t_0-t)},\\\\\n&a=a_0 (t_0-t)^{\\frac{d_2}{2d_1}},\n\\end{split}\n\\]\nwhere $t_0$ and $a_0$ are integration constants.\n\nWhen $d=6$, $d_1$ vanishes. In this case, $\\beta$ is given by a constant, say $\\beta_0$. Then, \n\\eq{eq:minia} gives\n\\[\na=a_0 \\exp \\left[ -\\frac{d_2 \\beta_0}{2} t \\right].\n\\]\nThus, we see that, in the critical case $d=6$, the solution is given by de Sitter spacetime.\n\nAs is well known, de Sitter spacetime has the invariance of a conformal symmetry $SO(d+1,1)$. \nIn statistical physics, the appearance of a conformal symmetry is the sign that a system is on a critical point. \nThis suggests that CTM at $d=6$ is on a critical point in some sense. In fact, as shown in Section~\\ref{sec:conttheory}, \nfor the reality of $B$, the sign of the kinetic term of the scalar field must change its sign at $d=6$.\nIn $d>6$, it gets the wrong sign, and the the scalar field becomes unstable in the direction \nof larger spatial fluctuations.\nThis means that $d=6$ can be thought of as a phase transition point \nbetween a stable phase at $d<6$ and another phase at $d>6$.\nConsidering the instability in the direction of larger spatial fluctuations, \nthe latter phase probably contradicts our assumption of a continuous\nspace. The understanding of the phase transition should be pursued further.\n\n\\section{Summary and future prospects}\n\\label{sec:summary}\nIn this paper, we have analyzed the equation of motion (EOM) of the canonical tensor model (CTM)\nin a formal continuum limit by employing a derivative expansion of its tensor up to the fourth order. \nWe have shown that, up to the order, the EOM of CTM in the continuum limit agrees with \nthat of a coupled system of gravity and a scalar field obtained in the framework of the Hamilton-Jacobi methodology. \nThe action of the gravitational system is composed of the curvature term, the scalar field kinetic term\nand an exponential potential of the scalar field. The system is classically invariant under a dilatational transformation. \nThe action is physically valid in the range of the spatial dimensions, $2\\leq d \\leq 6$,\nand, in $d>6$, the system is unstable due to the wrong sign of the kinetic term of the scalar field.\nAt the critical case $d=6$, de Sitter spacetime is a solution to the EOM, while, in $2\\leq d < 6$,\nthe time evolution of the scale factor of a flat space has a power-law behavior.\n \nThe most significant achievement of this paper is to have concretely shown that CTM indeed derives a \ngeneral relativistic system in a formal continuum limit. This was conjectured in our previous paper \\cite{Sasakura:2015pxa}\nfrom the observation that\nthe constraint algebra of CTM in the continuum limit agrees with that of the ADM formalism, \nbut no concrete correspondences were given. \nOn the other hand, in this paper, we have obtained the one-to-one correspondence of the fields \nbetween CTM in the continuum limit up to the fourth order \nand the gravitational system so that the two systems have a common EOM.\nThe action of the corresponding gravitational system has also been obtained.\n\nAn interesting question arising from our result is what is the meaning of the criticality at $d=6$.\nThe existence of de Sitter spacetime solution implies that the system has a conformal symmetry on this background\nin the dimension.\nOn the other hand, in our previous papers \\cite{Sasakura:2015xxa,Sasakura:2014zwa,Sasakura:2014yoa}, it was shown \nthat the Hamiltonian vector flows of CTM can be regarded as RG flows of statistical systems on random networks.\nThese two aspects of CTM suggest that a statistical system at criticality described by a \nsix-dimensional conformal field theory is associated to CTM \\cite{Strominger:2001pn,Strominger:2001gp}.\nIt would be interesting to identify the conformal field theory in a concrete manner.\n\nAnother interesting direction of study would be to extend the derivative expansion to higher orders,\nwhich includes higher spin fields than two with higher spin gauge symmetries. \nThere are general interests in pursuing higher spin gauge theories (See \\cite{Giombi:2016ejx} for a recent review.).\nSince our approach has a significant difference from the other ones in the sense \nthat we take a formal continuum limit of a consistent \ndiscretized theory in the canonical formalism, we would expect that our model may shed some new lights on the subject. \nFor that purpose, it would be necessary to set up a new efficient methodology for the analysis instead of relying on machine \npowers as in this paper.\n\nThe EOM of CTM is a set of first-order differential equations in time, and has been related \nto a gravitational system through the Hamilton-Jacobi equation.\nWhile the gravitational system contains the phenomena of second-order differential equations like wave propagations,\nit is not clear how to realize such phenomena in the framework of CTM. \nIt would be interesting to improve the canonical formalism of the tensor model in that direction.\n\n\\vspace{1cm}\n\\centerline{\\bf Acknowledgements} \nThe work of N.S. is supported in part by JSPS KAKENHI Grant Number 15K05050. \nThe work of Y.S. is funded under CUniverse research promotion project \nby Chulalongkorn University (grant reference CUAASC).\nN.S. would like to thank the great hospitality and the stimulating discussions with the members\nof Department of Physics of Chulalongkorn University, while he stayed there and part of this work was done. \nY.S. would like to thank the wonderful members in Nagoya University, Japan, \nwhere part of this work was done, for the kind hospitality and fruitful discussions. \nY.S. would like to appreciate Jan Ambj\\o rn for discussions about the dimension in the formal continuum limit. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\noindent\nLet $G$ be a nontrivial finite group.\nWe consider the minimal number of ramified primes in \nGalois extensions $L\/K$ with Galois group $G$,\nin various situations:\n\nIn the case $K=\\mathbb{Q}$,\nthe primes are as usual equivalence classes of absolute values,\ncorresponding to the prime numbers and the usual absolute value as the prime at infinity.\nLet $r_K(G)$ denote the minimal number of ramified primes\\footnote{the infinite prime is said to ramify in $L\/\\mathbb{Q}$\nif $L$ is not totally real}\n in a Galois extension $L\/K$ with Galois group $G$.\nBased on class field theory, Boston and Markin came to the following conjecture:\n\\begin{conjecture}[Boston--Markin \\cite{BM}]\\label{conj:BM}\nFor every nontrivial finite group $G$,\n$$\n r_\\mathbb{Q}(G) = \\max\\left\\{d(G^{\\rm ab}),1\\right\\},\n$$\nwhere $d(G)$ is the smallest cardinality of a set of \ngenerators of $G$. \n\\end{conjecture}\n\nWhile this \nwas verified by Boston and Markin for abelian groups and for groups of order up to 32,\nand since then was proven for certain solvable groups \\cite{Plans,KisilevskySonn,KisilevskySonn2010,KisilevskyNeftinSonn},\nit is widely open for most other groups.\nFor example, for the symmetric group $G=S_n$, $n>1$, this conjecture predicts realizations\nwith only one ramified prime.\nWhile this can of course be verified for small $n$,\nthe best general bound to date is the recent result of Bary-Soroker and Schlank \\cite{BSS}, who obtain $r_\\mathbb{Q}(S_n)\\leq 4$ for all $n$.\n\nIn the case where $K=k(T)$ is a rational function field over a field $k$, the primes are equivalence classes of ultrametric absolute values on $K$ trivial on $k$,\nwhich correspond to the monic irreducible polynomials $P\\in k[T]$ \nand the degree valuation as the prime at infinity.\nAs before we let $r_K(G)$ be\nthe minimal number of ramified primes\nin Galois extensions $L\/K$ with Galois group $G$,\nwhere in addition we require that $L\/K$ is {\\em geometric}, i.e.~$k$ is algebraically closed in $L$.\nThe geometric situation $k=\\mathbb{C}$ is well understood:\n\n\\begin{theorem}[Riemann Existence Theorem]\\label{thm:RET}\nFor every nontrivial finite group $G$,\n$$\n r_{\\mathbb{C}(T)}(G) = d(G)+1.\n$$\n\\end{theorem}\nIn positive characteristic, wild ramification\nleads to smaller bounds:\n\n\\begin{theorem}[Abhyankar conjecture for the line, \\cite{Ray94,Har94}]\\label{thm:Abhyankar}\nFor every prime number $p$ and every nontrivial finite group $G$,\n$$\n r_{\\overline{\\mathbb{F}}_p(T)}(G) = d(G\/p(G)) + 1,\n$$\nwhere $p(G)$ is the subgroup of $G$ generated by the $p$-Sylow subgroups of $G$.\n\\end{theorem}\n\n\nFor the global field $K=\\mathbb{F}_q(T)$,\nin analogy with Conjecture \\ref{conj:BM},\nwe thus conjecture the following:\n\n\\begin{conjecture}\\label{conj}\\label{bm}\nFor every prime power $q=p^\\nu$ and every nontrivial finite group $G$,\n$$\n r_{\\mathbb{F}_{q}(T)}(G) = \\max\\left\\{d((G\/p(G))^{\\rm ab}),1\\right\\}\n$$\n\\end{conjecture}\n\nWe note that $(G\/p(G))^{\\rm ab}=G^{\\rm ab}\/p(G^{\\rm ab})$,\nand that the close analogy with the case of $K=\\mathbb{Q}$ builds in an essential way on our restriction \nthat \nthe extension $L$ of $K=\\mathbb{F}_q(T)$ with Galois group $G$ is geometric.\nIn Section \\ref{sec:abelian} we prove this conjecture for abelian groups,\nthereby showing that the right hand side is always a lower bound for $r_{\\mathbb{F}_{q}(T)}(G)$.\nThere we also discuss the contribution of De Witt \\cite{DeWitt},\nwho proves Conjecture \\ref{conj} for certain solvable groups.\n\n\\begin{remark}\\label{rem:joachim} \nAnother potential constraint on $r_{\\mathbb{F}_{q}(T)}(G)$ was pointed out to us by Joachim K\\\"onig. \nFor a finite geometric Galois extension $L$ of $K=\\mathbb{F}_q(T)$,\nthe conjugacy classes of inertia groups generate ${\\rm Gal}(L\/K)$,\nand each inertia group is cyclic-by-$p$, i.e.~an extension of a cyclic group by a $p$-group (see \\cite[Corollary IV.2.4]{Ser79}). Thus Conjecture \\ref{conj} is possibly true only if any finite group $G$ is generated by $\\max\\{d((G\/p(G))^{\\rm ab}),1\\}$ conjugacy classes of cyclic-by-$p$ groups. \nWe give the proof of this elementary but nontrivial group-theoretic fact also in Section \\ref{sec:generators}. \nA similar remark applies to Conjecture \\ref{conj:BM}, where Lemma~\\ref{lem:gen_by_cyc} ensures there is no group-theoretic obstruction.\n\\end{remark}\n\nIn the main part of this work (Sections \\ref{sec:tworamified} and \\ref{sec:pleqn}) we prove various results for certain non-solvable groups, both conditional and unconditional,\nwith a focus on symmetric and alternating groups.\nNote that Conjecture \\ref{conj}\npredicts \n$$\n r_{\\mathbb{F}_q(T)}(S_n)=1,\\quad r_{\\mathbb{F}_q(T)}(A_n)=1\n$$\nfor every $n>2$ and every prime power $q=p^\\nu$.\nFor example, we obtain the following for $S_n$ and $A_n$:\n\n\\begin{theorem}\\label{thm:Sn}\nLet $n\\geq 2$ and $q=p^\\nu$ a prime power.\nThen \n\\begin{enumerate}\n\\item $r_{\\mathbb{F}_q(T)}(S_n)\\leq 2$, and \n\\item $r_{\\mathbb{F}_q(T)}(S_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $p(2n-3)^2$\n\\item The function field analogue of Schinzel's hypothesis H (Conjecture \\ref{conj:SchinzelFF}) holds for $\\mathbb{F}_q(T)$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:An}\nLet $n\\geq 3$ and $q=p^\\nu$ a prime power.\n\\begin{enumerate}\n\\item If $p>2$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$, then $r_{\\mathbb{F}_q(T)}(A_n)\\leq 2$, and \n\\item $r_{\\mathbb{F}_q(T)}(A_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $22$ with $n\\neq p+1$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$ or $p=3$.\n\\item $A_n$ if $p=2$ with either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $10\\neq n\\ge 8,n\\equiv 0,1,2,6,7\\pmod 8.$\n\\end{enumerate}\nThen $r_{\\mathbb{F}_q(T)}(G)=\\max\\{d((G\/p(G))^{\\rm ab}),1\\}$, i.e. Conjecture \\ref{bm} holds for $G$.\n\\end{theorem}\n\nEn route to proving Theorem~\\ref{thm:main1} we will obtain evidence for the following conjecture of Abhyankar (see \\cite[\\S 16]{Abh01},\\cite[\\S 5]{HOPS}, a weaker form was stated in \\cite[Conjecture 9.2(C)]{Abh95})\nwhich is an analogue over $\\mathbb{F}_q$ of the Abhyankar conjecture for the affine line (which is the case $G=p(G)$ of Theorem~\\ref{thm:Abhyankar}):\n\n\\begin{conjecture}[Abhyankar's arithmetic conjecture for the affine line]\\label{conj:abhyankar} \nLet $G$ be a finite group which is cyclic-by-quasi-$p$, meaning that $G\/p(G)$ is cyclic (see Definition~\\ref{def:quasip} below).\nThen for every power $q$ of $p$ there exists a Galois extension $L\/\\mathbb{F}_q(T)$ (not necessarily geometric)\nramified only over the infinite prime\nsuch that ${\\rm Gal}(L\/\\mathbb{F}_q(T))=G$.\n\\end{conjecture}\n\n\nWe will show that Conjecture \\ref{conj:abhyankar} holds for the alternating group $A_n$ provided $n\\ge p>2,n\\neq p+1$ \n(note that for $n>3$ the group $A_n\/p(A_n)$ is cyclic only if $n\\ge p$).\n\n\\begin{theorem}\\label{thm:abhyankar} Assume $n\\ge p>2$. If $n=p+1$ assume additionally that $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$. Then there exists a Galois extension $L\/\\mathbb{F}_q(T)$ ramified only over the infinite prime\nwith ${\\rm Gal}(L\/\\mathbb{F}_q(T))=A_n$.\n\\end{theorem}\nTheorem~\\ref{thm:abhyankar} will follow from Theorem~\\ref{thmlist} below. For some values of $q,n$ the theorem above is implicit in the work of Abhyankar \\cite{Abh92,Abh93}, but not in the above generality.\n\n\n\n\nTo obtain these results we apply a variety of tools and recent results\nfrom Galois theory, finite group theory (including the Classification of Finite Simple Groups for some of the results), analytic number theory\nand number theory over function fields,\nand we also employed the help of a computer to eliminate some small exceptional cases.\nSee the summary in Section \\ref{sec:summary} \nfor the proof of Theorems \\ref{thm:Sn} and \\ref{thm:An},\nand the corresponding sections for the full strength and full generality of the results.\nUsing our results for $\\mathbb{F}_q(T)$ we also obtain a conditional result for $\\mathbb{Q}$ (Theorem~\\ref{thm:S_n_over_Q}):\n\n\\begin{theorem}\nSchinzel's hypothesis H (Conjecture \\ref{conj:Schinzel}) implies that\n$r_\\mathbb{Q}(S_n)=1$ for every $n>1$,\ni.e.~Conjecture \\ref{conj:BM} holds for all symmetric groups.\n\\end{theorem}\n\nThis improves upon a result of Plans (see Remark \\ref{remark:plans} below), which shows that under Schinzel's hypothesis H there exists a Galois extension $L\/\\mathbb Q$ with Galois group $S_n$ and ramified over a single finite prime and the infinite prime, whereas our construction has no ramification at the infinite prime.\n\n\n\\section{Preliminaries and notation}\n\\noindent\nWe start by collecting a few definitions and statements that we will use throughout the paper.\n\n\\subsection{Function fields}\n\nFor a thorough introduction to the subject see \\cite{Ros02,Stichtenoth}. Here we only recall some terminology, notation and a few basic facts. \nLet $k$ be a base field.\nA function field (of one variable) over $k$ is a finite extension of the rational function field $k(T)$. \nA \\emph{global function field} is a function field over a finite base field $k=\\mathbb{F}_q$ (this will be our main case of interest).\n\nA function field $K\/k$ is {\\em regular}\nif $K$ is linearly disjoint over $k$ from the algebraic closure $\\bar{k}$ of $k$,\nwhich in case that $k$ is perfect simply means that\n$k$ is relatively algebraically closed in $K$.\nIf $K\/k$ is a regular function field, there exists a unique geometrically integral non-singular projective curve $C\/k$ such that $K$ is the function field of $C$. Conversely the function field of a curve as above is a regular function field over $k$.\n\nLet $K\/k$ be a regular function field and $C$ the corresponding curve. \nA \\emph{prime} $P$ of $K\/k$ is an\nequivalence class of nontrivial absolute values on $K$ trivial on $k$. Each such absolute value corresponds to a discrete valuation $v$,\nand it corresponds to a prime divisor on $C$. \nIf $P$ is a prime of $K$, the residue field $k(P)$ at $P$ is a finite extension of $k$ and $\\deg P=[k(P):k]$ is called the \\emph{degree} of $P$. By a \\emph{point} or \\emph{geometric point} of $C$ we will always mean a closed point of $C\\times_k\\bar k$. \nFor an extension $L\/k$ we denote by $C(L)$ the set of $L$-rational points on $C$.\n\n\\renewcommand{\\div}{\\mathrm{div}}\n\nThe rational function field $k(T)$ corresponds to the projective line $\\P^1_k$ and has primes of two types: the \\emph{finite primes} of the form $\\div(f)_0$ (the zero divisor of a rational function $f$) where $f\\in k[X]$ is an irreducible polynomial (such a prime has degree $\\deg f$) and the \\emph{infinite prime} $\\infty=\\div(1\/T)_0$ which has degree 1.\n\nLet $L\/K$ be a finite extension of function fields over $k$. It is called \\emph{regular}, or \\emph{geometric}, if $L$ is also a regular function field over $k$. Regular extensions correspond to finite covers of the associated curves. \nIf $P$ is a prime of $K$ and $Q$ a prime of $L$ lying over $P$, we denote by $e(Q\/P)$ the ramification index of $Q$ over $P$. \nWe say that $L\/K$ is ramified (resp. wildly ramified) at $Q$ if $e(Q\/P)>1$ (resp.\\ $\\mathrm{char}(k)|e(Q\/P)$). We say that $L\/K$ is ramified (resp. wildly ramified) \\emph{over} $P$ if there exists a prime $Q$ of $L$ lying over $P$ at which the extension is ramified (resp. wildly ramified).\nRamification which is not wild is said to be \\emph{tame}.\nIf $L\/K$ is tamely ramified over at least one prime, then $L\/K$ is separable.\n\nIf $C_L$ and $C_K$ are the underlying curves then the extension $L\/K$ corresponds to a finite morphism $w\\colon C_L\\to C_K$. \nThis correspondence defines an equivalence of categories between regular function fields over $k$ and absolutely irreducible non-singular projective curves defined over $k$. \nIf $P$ is a prime divisor of $C_L$, and $Q=w(P)$ (a prime divisor of $C_K$), we say that $w$ is ramified (resp. tamely ramified) at $P$ whenever the extension $L\/K$ is ramified (resp. tamely ramified) at the corresponding prime of $L$ which we identify with $P$. \nIn this case we also call the geometric points corresponding to $P$ {\\em ramification points} of $w$,\nand the geometric points corresponding to $Q$ {\\em branch points} of $w$.\n\n\\subsection{Ramification theory}\n\nLet $K$ be a field. For a separable polynomial $f\\in K[X]$ of degree $n$ we define its Galois group ${\\rm Gal}(f\/K)$ to be the Galois group of its splitting field over $K$.\nWe will always interpret the Galois group ${\\rm Gal}(f\/K)$ as a subgroup of the symmetric group $S_n$ via its action on the roots of $f$. The embedding into $S_n$ is well-defined up to conjugation.\n\n\\begin{lemma}\\label{lemcycle} \nLet $K$ be a function field over an algebraically closed field $k$\nof characteristic $p\\geq 0$,\nlet $f\\in K[X]$ be a separable irreducible polynomial and let $L=K(\\alpha)$ where $\\alpha$ is a root of $f$. \nLet $P$ be a prime (finite or infinite) of $K$ and let $Q_1,\\ldots,Q_r$ be the primes of $L$ lying over $P$. \nLet $e_i=e(Q_i\/P)$ be the ramification indices.\n\\begin{enumerate}\\item[(i)]\nAssume that $p\\nmid e_i$ for all $i$. \nThen ${\\rm Gal}(f\/K)$ contains a permutation which is a product of $r$ disjoint cycles of lengths $e_1,\\ldots,e_r$.\n\\item[(ii)]\nAssume that\n$p\\nmid e_i$ and $(e_i,e_1)=1$ for $2\\le i\\le r$, \nand either $e_1=p$ or $p\\nmid e_1$. \nThen ${\\rm Gal}(f\/K)$ contains a cycle of length $e_1$.\\end{enumerate}\\end{lemma}\n\n\\begin{proof} See \\cite[\\S 3, Generalized Cycle Lemma]{Abh93}.\\end{proof}\n\n\\begin{lemma}[Abhyankar's Lemma] \\label{lem:abhyankar} \nLet $k$ be perfect, $K\/k$ a function field, and\n$L,M$ finite separable extensions of $K$. \nLet $P$ be a prime of $K$, $Q$ a prime of $L$ lying over $P$, $P'$ a prime of $M$ lying over $P$. \nAssume that \n\\begin{enumerate}\n\\item $Q$ is tamely ramified over $P$, and\n\\item $e(Q\/P)$ divides $e(P'\/P)$.\n\\end{enumerate}\nThen every prime $Q'$ of $LM$ lying over both $Q$ and $P'$ is unramified over $M$, i.e.~$e(Q'\/P')=1$.\n\\end{lemma}\n\n\\begin{proof} \nThis follows from \\cite[Theorem 3.9.1]{Stichtenoth}\nby the multiplicativity of the ramification index.\n\\end{proof}\n\n\\subsection{Group theory}\nWe denote by $C_n$, $D_n$, $S_n$, $A_n$ \nthe cyclic, dihedral, symmetric respectively alternating group of degree $n$.\nWe will need the following group-theoretic result of Jones, which relies on the Classification of Finite Simple Groups (CFSG) for its proof.\n\n\\begin{theorem}[Jones]\\label{thm:jones} Let $G\\leqslant S_n$ be a primitive permutation group containing a cycle of length $l>1$.\n\\begin{enumerate}\\item[(i)]If $l\\le n-3$ then $G\\geqslant A_n$.\n\\item[(ii)]If $l=n-2$ is prime then either $G\\geqslant A_n$ or $l=2^k-1$ is a Mersenne prime and ${PGL}_2(2^k)\\leqslant G\\leqslant{P\\Gamma L}_2(2^k)$ with its standard action on $\\P^1(\\mathbb{F}_{2^k})$.\n\\end{enumerate}\n\\end{theorem} \n\n\\begin{proof} This is a direct consequence of \\cite[Corollary 1.3]{Jon14}.\\end{proof}\n\n\n\\begin{remark}\\label{remark:jordan}\nIn the case that $l$ is prime Theorem~\\ref{thm:jones}(i) is a classical (and elementary) result of Jordan \\cite[Theorem 3.3E]{DiMo96}. Most (though not all) of the applications in Section~\\ref{sec:pleqn} only require this weaker version. In Section~\\ref{sec:tworamified} Jones's theorem is used in its full form.\n\\end{remark}\n\nWe will also make use of the following elementary fact:\n\n\\begin{lemma}\\label{lem:primtrans}\nLet $G\\leqslant S_n$ be a primitive group containing a transposition. Then $G=S_n$.\n\\end{lemma}\n\n\\begin{proof} See \\cite[Theorem 13.3]{Wie64}.\\end{proof}\n\n\n\n\\subsection{Discriminants}\n\nLet $K$ be a field. \nFor a polynomial $f\\in K[X]$ of degree $n$ we denote its leading coefficient by $\\mathrm{lc}(f)$,\nand if $\\alpha_1,\\dots,\\alpha_n\\in\\overline{K}$ are the zeros of $f$ (listed with multiplicity), the discriminant of $f$ is defined by\n\\begin{equation}\\label{eq:discdefinition}\n\\disc(f) \\;=\\; \\mathrm{lc}(f)^{2n-2}\\prod_{i\\deg c$, $f'\\neq0$, and that $c$ is separable. \nThen the finite critical values of $w$ are the roots of $\\disc_X(f-Uc)$, more precisely\n$$\n\\disc_X(f(X)-Uc(X))=a\\cdot\\prod_{(f'c-fc')(\\alpha)=0}(U-w(\\alpha)),\n$$\nwith $a\\in k^\\times$ and the product over the roots of $f'c-fc'$ in $\\bar k$ with multiplicity.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n{\\bf (i).} We identify $U=w(X)$, viewing $U$ as an element of $k(X)$. This gives the extension of function fields $k(X)\/k(U)$ corresponding to the rational map $w$. Let $\\alpha\\in\\bar k$ be such that $c(\\alpha)\\neq 0$ and denote $\\beta=w(\\alpha)$. Then $X-\\alpha$ is a local parameter for $\\P^1_X$ at the point $\\alpha$ and $U-\\beta$ is a local parameter for $\\P^1_U$ at the point $\\beta$. \nWe have $U-\\beta=w(X)-\\beta=(X-\\alpha)^eu(X)$ \nwith $u\\in k(X)$ which has no pole at $\\alpha$. \nTaking derivatives we obtain\n$$\n w'(X)=e(X-\\alpha)^{e-1}u(X)+(X-\\alpha)^eu'(X).\n$$ \nAs $\\alpha$ is neither a pole nor a zero of $u$, and hence also not a pole of $u'$,\nthe multiplicity $\\nu$ of the zero $\\alpha$ of $w'$ \nis $\\nu=e-1$ if ${\\rm char}(k)\\nmid e$, and $\\nu\\geq e$ if ${\\rm char}(k)\\mid e$.\n\n{\\bf (ii).} Denote $h=(f',c')\\in\\bar k[X]$ and write $f'=hu,c'=hv,u,v\\in\\bar k[X]$. \nAs $f'\\neq 0$, also $u\\neq 0$.\nIf $\\deg c=0$ the \nassertion of (ii) is immediate from (i) and (\\ref{eq:disc2}), so let us assume $\\deg c>0$ and then \n$c',v\\neq 0$ by our assumption that $c$ is separable. Using the properties of resultants and \ndiscriminants (\\ref{eq:resdefinition}),(\\ref{eq:disc}),(\\ref{eq:resbimult}),(\\ref{eq:ressym}),(\\ref{eq:resalt}) and using $\\sim$ to denote\nequality up to a non-zero multiplicative constant (in $k$),\nwe calculate:\n\\begin{eqnarray*}\n\\disc_X(f-Uc)&\\stackrel{(\\ref{eq:disc})}{\\sim}&\\mathrm{Res}(f'-Uc',f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=& \\mathrm{Res}(h,f-Uc)\\cdot\\mathrm{Res}(u-Uv,f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=&\\mathrm{Res}(h,f-Uc)\\cdot\\frac{\\mathrm{Res}(u-Uv,fv-Ucv)}{\\mathrm{Res}(u-Uv,v)}\\\\\n&\\stackrel{(\\ref{eq:ressym}),(\\ref{eq:resalt})}\\sim&\\mathrm{Res}(h,f-Uc)\\cdot\\frac{\\mathrm{Res}(u-Uv,fv-uc)}{\\mathrm{Res}(u,v)}\\\\\n&\\stackrel{(\\ref{eq:ressym})}\\sim&\\mathrm{Res}(h,f-Uc)\\cdot\\mathrm{Res}(uc-fv,u-Uv)\\\\\n&\\stackrel{(\\ref{eq:resdefinition})}\\sim&\\prod_{h(\\alpha)=0}\\left(c(\\alpha)U-f(\\alpha)\\right)\\prod_{(uc-fv)(\\alpha)=0}(u(\\alpha)-Uv(\\alpha))\\\\\n&\\sim&\\prod_{h(\\alpha)=0}\\left(U-\\frac{f(\\alpha)}{c(\\alpha)}\\right)\\prod_{(uc-fv)(\\alpha)=0}\\left(U-\\frac{f'(\\alpha)}{c'(\\alpha)}\\right)\\\\\n&\\sim&\\prod_{(f'c-fc')(\\alpha)=0}(U-w(\\alpha)).\n\\end{eqnarray*}\nIn the last line we used the fact that for a root $\\alpha$ of $(f'c-fc')\/h$ we have $f'(\\alpha)\/c'(\\alpha)=w(\\alpha)$.\n\\end{proof}\n\n\\subsection{Monodromy of rational functions}\\label{sec:monodromy}\nThe definitions and results in this subsection will be used only in Section \\ref{sec:two}.\nLet $k$ be a field of characteristic $p\\geq0$. \nThe {\\em (arithmetic) monodromy} $\\mathrm{Mon}(w)$ of a non-constant rational function $w\\in k(X)$ \nfor which the extension $k(X)\/k(w)$ is separable\nis the Galois group of the Galois closure of $k(X)\/k(w)$.\nWhen ambiguity about the base field may arise we write $\\mathrm{Mon}_k(w)$. \nWe usually view $\\mathrm{Mon}(w)$ as a subgroup of $S_n$ with $n=\\deg w$ via its action on the generic fiber of the corresponding morphism $w\\colon \\P^1\\to\\P^1$.\nA rational function $w\\in k(X)$ is called \\emph{indecomposable} if it cannot be written as a composition $w=u\\circ v$ with $u,v\\in\nk(X)$ where $\\deg u,\\deg v>1$. \n\n\n\\begin{lemma}\\label{lem:indecomposable_primitive}\nLet $w\\in k(X)$ with $k(X)\/k(w)$ separable.\nThen $w$ is indecomposable if and only if\n$\\mathrm{Mon}(w)\\leqslant S_n$ is primitive.\n\\end{lemma}\n\n\\begin{proof}\nIf $w=u\\circ v$ with $\\deg u>1$, $\\deg v>1$, \nand $\\alpha_1,\\dots,\\alpha_m\\in\\overline{k(T)}$ are the roots of $u-T$,\nthen $v^{-1}(\\alpha_1),\\dots,v^{-1}(\\alpha_m)$ is a nontrivial partition of the roots of $w-T$,\nwhich is preserved by $\\mathrm{Mon}(w)$.\nThe converse implication is proven for example in \\cite[Lemma 2]{Fried} in the case of polynomials,\nand the case of rational functions can be proven similarly using L\\\"uroth's theorem.\n\\end{proof}\n\n\\begin{definition}\\label{def:ramification type}\nLet $w\\colon \\P^1\\to\\P^1$ be a tamely ramified rational function defined over an algebraically closed field $k$. \nThe \\emph{ramification type} of $w$ (called \\emph{passport} in \\cite{AdZv15}) is the unordered tuple of partitions $(\\lambda_1,\\ldots,\\lambda_r)$, where $b_1,\\ldots,b_r$ are the distinct branch points of $w$ and $\\lambda_i$ is the partition of $n=\\deg w$ given by the ramification indices over $b_i$.\nWe write partitions of $n$ as $n_1^{e_1}\\dots n_r^{e_r}$ with $0\\leq n_1<\\dots1$ over the algebraically closed field $k$\nwith monodromy group $\\mathrm{Mon}(w)\\leqslant S_n$.\nAssume that $w$ has at most two critical values $a,b$,\nand that $w^{-1}(a)$ contains only one critical point $a'$.\nIf the ramification index of $a'$ is prime, and all ramification over $b$ is \ntame, then $\\mathrm{Mon}(w)$ is primitive.\n\\end{lemma}\n\n\\begin{proof}\nLet $L$ be the Galois closure of the extension $K(X)\/K(T)$ induced by $w$.\nLet $K(T)\\subseteq M\\subsetneq K(X)$ be an intermediate extension. \nSince there is only one ramified prime of $K(X)$ over $T=a$ and it has prime ramification index, \nthe extension $M\/K(T)$ has to be unramified over $T=a$. \nHowever, it is also tamely ramified over $T=b$ and unramified elsewhere, \nhence $M=K(T)$ by the Riemann-Hurwitz formula.\nSo $K(X)\/K(T)$ is a minimal proper extension,\nhence by Galois theory ${\\rm Gal}(L\/K(X))$ is maximal in ${\\rm Gal}(L\/K(T))$,\nwhich means that $\\mathrm{Mon}(w)={\\rm Gal}(L\/K(T))\\leqslant S_n$ is primitive.\n\\end{proof}\n\n\\section{Abelian groups}\n\\label{sec:abelian}\n\\label{sec:generators}\n\n\\noindent\nIn this section we first prove the group-theoretic fact mentioned in Remark \\ref{rem:joachim} \nand then verify Conjecture \\ref{conj} for abelian groups. \n\nIf $G$ is a group and $a\\in G$ is an element we denote by $a^G$ the conjugacy class of $a$. For a subset $S\\subseteq G$ we denote $S^G=\\bigcup_{a\\in S}a^G$.\nRecall that for a finite group $G$ we denote by $p(G)$ the subgroup generated by its $p$-Sylow subgroups.\n A \\emph{cyclic-by-$p$} group is an extension of a cyclic group by a $p$-group. \n\n\\begin{lemma}\\label{lem:gen_by_cyc} \nLet $G$ be a finite group. There exist $x_1,\\ldots,x_r\\in G$ with $r=\\max\\{d(G^\\mathrm{ab}),1\\}$ such that $G=\\langle x_1^G,\\ldots,x_r^G\\rangle$.\\end{lemma}\n\n\\begin{proof} \nThis is a special case of the main theorem in \\cite{Kut76},\nsee also \\cite{Bae64}.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:gen_by_cyc_p} \nLet $G$ be a finite group and $p$ a prime number. There exist subgroups $Q_1,\\ldots,Q_r$ of $G$ such that $r=\\max\\{d((G\/p(G))^\\mathrm{ab}),1\\}$, each $Q_i$ is cyclic-by-$p$ and $G=\\langle Q_1^G,\\ldots,Q_r^G\\rangle$.\n\\end{proposition}\n\n\n\\begin{proof} \nLet $H=G\/p(G)$, so that $r=\\max\\{d(H^\\mathrm{ab}),1\\}$. \nBy Lemma~\\ref{lem:gen_by_cyc} there exist $y_1,\\ldots,y_r\\in H$ with $H=\\langle y_1^H,\\ldots,y_r^H\\rangle$. \nLet $P\\leqslant G$ be a $p$-Sylow subgroup. If $P=1$, i.e. $(|G|,p)=1$, then the claim holds with $Q_i=\\langle y_i\\rangle$, so we may assume $P\\neq 1$. By Frattini's argument \\cite[Theorem 3.7]{Gor80} we have $p(G)N_G(P)=G$, so we can choose $x_1,\\ldots,x_r\\in N_G(P)$ such that $y_i = p(G)x_i$.\nDenote $Q_i=\\langle x_i\\rangle P$. Each $Q_i$ is a subgroup of $G$ since $x_i\\in N_G(P)$, it is cyclic-by-$p$ since it is an extension of $\\langle x_i\\rangle\/(\\langle x_i\\rangle\\cap P)$ by $P$, and finally since\n$p(G)=\\langle P^G\\rangle$ and $G=\\langle p(G),x_1^G,\\ldots,x_r^G\\rangle$ we have $G=\\langle P^G,x_1^G,\\ldots,x_r^G\\rangle=\\langle Q_1^G,\\ldots Q_r^G\\rangle$, as required.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm:abelian}\nConjecture \\ref{conj} holds for $G$ abelian.\n\\end{theorem}\n\n\n\\begin{proof}\n\tLet $q=p^\\nu$ and let $G$ be a nontrivial finite abelian group.\n\tThe claim is that $r_{\\mathbb{F}_q(T)}(G)=1$ if $G$ is a $p$-group and $r_{\\mathbb{F}_q(T)}(G)=d(G\/p(G))$ otherwise.\n\t\n\tWe first prove the lower bound.\n\tLet $L\/\\mathbb{F}_q(T)$ be an abelian Galois extension with Galois group $G$ and $\\mathbb{F}_q$ algebraically closed in $L$.\n\tIf $G$ is a $p$-group, the lower bound $r_{\\mathbb{F}_q(T)}(G)\\geq 1$ follows from the Riemann-Hurwitz formula.\n If $G$ is not a $p$-group, then $G\/p(G)$ is nontrivial, \n and since $r_{\\mathbb{F}_q(T)}(G)\\geqr_{\\mathbb{F}_q(T)}(G\/p(G))$ we can assume that $p(G)=1$. \n Let $P_1,\\ldots, P_k$ be all the finite primes of $\\mathbb{F}_q(T)$ ramified in $L$, and let $I_1,\\ldots, I_k$ be the corresponding inertia groups (note that $I_i$ are well defined since the extension is abelian). Then each $I_i$ is cyclic, as $P_i$ is tamely ramified in $L$. \nAs also the infinite prime of $\\mathbb{F}_q(T)$ is tamely ramified in $L$,\n$G=\\left< I_1,\\ldots, I_k\\right>$, cf.~\\cite[Proposition 4.4.6]{Serre}.\nIn particular, $r_{\\mathbb{F}_q(T)}(G)\\geq k\\geq d(G)$.\n\n\tTo show that the lower bound is tight, we construct a suitable Galois extension. \n\tWrite $G= G\/p(G) \\times p(G)$, let $k$ and $p^\\alpha$ be the exponents of $G\/p(G)$ respectively $p(G)$,\n\tand let $d=\\max\\{d(G\/p(G)),1\\}$. \n\tChoose distinct primes $P_1, \\ldots, P_d$ such that $k(q-1)\\mid q^{\\deg P_i}-1$ (i.e.\\ such that the multiplicative order of $q$ modulo $k(q-1)$ divides $\\deg P_i$). \n\n\tLet $R_i$ be the completion of $\\mathbb{F}_q[T]$ at $P_i$ and $E_i$ its fraction field. \n\tBy \\cite[Proposition~II.5.7]{Neukirch}, $E_i^\\times = \\left<\\pi\\right> \\times(\\mathbb{F}_{q^{\\deg P_i}})^\\times \\times G_i$, where \n\t$\\pi$ is a uniformizer and\n\t\\[\n\t\tG_i = 1+ \\pi\\mathbb{F}_{q^{\\deg P_{i}}}[[\\pi]] \\cong \\mathbb{Z}_p^{\\mathbb{N}}.\n\t\\]\nIn particular, $R_i^\\times = (\\mathbb{F}_{q^{\\deg P_i}})^\\times \\times G_i$. \n\tBy class field theory (take inverse limit in \\cite[Proposition~2.2, Theorem~2.3, and Theorem~3.2]{Hayes}) the maximal abelian extension of $\\mathbb{F}_q(T)$ which is regular over $\\mathbb{F}_q$, tamely ramified at infinity, and unramified outside $\\{P_i,\\infty\\}$ has Galois group $R_i^\\times$ and the tame inertia group at $\\infty$ corresponds to $\\mathbb{F}_{q}^\\times$. Hence $R_i^\\times\/\\mathbb{F}_q^\\times$ is the Galois group of the maximal abelian extension that is regular over $\\mathbb{F}_q$ and unramified outside $\\{P_i\\}$. As $\\mathbb{F}_q(T)$ has no unramified extensions regular over $\\mathbb{F}_q$, those extensions for different $i$ are linearly disjoint, so the Galois group of the maximal abelian extension that is regular over $\\mathbb{F}_q$ and unramified outside $\\{P_1,\\ldots, P_d\\}$ is isomorphic to \n\t\\[\n\t\t\\prod_{i=1}^d R_i^\\times\/\\mathbb{F}_q^\\times \\cong \\prod_{i=1}^d C_{(q^{\\deg P_i}-1)\/(q-1)} \\times \\mathbb{Z}_p^{\\mathbb{N}}.\n\t\\]\n\tBy the assumption that $k\\mid (q^{\\deg P_i}-1)\/(q-1)$, we get that $G\/p(G)$ is a quotient of $ \\prod_i C_{(q^{\\deg P_i}-1)\/(q-1)} $ (this is vacantly true if $G\/p(G)=1$). Obviously, $p(G)$ is a quotient of $\\mathbb{Z}_p^{\\mathbb{N}}$. Hence $G=G\/p(G)\\times p(G)$ is a quotient of $\\prod_i R_i^\\times\/\\mathbb{F}_q^\\times$. This implies we can realize $G$ as the Galois group of a geometric Galois extension of $\\mathbb{F}_q(T)$ unramified outside $\\{P_1,\\ldots, P_d\\}$, as needed.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:lowerbound}\nFor every prime power $q=p^\\nu$ and every nontrivial finite group $G$,\n$$\n r_{\\mathbb{F}_{q}(T)}(G) \\geq \\max\\left\\{d((G\/p(G))^{\\rm ab}),1\\right\\}.\n$$\n\\end{corollary}\n\n\\begin{remark}\nFor $K=\\mathbb{F}_p(T)$, a conjecture \nwas posed in \\cite[Conjecture 1.1]{DeWitt} and claimed to be proven\nfor abelian finite groups,\nnamely that\n$$\n r_{\\mathbb{F}_p(T)}(G) = \\begin{cases} d((G\/p(G))^{\\rm ab})+1,&\\mbox{if }p\\,|\\,|G^{\\rm ab}|\\\\\\max\\{d((G\/p(G))^{\\rm ab}),1\\},\n &\\mbox{otherwise}\\end{cases}.\n$$\nHowever, already \nin the case $G=\\mathbb{Z}\/2p\\mathbb{Z}$, $p>2$,\nthe number obtained there is too big:\nFor example, $G=\\mathbb{Z}\/6\\mathbb{Z}$ can be realized over $\\mathbb{F}_3(T)$ with only one ramified prime of degree $2$,\nfor example by $\\mathbb{F}_3(T,x,y)$ with $x^2=y^3-y=(T^2+1)^{-1}$.\nThe incorrect lower bound in the case $p||G^{\\rm ab}|$ in \\cite{DeWitt} \npresumably comes from the false assumption there that an extension with such group $G$ must be wildly ramified at infinity. \n\n\nIn any case, at least for $q=p$, \\cite[Theorem 2.6]{DeWitt} proves\nConjecture \\ref{conj} for abelian groups of order prime to $p$\n(and thus proves a special case of Theorem~\\ref{thm:abelian}).\nMoreover, \n\\cite[Theorem 3.6]{DeWitt} \nproves Conjecture \\ref{conj}\nfor all semiabelian $\\ell$-groups $G$, with $\\ell$ a prime number different from $p$\n(building on results from \\cite{KisilevskySonn} and \\cite{KisilevskyNeftinSonn}),\nand \\cite[Corollary 6.4]{DeWitt} proves Conjecture \\ref{conj}\nfor all $\\ell$-groups with $\\ell$ a prime number dividing $q-1$\n(both results are stated for $q=p$, but the proofs go through also for general $q$).\n\nWe note however that the proof of \\cite[Theorem 6.9]{DeWitt}, which claims to prove the above conjecture for all nilpotent groups,\nseems to be incomplete regarding both $p$-groups (\\cite[Theorem 2.5]{DeWitt}) \nand $\\ell$-groups with $\\ell$ coprime to $p-1$ (\\cite[Corollary 6.8]{DeWitt}).\n\\end{remark}\n\n\n\n\\section{$S_n$ and $A_n$ for $p>n$}\n\n\\label{sec:tworamified}\n\n\\noindent\nIn this section we obtain Galois extensions of group $S_n$ and $A_n$ with ramification over at most two primes\nin the case where $n$ is smaller than the characteristic $p$.\nWe start with splitting fields of polynomials\nof the form $f(X)-T$ in Section \\ref{sec:Morse}\nand $f(X)-Tc(X)$ in Section \\ref{sec:two},\nwhich produce extensions with one ramified finite prime \nand tame ramification over infinity.\nIn Section \\ref{sec:oneramified}\nwe then explain how to eliminate the tame ramification at infinity\nin many cases.\nFinally, in Section \\ref{sec:twinprimes}\nwe present yet another approach to obtain extensions with two ramified primes,\nby working with trinomials and applying recent results on small gaps between primes in function fields.\n\nWe recall that \nif $f\\in\\mathbb{F}_q[T,X]$ is monic in $X$,\nthen the only primes of $K=\\mathbb{F}_q(T)$ that possibly ramify in \nthe splitting field $L$ of $f$ over $K$\nare the divisors of the discriminant $\\disc_X(f)$, and possibly the infinite prime of $K$ (Lemma~\\ref{lem:Dedekind}).\nIn particular, if $\\disc_X(f)$ is irreducible or, more generally, a prime power, at most two primes of $K$ ramify in $L$.\n\n\\subsection{Two ramified primes via Morse polynomials}\n\\label{sec:Morse}\n\n\nWe start with splitting fields of polynomials of the simple from $f(X)-T$,\nwhere in some cases the classical theory of Morse polynomials\nproduces suitable extensions of group $S_n$ with two ramified primes.\nLet $k$ be a field of characteristic $p\\neq2$,\nand recall that\na polynomial $f\\in k[X]$ of degree $n$ not divisible by $p$ is {\\em Morse} if it has exactly $n-1$ distinct critical values, i.e.\nthe roots $\\alpha_1,\\dots,\\alpha_{n-1}$ of $f'$ in $\\bar{k}$ are simple and\n$f(\\alpha_i)\\neq f(\\alpha_j)$ for $i\\neq j$.\nIf $f$ is Morse, then\n${\\rm Gal}(f(X)-T\/\\overline{k}(T))=S_n$\nby \\cite[Theorem 4.4.5]{Serre},\nso the splitting field of $f(X)-T$ over $k(T)$\nis geometric with Galois group $S_n$.\n\n\\begin{lemma}\\label{lem:f'Morse_disc_irred}\nLet $f\\in k[X]$ be Morse of degree $n$ with $p\\nmid n$. \nThen $f'$ is irreducible if and only if $D(T):=\\disc_X(f(X)-T)\\in k[T]$ is irreducible.\n\\end{lemma}\n\n\\begin{proof}\nLet $A$ be the set of roots of $f'$, and $G={\\rm Gal}(k(A)\/k)$.\nBy (\\ref{eq:disc2}) or Lemma~\\ref{lem:crit}, $D\\sim\\prod_{a\\in A}(T-f(a))$\n(where as before $\\sim$ denotes equality up to a non-zero constant).\nThe fact that $f$ is Morse implies that $f$ is injective on $A$,\nso $f$ induces an isomorphism of $G$-sets $A\\rightarrow f(A)$.\nIn particular, $k(f(a))=k(a)$ for every $a\\in A$,\nhence $D\\sim\\prod_{a\\in A}(T-f(a))$ is irreducible\nif and only if $f'\\sim\\prod_{a\\in A}(X-a)$ is.\n\\end{proof}\n\n\n\n\\begin{proposition}\\label{thm:S_nlargeq}\\label{thm:large_q}\n Let $n\\geq3$ and $q=p^\\nu$ with $p>n$.\n There are $\\frac{q^n}{n-1} + O_n(q^{n-1})$ many \n monic Morse $f\\in\\mathbb{F}_q[X]$ of degree $n$\n such that the splitting field of $f(X)-T$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $S_n$,\n and $\\disc_X(f(X)-T)$ is irreducible.\n\\end{proposition}\n\n\\begin{proof}\nLet $M_n\\cong \\mathbb{A}^n$ be the space of monic polynomials of degree $n$ (identified with the $n$-tuple of non-leading coefficients) considered as a variety over $\\mathbb{F}_q$.\nThe subset $\\mathcal{M}\\subseteq M_n$ of monic Morse polynomials of degree $n$\nis Zariski-open and dense \nwith complement $M_n\\setminus\\mathcal{M}$ the zero set of a polynomial in the coefficients $a_0,\\dots,a_{n-1}$\nof degree $O_n(1)$, see \\cite[Proposition 4.3 and (2) in its proof]{Geyer}.\nThus \nthe elementary estimate \\cite[Lemma 1]{LangWeil} gives that\nthe set $\\mathcal{M}(\\mathbb{F}_q)$ of $f\\in M_n(\\mathbb{F}_q)$ that are Morse\nsatisfies\n$|\\mathcal{M}(\\mathbb{F}_q)|= q^n+O_n(q^{n-1})$.\n\nLet $\\mathcal{P}$ denote the set of $f\\in M_n(\\mathbb{F}_q)$\nfor which $f'$ is irreducible.\nAs $p>n$, the map $M_n(\\mathbb{F}_q)\\rightarrow M_{n-1}(\\mathbb{F}_q)$, $f\\mapsto \\frac{1}{n}f'$\nis surjective with fibers of size $q$,\nso by the Prime Polynomial Theorem,\n$|\\mathcal{P}|=\\frac{q^n}{n-1}+O_n(q^{(n-1)\/2})$.\nIt follows that $|\\mathcal{M}(\\mathbb{F}_q)\\cap\\mathcal{P}|=\\frac{q^n}{n-1} + O_n(q^{n-1})$.\nNow for every $f\\in\\mathcal{M}(\\mathbb{F}_q)$,\nthe splitting field of $f-T$ is geometric with Galois group $S_n$ (see above),\nand\n$\\disc_X(f-T)$ is irreducible\nif and only if $f\\in\\mathcal{P}$ (Lemma~\\ref{lem:f'Morse_disc_irred}),\nso the claim follows.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:f'irred_Morse}\nLet $f\\in\\mathbb{F}_q[X]$ of degree $n$ where $q=p^\\nu$, $p\\nmid n$ and $n-1$ is prime.\nIf $f'$ is irreducible, then $f$ is Morse.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $f'$ is irreducible\nand that there exist $a\\neq b$ in $\\overline{\\mathbb{F}}_q$ with $f'(a)=f'(b)=0$ and $f(a)=f(b)$. \nAs $f'$ is irreducible of degree $n-1$, we have $a,b\\in\\mathbb{F}_{q^{n-1}}$\nand there exists $1\\neq\\sigma\\in{\\rm Gal}(\\mathbb{F}_{q^{n-1}}\/\\mathbb{F}_q)$ with $b=a^\\sigma$.\nThus $c:=f(a)=f(b)=f(a)^\\sigma$ is in the fixed field of $\\sigma$, which is $\\mathbb{F}_q$ due to the assumption that\n$n-1$ is prime.\nSo $f-c\\in\\mathbb{F}_q[X]$ has root $a$, hence $f'|f-c$.\nDeriving gives $f'|f''$, which contradicts the separability of $f'$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:Morse2}\nLet $p\\geq3$ and $q=p^\\nu$. Suppose that $n-1$ is prime and $n\\in\\{2,\\dots,p-1\\}\\cup\\{p+1\\}$.\nThen there exists $f\\in\\mathbb{F}_q[X]$ of degree $n$ such that\nthe splitting field of $f(X)-T$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $S_n$,\nand $\\disc_X(f(X)-T)$ is irreducible.\n\\end{proposition}\n\n\\begin{proof}\nThere exists $f\\in\\mathbb{F}_q[X]$ of degree $n$ with $f'$ irreducible:\nIf $n0$.\n\\begin{enumerate}\n\\item[(i)] Assume that $p\\neq 2$, $m\\equiv n\\pmod 2$ and $g=f'c-fc'$ is squarefree. \nThen $\\mathrm{Mon}(w)=S_n$ or one of the following:\n\\begin{itemize}\n\\item $n=6,m=2,\\mathrm{Mon}(w)=PGL_2(5)$ with ram.~type $(1^22^2,1^22^2,2^3,1^24^1)$.\n\\item $n=8,m=2,\\mathrm{Mon}(w)=PGL_2(7)$ with ram.~type $(1^22^3,1^22^3,1^22^3,1^26^1)$.\n\\item $n=9,m=1,\\mathrm{Mon}(w)=AGL_2(3)$ with ram.~type $(1^32^3,1^32^3,1^32^3,1^18^1)$.\n\\item $n=10,m=2,\\mathrm{Mon}(w)=P\\Gamma L_2(9)$ with ram.~type $(1^42^3,1^42^3,2^5,1^28^1)$.\n\\end{itemize}\n\\item[(ii)] Assume that $p\\notin\\{2,3\\}$, $m\\not\\equiv n\\pmod 2$, and $f'c-fc'=g^2$ where $g\\in k[X]$ is squarefree. \nThen $\\mathrm{Mon}(w)=A_n$ or one of the following:\n\\begin{itemize}\n\\item $n=8,m=1,\\mathrm{Mon}(w)=L_2(7)$ or $A\\Gamma L_1(8)$ with ram.~type $(1^23^2,1^23^2,1^17^1)$.\n\\item $n=9,m=0,\\mathrm{Mon}(w)=P\\Gamma L_2(8)$ with ram.~type $(1^33^2,1^33^2,9^1)$.\n\\item $n=12,m=1,\\mathrm{Mon}(w)=M_{12}$ with ram.~type $(1^33^3,1^33^3,1^111^1)$.\n\\item $n=24,m=1,\\mathrm{Mon}(w)=M_{24}$ with ram.~type $(1^63^6,1^63^6,1^123^1)$.\n\\end{itemize}\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\begin{proof} \n\nThe classification of monodromy groups of indecomposable rational functions with precisely one multiple pole over $\\mathbb{C}$ is given in \\cite[\\S 2.2 and Theorem 12]{AdZv15} (case of 3 critical values) and \\cite[Theorem 1]{Adr17} (case of 4 or more critical values). \nBy Lemma~\\ref{lem:sga} the possible monodromy and ramification type pairs for a tamely ramified rational function over an algebraically closed field in arbitrary characteristic can only be the ones occurring over $\\mathbb{C}$. \nThe lists of possible monodromy groups different from $A_n,S_n,C_n,D_n$ together with the corresponding ramification data appear in \\cite[\\S 3]{AdZv15} and \\cite[Table 1]{Adr17}. We will refer to these henceforth as \\emph{the tables}.\n\nThe case $C_n$ only occurs if $n$ is prime with ram.~type $(n^1,n^1)$,\nand the case $D_n$ occurs only when $n>2$ is prime with ram.~type \n$(1^12^{(n-1)\/2}, 1^12^{(n-1)\/2}, n^1)$, see \\cite[Section 2.2]{AdZv15}\nwhere this can be read off from the corresponding dessins d'enfants. \nIn both cases there is a totally ramified branch point (i.e.\\ after applying a fractional-linear transformation $w$ becomes a polynomial of degree $n$).\n\nConsider the rational map $w=\\frac fc\\colon\\P^1\\to\\P^1$. \nBy our assumptions it has degree $n$, simple poles at the roots of $c$ and a pole of multiplicity $n-m$ at infinity. \nNote that $\\deg(f'c-fc')=n+m-1$ as $m\\not\\equiv n\\pmod p$. \n\\\\\n\n$(i)$.\nAssume that $p\\neq 2$, $m\\equiv n\\pmod 2$ and $g=f'c-fc'$ is squarefree.\nAssume further that $\\mathrm{Mon}(w)\\neq S_n$, in particular $n>2$. \nAs all zeros of $g$ are simple and $p\\neq 2$, \nthe ramification points of $w$ are\nthe zeros of $g$ with ramification index 2 (Lemma~\\ref{lem:crit}) \nand $\\infty$ with ramification index $n-m$. \nIn particular, $w$ is tamely ramified (as $p\\neq 2$ and $p\\nmid n-m$)\nand the ramification type of $w$ has the form \n$$\n \\left(1^{a_1}2^{b_1},\\ldots,1^{a_r}2^{b_r},1^{m}(n-m)^1\\right), \n$$\nwhere $r$ is the number of finite critical values of $w$\nand $a_i+2b_i=n$ for all $i$. \nAs $\\sum_{i=1}^rb_i=\\deg g\\geq n-1>n\/2$, we see that $r\\geq2$.\n\nNote that $\\mathrm{Mon}(w)$ cannot be $A_n$ since the last entry in the ramification type corresponds to an odd permutation, \nit cannot be $C_n$ with $n$ prime since then it would have only one finite critical value and it cannot be $D_n$ with $n$ prime since this case occurs only if $w$ has a totally ramified branch point (i.e. $n^1$ in its ramification type), but if $n>2$ is prime then $m$ must be odd and thus $w$ is not a polynomial. \nTherefore $\\mathrm{Mon}(w)$ must be one of the entries in the tables.\n\nGoing through the tables we see that the only entries with a ramification type as above (with $n-m$ even) are:\n\\begin{itemize}\n\\item {\\bf 4\/6.2} with $\\mathrm{Mon}(w)=PGL_2(5)$ and ramification type $(1^22^2,1^22^2,2^3,1^24^1)$.\n\\item {\\bf 4\/8.4} with $\\mathrm{Mon}(w)=PGL_2(7)$ and ramification type $(1^22^3,1^22^3,1^22^3,1^26^1)$.\n\\item {\\bf 4\/9.1} with $\\mathrm{Mon}(w)=AGL_2(3)$ and ramification type $(1^32^3,1^32^3,1^32^3,1^18^1)$.\n\\item {\\bf 4\/10.1} with $\\mathrm{Mon}(w)=P\\Gamma L_2(9)$ and ramification type $(1^42^3,1^42^3,2^5,1^28^1)$.\n\\end{itemize}\n\n$(ii)$.\nNow assume instead that $p\\notin\\{2,3\\}$, $m\\not\\equiv n\\pmod 2$ and $f'c-fc'=g^2$ with $g$ squarefree.\nAs all zeros of $g^2$ have multiplicity $2$ and $p\\notin\\{2,3\\}$, \nthe ramification points of $w$ are\nthe zeros of $g$ with ramification index 3 (Lemma~\\ref{lem:crit}) \nand $\\infty$ with ramification index $n-m$. \nIn particular, $w$ is tamely ramified (as $p\\neq 3$ and $p\\nmid n-m$) and has a ramification type of the form\n$$\n \\left(1^{a_1}3^{b_1},\\ldots,1^{a_r}3^{b_r},1^{m}(n-m)^1\\right), \n$$\nwhere $r$ is the number of finite critical values of $w$. \nSince now $n-m$ is odd, all entries in the ramification type correspond to even permutations.\nSo since $\\mathrm{Mon}(w)$ is generated by the inertia groups, $\\mathrm{Mon}(w)\\leqslant A_n$, which excludes the case $S_n$. \nIf $n=3$ (and $m=0$), then $r=1$ and $\\mathrm{Mon}(w)=A_3=C_3$.\nIf $n>3$, then similarly to part (i) one can argue that $r\\geq2$\nand that one can exclude the cases $C_n$ and $D_n$ ($D_n$ comes with ramification index 2 at the finite critical points).\n\nNow once again we go over the tables and list the entries with a ramification type of the above form (with $n-m$ odd):\n\\begin{itemize}\n\\item {\\bf 8.1} with $\\mathrm{Mon}(w)=A\\Gamma L_1(8)$ and ramification type $(1^23^2,1^23^2,1^17^1)$.\n\\item {\\bf 8.9} with $\\mathrm{Mon}(w)=L_2(7)$ and ramification type $(1^23^2,1^23^2,1^17^1)$.\n\\item {\\bf 9.7} with $\\mathrm{Mon}(w)=P\\Gamma L_2(8)$ and ramification type $(1^33^2,1^33^2,9^1)$.\n\\item {\\bf 12.10} with $\\mathrm{Mon}(w)=M_{12}$ and ramification type $(1^33^3,1^33^3,1^111^1)$.\n\\item {\\bf 24.5} with $\\mathrm{Mon}(w)=M_{24}$ and ramification type $(1^63^6,1^63^6,1^123^1)$.\\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\begin{remark} \nWhile the above proposition relies on the CFSG in general, \nif we assume that $n-m$ is small, much more elementary group theory is sufficient. \nFor example if $n-m\\le 15$ the result can be derived from the classification by Manning of primitive groups of class $\\le 15$ (see \\cite[\\S II.15]{Wie64} for the list of references).\n\\end{remark}\n\n\n\n\n\n\\begin{lemma}\\label{lem:indec} \nLet $w\\colon\\P^1\\to\\P^1$ be a tamely ramified rational function defined over a field $k$. \nAssume that $w$ has a multiple pole at $\\infty$ and all other poles over $\\overline{k}$ are simple. Assume further that the numerator $g\\in k[X]$ of the derivative $w'$ is irreducible or the square of an irreducible polynomial.\nThen $w$ is indecomposable over $k$.\n\\end{lemma}\n\n\\begin{proof} It will be convenient to introduce two new variable symbols $T,U$ and consider three separate copies of $\\P^1_k$ which we will denote $\\P^1_X,\\P^1_T,\\P^1_U$. \nFor the sake of contradiction, \nassume that $w=v\\circ u$ is the composition of two rational functions\n$u,v\\in k(X)$ with $\\deg u,\\deg v>1$. \nSince $w(\\infty)=\\infty$, assume without loss of generality that $u(\\infty)=\\infty$ and $v(\\infty)=\\infty$.\nConsider the corresponding coverings\n$$\n \\P^1_X\\xrightarrow{u}\\P^1_T\\xrightarrow{v}\\P^1_U.\n$$\nFor a divisor $D=\\sum_P n_PP$ on $\\P^1_X$ we denote \n$$\n D^0=\\sum_{w(P)\\neq\\infty}n_PP,\\quad D^\\infty=\\sum_{w(P)=\\infty}n_PP\n$$ \nand similarly for a divisor $D$ on $\\P^1_T$ we denote\n$$\n D^0=\\sum_{v(P)\\neq\\infty}n_PP,\\quad D^\\infty=\\sum_{v(P)=\\infty}n_PP.\n$$\nConsider the differents $\\mathfrak d_w$, $\\mathfrak{d}_u$, $\\mathfrak{d}_v$ of $w$, $u$ and $v$\nas divisors on $\\P^1_X$, $\\P^1_X$ and $\\P^1_T$, respectively.\nThe Riemann-Hurwitz formula implies that\n$\\deg\\mathfrak{d}_u=2\\deg u-2$ and\n$\\deg\\mathfrak{d}_v=2\\deg v-2$.\nSince all finite poles of $w$ are simple,\nthe same holds for the finite poles of $u$ and $v$.\nMoreover, since $w$ is tamely ramified, so are $u$ and $v$,\nwhich therefore each ramify over at least two geometric primes.\nThus, $\\mathfrak{d}_u^0> 0$ and $\\mathfrak{d}_v^0>0$.\nBy the basic properties of differents we have \n$\\mathfrak d_w=u^*\\mathfrak d_v+\\mathfrak d_u$.\nTherefore, as all divisors in this relation are effective,\n\\begin{equation}\\label{eq:indec1}\n \\mathfrak d_w^0=u^*\\mathfrak d_v^0+\\mathfrak d_u^0.\n\\end{equation}\nNote that $\\mathfrak d_w^0$ is precisely the zero divisor of the numerator of $w'$,\nwhich by assumption is of the form\n$\\mathfrak d_w^0=Q$ or $\\mathfrak d_w^0=2Q$ for some prime divisor $Q$.\nThe first case is already excluded by (\\ref{eq:indec1}), \nso assume that $\\mathfrak d_w^0=2Q$. \nThen (\\ref{eq:indec1}) gives that\n$u^*\\mathfrak d_v^0=\\mathfrak d_u^0=Q$.\nWe have\n$$\n (\\deg u)(\\deg\\mathfrak d_v^0)=\\deg u^*\\mathfrak{d}_v^0=\\deg Q=\\deg \\mathfrak d_u^0\\le\\deg\\mathfrak d_u=2\\deg u-2,\n$$\nwhich implies $(\\deg u)(\\deg\\mathfrak{d}_v^0-2)\\leq -2$,\nhence $\\deg u=2$ and $\\deg\\mathfrak d_v^0=1$. \nSince $\\deg\\mathfrak{d}_v^0+\\deg\\mathfrak{d}_v^\\infty=2\\deg v-2$, the latter shows that $\\mathfrak d_v^\\infty>0$.\nSince $\\deg\\mathfrak{d}_u^0+\\deg\\mathfrak{d}_u^\\infty=2\\deg u-2=2$\nand $\\deg\\mathfrak{d}_u^0=\\deg Q=(\\deg u)(\\deg\\mathfrak{d}_v^0)\\geq 2$ it also follows that $\\mathfrak{d}_u^\\infty=0$.\nThus $v$ ramifies over $\\infty$ but $u$ does not,\nwhich implies that $w=v\\circ u$ has $\\deg u=2$ many\nmultiple poles over $\\overline{k}$, contradicting our assumption.\n\\end{proof}\n\n\\begin{proposition} \\label{prop:monfq}\nLet $p$ be prime, $q$ a power of $p$, and $m\\ge 0$ and $n\\ge m+2$ integers with $n\\not\\equiv m\\pmod p$. \nLet $f,c\\in\\mathbb{F}_q[X]$ be polynomials with ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m$, $(f,c)=1$ and $c$ squarefree.\nLet $w=\\frac{f}{c}$.\n\\begin{enumerate}\n\\item[(i)] Assume $p>2$, $m\\equiv n\\pmod 2$, $m\\neq 2$ and $g=f'c-fc'$ is irreducible. Then $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$ or \n\\begin{itemize}\n\\item $n=9,m=1$, $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=AGL_2(3)$ with ram.~type $(1^32^3,1^32^3,1^32^3,1^18^1)$\n\\end{itemize}\n\n\\item[(ii)] Assume $p>3$, $m\\not\\equiv n\\pmod 2$, and $f'c-fc'=g^2$ where $g\\in \\mathbb{F}_q[X]$ is irreducible. Then $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$ or one of the following:\n\\begin{itemize}\n\\item $n=8,m=1,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A\\Gamma L_1(8)$ or $L_2(7)$ and ram.~type $(1^23^2,1^23^2,1^17^1)$.\n\\item $n=9,m=0,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=P\\Gamma L_2(8)$ and ram.~type $(1^33^2,1^33^2,9^1)$.\n\\item $n=12,m=1,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=M_{12}$ and ram.~type $(1^33^3,1^33^3,1^111^1)$.\n\\item $n=24,m=1,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=M_{24}$ and ram.~type $(1^63^6,1^63^6,1^123^1)$.\n\\end{itemize}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} \nNote that $\\mathrm{Mon}_{\\overline{\\F}_q}(w)\\unlhd\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ are transitive\nsubgroups of $S_n$.\nBy Lemma~\\ref{lem:indec}, $w$ is indecomposable over $\\mathbb{F}_q$,\nhence $\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ is primitive (Lemma \\ref{lem:indecomposable_primitive}),\nand Lemma~\\ref{lemcycle}(ii) applied to the infinite prime\nshows that $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ contains an $(n-m)$-cycle.\nWe treat parts $(i)$ and $(ii)$ simultaneously but distinguish several cases according to the value of $m$:\n\nCase $m=0$: In this case $w$ \nis a polynomial. By \\cite[Theorem 3.5]{FrMa69} a polynomial which is indecomposable over $\\mathbb{F}_q$ of degree coprime with $p$ is also indecomposable over $\\overline{\\F}_q$ and therefore in our case $w$ is indecomposable over $\\overline{\\F}_q$. \nThe conditions of Proposition~\\ref{prop:mon}(i) resp.~(ii) hold and therefore \n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$ (none of the exceptional cases has $m=0$) resp.~$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$ or the exceptional case with $m=0$ appearing in Proposition~\\ref{prop:mon}(ii).\n\nCase $m=1$:\n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is transitive and contains an $(n-1)$-cycle and is therefore primitive, hence $w$ is indecomposable over $\\overline{\\F}_q$ (Lemma \\ref{lem:indecomposable_primitive}). The conditions of Proposition~\\ref{prop:mon}(i) resp.~(ii) hold and therefore $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$ or the exceptional case with $m=1$ in Proposition~\\ref{prop:mon}(i) resp.~$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$ or one of the exceptional cases with $m=1$ listed in Proposition~\\ref{prop:mon}(ii).\n\nCase $m=2$: Note that this case is allowed only in part $(ii)$. \nSince $n\\ge 5$ is odd by assumption, $(n-2)$ is coprime with $n$ and so $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ must be primitive since it is transitive and contains an $(n-2)$-cycle. Hence $w$ is indecomposable over $\\overline{\\F}_q$ (Lemma~\\ref{lem:indecomposable_primitive}). The conditions of Proposition~\\ref{prop:mon}(ii) hold and therefore $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$.\n\nCase $m>2$: Here $\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ is primitive and contains an $(n-m)$-cycle, so by Theorem~\\ref{thm:jones}(i) it is $S_n$ or $A_n$.\nNote that $n\\geq m+2\\geq 5$, so $A_n$ is simple and $S_n$ has a unique nontrivial cyclic quotient.\nNow $\\mathrm{Mon}_{\\overline{\\F}_q}(w)\\unlhd\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ and $\\mathrm{Mon}_{\\mathbb{F}_q}(w)\/\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is cyclic,\nhence also $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is $S_n$ or $A_n$.\nUnder the assumptions of part $(i)$, the $(n-m)$-cycle is odd, and thus $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$.\nUnder the assumptions of part $(ii)$, the $(n-m)$-cycle is even, as are all the other inertia subgroups\n(the ramification indices at the finite ramified primes all equal $3$),\nso since $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is generated by the inertia subgroups,\n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$.\n\\end{proof}\n\nFor a squarefree polynomial $c\\in\\mathbb{F}_q[X]$ we denote \n$$\n \\mathcal{H}_c=\\left\\{R^p: R\\in\\left(\\mathbb{F}_q[X]\/c^2\\mathbb{F}_q[X]\\right)^\\times\\right\\}\\leqslant\\left(\\mathbb{F}_q[X]\/c^2\\mathbb{F}_q[X]\\right)^\\times.\n$$\nDenote also \n\\begin{equation}\\label{eq:def_pi_c}\n \\pi_c=\\prod_{i=1}^m(X-\\alpha_i)^2\\cdot\\sum_{i=1}^m\\frac 1{(X-\\alpha_i)^2}=\\sum_{i=1}^m\\prod_{j\\neq i}(X-\\alpha_j)^2\\in\\mathbb{F}_q[X],\n\\end{equation}\nwhere $\\alpha_1,\\dots,\\alpha_m\\in\\overline{\\F}_q$ are the roots of $c$. Note that $(\\pi_c,c)=1$.\nIn short we write $\\pi_c\\mathcal{H}_c$ for the coset $(\\pi_c+c^2\\mathbb{F}_q[X])\\mathcal{H}_c$ in $(\\mathbb{F}_q[X]\/c^2)^\\times$.\n\n\\begin{lemma} \\label{lem:der} \nLet $c\\in\\mathbb{F}_q[X],\\deg c=m$ be a squarefree polynomial and $g\\in\\mathbb{F}_q[X]$ another polynomial such that $(g,c)=1$ and $\\deg g\\ge 2m$. \nDenote $n=\\deg(g)-m+1$ and assume that $p=\\mathrm{char}(\\mathbb{F}_q)>n-m$. \nAssume further that $g\\bmod c^2\\in\\pi_c\\mathcal H_c$. \nThen there exists $f\\in\\mathbb{F}_q[X],\\deg f=n,(f,c)=1$ such that $g=f'c-fc'$.\n\\end{lemma}\n\n\\begin{proof} \nWithout loss of generality, $c$ is monic.\nWrite $g=uc^2+v,u,v\\in\\mathbb{F}_q[X],\\deg v<\\deg c^2=2m$. Let $\\alpha_1,\\ldots,\\alpha_m\\in\\overline{\\F}_q$ be the \nroots of $c$ (they are pairwise distinct since $c$ is squarefree). By assumption $v\\equiv\\pi_cw\\pmod{c^2}\n$, where $w$ is a $p$-th power modulo each $(X-\\alpha_i)^2$, the latter condition implying $w\\equiv \na_i\\pmod {(X-\\alpha_i)^2}$ for a (uniquely determined) $0\\neq a_i\\in\\overline{\\F}_q$. \nIt follows using (\\ref{eq:def_pi_c}) that $v\\equiv a_i\\prod_{j\\neq i}(X-\\alpha_j)^2\\pmod{(X-\\alpha_i)^2}$ for each $1\\le i\\le m$,\nhence $v\\equiv \\sum_{i=1}^ma_i\\prod_{j\\neq i}(X-\\alpha_j)^2\\pmod{c^2}$. \nSince both sides of this congruence are of degree less than $\\deg c^2$,\nwe get the following\npartial fraction decomposition over $\\overline{\\F}_q$:\n$$\n \\frac v{c^2}=\\sum_{i=1}^m\\frac{a_i}{(X-\\alpha_i)^2}.\n$$ \nWe therefore can write\n$$\n -\\sum_{i=1}^m\\frac{a_i}{X-\\alpha_i}=\\frac\\psi c\\quad\\mbox{ with }\\quad\\psi\\in\\overline{\\F}_q[X],\\deg\\psin-m$. \nLet $c\\in\\mathbb{F}_q[X],\\deg c=m$ be squarefree.\n\\begin{enumerate}\n\\item[(i)] Assume that $q^{\\frac {n-m-1}2}>2m+1$. \nThen there exists $f\\in\\mathbb{F}_q[X]$, $\\deg f=n$, $(f,c)=1$, such that $g=f'c-fc'$ is irreducible in $\\mathbb{F}_q[X]$.\n\\item[(ii)] Assume $n\\not\\equiv m\\pmod 2$ and $q^{\\frac {n-3m-1}4}>2m+1$. Then there exists $f\\in\\mathbb{F}_q[X]$, $\\deg f=n$, $(f,c)=1$, such that $f'c-fc'=g^2$ with $g\\in\\mathbb{F}_q[X]$ irreducible.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} \nIn case (i) let $N=n+m-1$ and $\\psi_c=\\pi_c$;\nin case (ii) let $N=\\frac{n+m-1}{2}$ and choose $\\psi_c\\in\\mathbb{F}_q[X]$ with $\\psi_c^2\\equiv\\pi_c\\pmod{c^2}$.\nThe latter exists since $\\pi_c$ is a square modulo $c$ (e.g.~$\\pi_c\\equiv\\delta^2\\pmod c$ with\n$\\delta=\\prod_{i=1}^m(X-\\alpha_i)\\cdot\\sum_{i=1}^m\\frac 1{(X-\\alpha_i)}$)\nand the kernel of $(\\mathbb{F}_q[X]\/c^2)^\\times\\rightarrow(\\mathbb{F}_q[X]\/c)^\\times$\nhas order $q^m$, which is odd.\n\nBy Lemma~\\ref{lem:der} it suffices to show that there is a monic irreducible polynomial $g\\in\\mathbb{F}_q[X]\n$ with $\\deg g=N$ such that $g\\bmod c^2\\in\\mathcal \\psi_c\\mathcal{H}_c$ (for case $(ii)$ note that $g\\bmod c^2\\in\n\\psi_c\\mathcal{H}_c$ implies $g^2\\bmod c^2\\in\\pi_c\\mathcal{H}_c$).\nDenote by $\\mathcal H_c^\\perp$ the group of Dirichlet characters modulo $c^2$ that are trivial on $\\mathcal H_c$ (this is the orthogonal group of $\\mathcal H_c$). \nWe have $|\\mathcal H_c^\\perp|=[(\\mathbb{F}_q[X]\/c^2)^\\times:\\mathcal H_c]=q^m$. \nBelow when we sum over $g$ we will always restrict $g$ to be monic,\nand $\\Lambda$ denotes the polynomial von Mangoldt function, \ni.e.~$\\Lambda(g)=\\deg P$ if $g=P^\\nu$ for a monic irreducible $P\\in\\mathbb{F}_q[X]$ and $\\nu\\geq1$, and $\\Lambda(g)=0$ otherwise. \nBy the second orthogonality relation \\cite[Corollary on p. 63]{Ser73} applied to the group $\\left(\\mathbb{F}_q[X]\/c^2\\right)^\\times\/\\mathcal H_c$ (the dual of this group is can be identified with $\\mathcal H_c^\\perp$) and the element $g\/\\psi_c$ we have\n\n$$\\sum_{\\chi\\in\\mathcal H_c^\\perp}\\overline{\\chi(\\psi_c)}\\chi(g)=\\left[\\begin{array}{ll}q^m,&\\mbox{if } g\\bmod c^2\\in\\psi_c\\mathcal H_c,\\\\\n0,&{\\mathrm {otherwise}},\\end{array}\\right.$$ and hence\n\n$$\n \\sum_{\\deg g=N\\atop{g \\bmod c^2\\in\\psi_c\\mathcal H_c}}\\Lambda(g)=\n\\frac 1{q^m}\\sum_{\\chi\\in\\mathcal H_c^\\perp}\\overline{\\chi(\\psi_c)}\\sum_{\\deg g=N}\\chi(g)\\Lambda(g).\n$$\nLet $\\chi_1$ denote the trivial character modulo $c^2$.\nFor all $\\chi\\neq\\chi_1$,\nWeil's Riemann Hypothesis for function fields gives that\\footnote{For example using the notation of \\cite[p.~41-42]{Ros02}:\n$\\sum_{\\deg g=N}\\chi(g)\\Lambda(g)=c_N(\\chi)=-\\sum_{k=1}^{M-1}\\alpha_k(\\chi)^N$\nwhere $M=\\deg c^2$ and $|\\alpha_k(\\chi)|\\leq\\sqrt{q}$.}\n$$\n \\left|\\sum_{\\deg g=N}\\chi(g)\\Lambda(g)\\right|\\le (2m-1)q^{\\frac{N}2}.\n$$\nThus together with the\nelementary relation $\\sum_{\\deg g=N}\\Lambda(g)=q^N$ (see \\cite[Proposition 2.1]{Ros02}) we obtain\n\\begin{multline*}\n\\sum_{\\deg g=N\\atop{g \\bmod c^2\\in\\psi_c\\mathcal H_c}}\\Lambda(g)\\ge\n\\frac{1}{q^m}\\sum_{\\deg g=N\\atop{(g,c)=1}}\\Lambda(g)-\\frac{q^m-1}{q^m}(2m-1)q^{\\frac{N}2}\\ge \n q^{N-m}-1-(2m-1)q^{\\frac{N}2}.\n\\end{multline*}\nBy the same elementary relation,\n$$\n \\sum_{\\deg g=N\\atop{g\\bmod c^2\\in\\mathcal \\psi_cH_c\\atop{g\\,\\mathrm{not\\,irreducible}}}}\\Lambda(g)\\le\\sum_{d|N\\atop{d\\neq N}}q^d\\le 2q^{\\frac{N}2}-1,\n$$ \n and therefore as long as\n$$\n q^{N-m}>(2m+1)q^{\\frac{N}2}\n$$ \nthere must be at least one irreducible $g$ with $\\deg g=N$ and $g\\bmod c^2\\in\\mathcal \\psi_c\\mathcal H_c$.\nTo conclude the proof note that $\\frac{N}{2}-m$ equals $\\frac{n-m-1}{2}$ in case $(i)$\nand $\\frac{n-3m-1}{4}$ in case $(ii)$.\n\\end{proof}\n\n{\n\\allowdisplaybreaks\n\n\\begin{theorem}\\label{thm:tame2} \nLet $p$ be a prime, $q$ a power of $p$, $0\\le m\\le n-2$ integers such that $n-m2$,\n\\item $m\\equiv n\\pmod 2$ and $m\\neq 2$, \n\\item $(n,m)\\neq (9,1)$, and\n\\item $q^{\\frac{n-m-1}2}>2m+1$,\n\\end{enumerate}\nor\n\\item[(ii)] $G=A_n$ and the following conditions hold\n\\begin{enumerate}\n\\item $p>3$,\n\\item $m\\not\\equiv n\\pmod 2$ and $(n,p)=1$,\n\\item $(n,m)\\neq (8,1),(9,0),(12,1),(24,1)$,\n\\item $q^{\\frac{n-3m-1}4}>2m+1$, and\n\\item $m\\ge 2$ or $\\left(\\frac q{n-m}\\right)=1$.\n\\end{enumerate}\n\\end{enumerate}\nThen there exist $f,c\\in\\mathbb{F}_q[X]$ with ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m$, and $c$ squarefree\nsuch that the splitting field of $f(X)-Tc(X)$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $G$\nand is ramified\nover only one finite prime $\\mathcal F$,\nwith $\\disc_X(f-Tc)$ a power of $\\mathcal F$ and ${\\rm deg}(\\disc_X(f-Tc))=n+m-1$.\nIn particular, $r_{\\mathbb{F}_q(T)}(G)\\leq 2$.\n\\end{theorem}\n\n\\begin{proof} \n\n$(i)$. First assume that $G=S_n$, $p>2$, $m\\neq 2$, $n\\equiv m\\pmod 2$, $(n,m)\\neq (9,1)$ and $q^{\\frac{n-m-1}2}>2m+1.$ \nTake any squarefree $c\\in\\mathbb{F}_q[X],\\deg c=m$. \nBy Proposition~\\ref{prop:g}(i) one can find an $f\\in\\mathbb{F}_q[X],\\deg f=n$ such that $g=f'c-fc'$ is irreducible of degree $n+m-1$. By Proposition~\\ref{prop:monfq}(i) the splitting field $K$ of $f(X)-Uc(X)$ over $\\mathbb{F}_q(U)$ is geometric with Galois group $S_n$. \nLet $w=\\frac{f}{c}\\in\\mathbb{F}_q(X)$.\n\nConsider the discriminant $D(U)=\\disc_X(f(X)-Uc(X))$. Let \n$$\n \\alpha_1,\\alpha_2=\\alpha_1^q,\\ldots,\\alpha_{n+m-1}=\\alpha_1^{q^{n+m-2}}\\in\\mathbb{F}_{q^{n+m-1}}\n$$ \nbe the roots of $g$. \nThen by Lemma~\\ref{lem:crit} the roots of $D$ are ${w(\\alpha_i)},1\\le i\\le n+m-1$ (including multiplicity). The \nFrobenius map ${\\rm Fr}_q$ acts cyclically on ${w(\\alpha_i)}$ and hence $D=\\mathcal{F}^r$ is a \npower of some prime $\\mathcal{F}\\in\\mathbb{F}_q[U]$.\nBy Lemma~\\ref{lem:crit}, $w$ is ramified only over $\\mathcal F$ and (possibly) $\\infty$, which concludes the proof in the case $G=S_n$.\n\n$(ii)$.\nNow assume that $G=A_n$, $p>3$, $n\\not\\equiv m\\pmod 2$, $(n,m)\\neq (8,1)$, $(9,0)$, $(12,1)$, $(24,1)$ and $q^{\\frac{n-3m-1}4}>2m+1$.\nLet $c\\in\\mathbb{F}_q[X],\\deg c=m$ be a monic squarefree polynomial. \nBy Proposition~\\ref{prop:g}(ii) there exists a monic irreducible $g\\in\\mathbb{F}_q[X]$ and a monic $f\\in\\mathbb{F}_q[X],\\deg f=n$ such that $f'c-fc'=(n-m)g^2$ (we can always adjust the leading coefficients this way). \nNote that $\\deg f'=n-1$ since $(n,p)=1$.\nLet $w=\\frac{f}{c}\\in\\mathbb{F}_q(X)$ and let $K$ be the splitting field of $f(X)-Uc(X)$ over $\\mathbb{F}_q(U)$.\nBy Proposition~\\ref{prop:monfq}(ii),\n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$, hence ${\\rm Gal}(K\/\\mathbb{F}_q(U))=\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ is either $A_n$ or $S_n$ and $K\/\\mathbb{F}_q(U)$ is geometric if it is $A_n$, which happens precisely when $D(U)=\\disc_X(f(X)-Uc(X))$ is a square in $\\mathbb{F}_q(U)$ (Lemma~\\ref{lem:discgalois}).\n\nTo shorten notation, we omit the variable $X$ in the following calculation. All discriminants and resultants are with respect to the variable $X$. Terms which are squares in $\\mathbb{F}_q(U)$ are of no consequence and are simply denoted by $\\square$.\nIf $\\deg c\\ge 1$ then using the properties of resultants and discriminants \nwe calculate:\n\\begin{eqnarray*}\nD(U)&\\stackrel{(\\ref{eq:disc})}=&(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(f'-Uc',f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=&(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,f-Uc)^{-1}\\mathrm{Res}(f'c-Uc'c,f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resalt})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,f)\\mathrm{Res}(f'c-fc'+c'(f-Uc),f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resalt}),(\\ref{eq:ressym})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,f)\\mathrm{Res}(f'c-fc',f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult}),(\\ref{eq:resdefinition})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,c')\\mathrm{Res}(c,c'f)(n-m)^n\\prod_{g(\\alpha)=0}(f(\\alpha)-Uc(\\alpha))^2\\\\\n&\\stackrel{(\\ref{eq:resalt})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,c')\\mathrm{Res}(c,c'f-cf')(n-m)^n\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,c')\\mathrm{Res}(c,m-n)\\mathrm{Res}(c,g)^2(n-m)^n\\\\\n&\\stackrel{(\\ref{eq:resdefinition})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2+m}\\mathrm{Res}(c,c')(n-m)^{n+m}\\\\\n&\\stackrel{(\\ref{eq:disc}),(\\ref{eq:ressym})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2+\\frac{m(m+1)}2}(n-m)\\disc(c).\n\\end{eqnarray*}\nIf $c=1$ a similar calculation gives the same final expression for $D(U)$.\n\nTherefore, $D(U)$ is a square in $\\mathbb{F}_q(U)$ if and only if \n$$\n \\delta:=(-1)^{\\frac{n(n-1)}2+\\frac{m(m+1)}2}(n-m)\\disc(c)\n$$ \nis a square in $\\mathbb{F}_q$.\nIf $m<2$, then $\\disc(c)=1$, and\nby quadratic reciprocity,\n$$\n \\left(\\frac{(-1)^{\\frac{n(n-1)}{2}+m}(n-m)}{p}\\right)\n =(-1)^{\\frac{p-1}{4}(n^2+m-1)}\\left(\\frac{p}{n-m}\\right)\n =\\left(\\frac{p}{n-m}\\right)\n$$\nas $n^2+m-1\\equiv 0\\pmod 4$ if either $m=0$ and $n\\equiv1\\pmod 2$, or $m=1$ and $n\\equiv0\\pmod 2$,\nthus $\\delta$ is a square in $\\mathbb{F}_q$ if and only if $\\left(\\frac q{n-m}\\right)=1$, which is precisely our assumption \n(e) for this case.\nIf $m\\geq2$, then\n$\\disc(c)$ is a square in $\\mathbb{F}_q$ if and only if\n$(-1)^m\\mu(c)=1$ where $\\mu(c)$\nis the polynomial M\\\"obius function, see \\cite[Lemma 4.1]{Conrad}.\nTherefore we can choose $c$ accordingly with an odd or an even number of irreducible factors\nto get the value we need so that $\\delta$ is a square in $\\mathbb{F}_q$. \n\nThis shows that $K\/\\mathbb{F}_q(U)$ is geometric with Galois group $A_n$.\nArguing as in $(i)$ one sees that $D=\\mathcal F^r$ with $\\mathcal F$ prime and that $w$ is ramified only over $\\mathcal F$ and possibly $\\infty$.\n\\end{proof}\n\n}\n\n\\subsection{Eliminating tame ramification at infinity}\n\\label{sec:oneramified}\n\nWe will now explain how to eliminate\nthe tame ramification at infinity of the extensions obtained in the previous subsections,\nconditional on a weak consequence of the function field analogue of Schinzel's hypothesis H\nthat can actually be proven in many cases.\nThe classical Hypothesis H states the following:\n\n\\begin{conjecture}[Schinzel's hypothesis H]\\label{conj:Schinzel}\nLet $f_1,\\dots,f_r\\in\\mathbb{Z}[X]$ irreducible.\nIf $f_1\\cdots f_r$ is not the zero function modulo any prime number $p$,\nthen there exist infinitely many $n\\in\\mathbb{Z}$ with $f_1(n),\\dots,f_r(n)$ simultaneously prime.\n\\end{conjecture}\n\nIn the function field setting, \nthe naive analogue fails due to inseparability issues, \nsee \\cite{CCG}.\nTaking this into account,\none conjectures:\n\n\\begin{conjecture}\\label{conj:SchinzelFF}\nLet $\\mathcal{F}_1,\\dots,\\mathcal{F}_r\\in\\mathbb{F}_q[T][X]$ irreducible and separable in $X$.\nIf $\\mathcal{F}_1\\cdots \\mathcal{F}_r$ is not the zero function modulo any prime polynomial $P\\in\\mathbb{F}_q[T]$,\nthen there exist infinitely many $h\\in\\mathbb{F}_q[T]$ with $\\mathcal{F}_1(h),\\dots,\\mathcal{F}_r(h)$ simultaneously irreducible in $\\mathbb{F}_q[T]$.\n\\end{conjecture}\n\nFor partial results on Conjecture \\ref{conj:SchinzelFF} see for example\n\\cite{Pollack,BS12,EntinBH,Entin,SawinShusterman}.\nWe will work with the following weaker assumption:\n\n\\begin{definition} \nWe say that property $H(q,d,e)$ holds if for any irreducible $\\mathcal{F}\\in\\mathbb{F}_q[X]$ with $\\deg(\\mathcal{F})|d$ there exists $h\\in\\mathbb{F}_q[T]$ with $e|\\deg(h)$ such that \n$\\mathcal{F}(h)$ is irreducible in $\\mathbb{F}_q[T]$. \n\\end{definition}\n\n\nWe write ${\\rm rad}(e)$ for the radical of $e$\nand $\\omega(e)$ for the number of distinct prime divisors of $e$.\nWe define ${\\rm rad}'(e)=\\frac{\\gcd(4,e)}{\\gcd(2,e)}{\\rm rad}(e)$,\ni.e.~${\\rm rad}'(e)$ equals $2{\\rm rad}(e)$ if $4|e$ \nand ${\\rm rad}(e)$ otherwise.\nWe denote by $v_\\ell$ the $\\ell$-adic valuation.\n\n\\begin{proposition}\\label{lem:Hqdm}\n$H(q,d,e)$ holds in each of the following cases:\n\\begin{enumerate}\n\\item Conjecture \\ref{conj:SchinzelFF} holds for $\\mathbb{F}_q$.\n\\item $q$ is sufficiently large with respect to $d$ and $e$.\n\\item ${\\rm rad}'(e)|q-1$ and $\\gcd(d,e)=1$.\n\\item ${\\rm rad}'(e)|q^\\ell-1$ for every prime divisor $\\ell$ of $d$, and $q\\geq(d-1)^2(2^{\\omega(e)}-1)^2$.\n\\item ${\\rm rad}(e)|q-1$ and $q\\geq(2^{\\max\\{v_2(e)-1,0\\}}d-1)^2(2^{\\omega(e)}-1)^2$.\n\\end{enumerate}\nMore precisely, \nfor any irreducible $\\mathcal{F}\\in\\mathbb{F}_q[X]$ with $\\deg(\\mathcal{F})|d$ there exists $h\\in\\mathbb{F}_q[T]$ with $e|\\deg(h)$ such that \n$\\mathcal{F}(h)$ is irreducible in $\\mathbb{F}_q[T]$,\nwhere\n\\begin{enumerate}[(a)]\n\\item there exists such $h$ with ${\\rm deg}(h)=e$ in cases (2), (3), (4) and (5),\n\\item there exists such $h$ that is monic in cases (1), (2), (4) and (5), and\n\\item there exists such $h$ that is monic and satisfies ${\\rm deg}(h)=e$ in cases (4) and (5), as well as in case (2) if $q$ is odd or $e\\geq 4$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nLet $e,d\\geq 1$ and $\\mathcal{F}\\in\\mathbb{F}_q[X]$ irreducible with ${\\rm deg}(\\mathcal{F})=n|d$ be given.\n\n(1): \nWithout loss of generality assume that $q-1|e$.\nThe polynomial\n$$\n g:=\\mathcal{F}(X^e+A_{e-1}X^{e-1}+\\dots+A_0)\\in\\mathbb{F}_q[A_0,\\dots,A_{e-1},X]\n$$ \nis separable in $X$ and irreducible:\nIndeed, $g$ is obtained from $\\mathcal{F}(A_0)$ by the invertible change of variables $A_0\\mapsto X^e+A_{e-1}X^{e-1}+\\dots+A_0$.\nLet $S$ be the set of primes $P\\in\\mathbb{F}_q[T]$ with ${\\rm deg}(P)\\leq\\log_q{\\rm deg}_Xg$\nand let $M=\\prod_{P\\in S}P$.\nBy \\cite[Theorem 1.1]{BaEn19_} there exist $a_0\\in\\mathbb{F}_q+M\\mathbb{F}_q[T]$ and $a_1,\\dots,a_{e-1}\\in\\mathbb{F}_q[T]$ with\n$\\tilde{g}(X):=g(a_0,\\dots,a_{e-1},X)\\in\\mathbb{F}_q[T][X]$ separable in $X$ and irreducible in $\\mathbb{F}_q(T)[X]$,\nhence also irreducible in $\\mathbb{F}_q[T,X]$ (since it is primitive in $X$).\nIn particular, $\\tilde{g}(0)=\\mathcal{F}(a_0)\\not\\equiv 0\\mbox{ mod }P$ for every $P\\in S$,\nand modulo a prime $P\\in\\mathbb{F}_q[T]$ not in $S$, \n$\\tilde{g}$ has at most ${\\rm deg}_X\\tilde{g}1$\nand note that the assumption implies that ${\\rm rad}'(e)|q^n-1$.\nLet $\\alpha\\in\\mathbb{F}_{q^n}$ be a root of $\\mathcal{F}$.\nCohen's result \\cite[Corollary 2.3]{Cohen} gives that\nthe number of $c\\in\\mathbb{F}_q$ for which $\\alpha+c$ is not an $\\ell$-th power in $\\mathbb{F}_{q^n}$ for any prime divisor $\\ell$ of $e$ is bigger than\n$\\frac{\\phi({\\rm rad}(e))}{{\\rm rad}(e)}(q-(n-1)(2^{\\omega(e)}-1)\\sqrt{q})$,\nin particular this number is positive as soon as\n$q\\geq(n-1)^2(2^{\\omega(e)}-1)^2$, which is satisfied by our assumption.\nAs in (3) we conclude that $T^e-(\\alpha+c)\\in\\mathbb{F}_{q^n}[T]$ is irreducible,\nhence $\\mathcal{F}(h)$ is irreducible for $h=T^e-c$.\n\n(5): Let $s=v_2(e)$. If $s\\leq 1$, then ${\\rm rad}'(e)={\\rm rad}(e)$, so the claim follows from (4).\nIf $s\\geq 2$, then $q$ is odd and we first apply (4) with $e=2$ to obtain a monic $h_1\\in\\mathbb{F}_q[T]$ of degree $2$ with $\\mathcal{F}_1(T):=\\mathcal{F}(h_1(T))$ irreducible of degree $2n$. \nWe iterate this until we obtain $\\mathcal{F}_{s-1}=\\mathcal{F}(h_{1}(\\dots h_{s-1}(T)))$\nirreducible of degree $2^{s-1}n$.\nThen ${\\rm rad}'(e\/2^{s-1})={\\rm rad}(e\/2^{s-1})$,\nso we can apply (4)\nto obtain a monic $h_s$ of degree $e\/2^{s-1}$ with $\\mathcal{F}(h_1(\\dots h_s(T)))$\nirreducible.\nThe polynomial $h=h_1\\circ\\dots\\circ h_s$ has degree $e$ and satisfies the claim.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:disjoint} \nLet $k$ be a perfect field and let $h\\in k[T]$ be a non-constant polynomial. \nDenote $U=h(T)$ and let $K$ be a Galois extension of $k(U)$ ramified only over $U=\\infty$ and the finite primes $\\mathcal{F}_1,\\dots,\\mathcal{F}_r\\in k[U]$, with tame ramification over $\\infty$. \nAssume that each $\\mathcal{F}_i(h(T))\\in k[T]$ is separable. \nThen the extensions $K,k(T)$ of $k(U)$ are linearly disjoint.\n\\end{lemma}\n\n\\begin{proof} \nDenote $D(U)=\\disc_T(h(T)-U)$. \nBy Lemma~\\ref{lem:crit} the finite branching primes of the extension $k(T)\/k(U)$ are exactly the primes dividing $D(U)$.\nIf $\\mathcal{F}_i|D$ for some $i$, then 0 would be a critical value of $\\mathcal{F}_i\\circ h\\colon\\P^1\\to\\P^1$, which means that $\\mathcal{F}_i(h)$ is not separable, contradicting our assumption. Thus $\\mathcal F_i\\nmid D$ for each $i$,\nso the finite branching loci of $k(T)\/k(U)$ and $K\/k(U)$ are disjoint,\nhence the extension $K\\cap k(T)\/k(U)$ is tamely ramified and ramified only over $\\infty$, and is therefore trivial.\nSince $K\/k(U)$ is Galois, this already implies that $K$ and $k(T)$ are linearly disjoint over $k(U)$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:eliminateinfty}\nLet $k$ be a perfect field of characteristic $p\\geq 0$, and let $f\\in k[U,X]$.\nAssume that the splitting field of $f$ over $k(U)$\nis ramified only over the primes $\\mathcal{F}_1,\\dots,\\mathcal{F}_r\\in k[U]$ \nand over the infinite prime with ramification index $e$ \nnot divisible by $p$.\nIf $h\\in k[T]$ is such that $e|{\\rm deg}(h)$,\nthen the splitting field of $f(h(T),X)$ over $k(T)$\nis ramified at most over the prime factors of $\\mathcal{F}_1(h(T)),\\dots,\\mathcal{F}_r(h(T))$.\n\\end{lemma}\n\n\\begin{proof}\nWe identify $U$ with $h(T)$ and $k(U)=k(h(T))\\subseteq k(T)$. \nLet $K$ be the splitting field of $f(U,X)$ over $k(U)$,\nand consider the splitting field $L$ of $f(h(T),X)$ over $k(T)$, which is the compositum of $K$ and $k(T)$. \nSince $k(T)\/k(U)$ is totally ramified over $\\infty$ of degree ${\\rm deg}(h)$ divisible by $e$,\nby Abhyankar's Lemma (Lemma~\\ref{lem:abhyankar}) the base change from $k(U)$ to $k(T)$ eliminates the ramification over infinity. \nEvery finite prime of $k(T)$ that ramifies in $L$ lies over a \nfinite prime of $k(U)$ that ramifies in $K$,\ni.e.~is a prime factor of some $\\mathcal{F}_i(h)$.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:tame} \nLet $p$ be a prime, $q$ a power of $p$, $0\\le m\\le n-2$ integers such that $n-m2$,\n\\item $m\\equiv n\\pmod 2$ and $m\\neq 2$,\n\\item $(n,m)\\neq (9,1)$,\n\\item $q^{\\frac{n-m-1}2}>2m+1$, and\n\\item $H(q,n+m-1,n-m)$,\n\\end{enumerate}\nor\n\\item[(ii)] $G=A_n$ and the following conditions hold\n\\begin{enumerate}\n\\item $p>3$,\n\\item $m\\not\\equiv n\\pmod 2$ and $(n,p)=1$,\n\\item $(n,m)\\neq (8,1),(9,0),(12,1),(24,1)$,\n\\item $q^{\\frac{n-3m-1}4}>2m+1$, \n\\item $m\\ge 2$ or $\\left(\\frac q{n-m}\\right)=1$, and\n\\item $H\\left (q,\\frac{n+m-1}2,n-m\\right)$.\n\\end{enumerate}\n\\end{enumerate}\nThen there exist $f,c\\in\\mathbb{F}_q[X]$ and $h\\in\\mathbb{F}_q[T]$\nwith ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m$ and $(n-m)|{\\rm deg}(h)$\nsuch that the splitting field of $f-hc$ over $\\mathbb{F}_q(T)$\nis geometric with Galois group $G$\nand is ramified over only one prime of $\\mathbb{F}_q(T)$.\nIn particular, $r_{\\mathbb{F}_q(T)}(G)=1$.\n\\end{theorem}\n\n\\begin{proof}\nIn $(i)$,\nTheorem~\\ref{thm:tame2}$(i)$\nprovides $f,c$ of the right degree such that the splitting field $K$ of \n$f-Uc$ over $\\mathbb{F}_q(U)$ is geometric with Galois group $S_n$\nand is ramified over only one finite prime\n$\\mathcal{F}$ of degree dividing $n+m-1$\nand over the infinite prime, with ramification index $n-m$.\nThus $H(q,n+m-1,n-m)$ provides a suitable $h$ with $\\mathcal{F}(h)$ irreducible,\nhence Lemma~\\ref{lem:disjoint} gives that \nthe splitting field $K\\mathbb{F}_q(T)$ of $f-hc$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $S_n$,\nand Lemma~\\ref{lem:eliminateinfty} gives that it is ramified only over $\\mathcal{F}(h)$.\n\nIn $(ii)$,\nTheorem~\\ref{thm:tame2}$(ii)$ similarly\nprovides $f,c$ of the right degree such that the splitting field of \n$f-Uc$ over $\\mathbb{F}_q(U)$ is geometric with Galois group $A_n$\nand is ramified over only one finite prime\n$\\mathcal{F}$, with $\\disc_X(f-Uc)=\\mathcal{F}^r$ of degree $n+m-1$,\nand over the infinite prime, with ramification index $n-m$.\nAs $G=A_n$ implies that $\\disc_X(f-Uc)$ is a square, $r$ is even,\nand so in particular ${\\rm deg}(\\mathcal{F})$ divides $\\frac{n+m-1}{2}$.\nThus $H(q,\\frac{n+m-1}2,n-m)$ \nprovides a suitable $h$ with $\\mathcal{F}(h)$ irreducible,\nand we conclude again with Lemma~\\ref{lem:disjoint} and Lemma~\\ref{lem:eliminateinfty}.\n\\end{proof}\n\n\\begin{remark}\nNote that in the cases (2)-(5) of Proposition~\\ref{lem:Hqdm},\none can obtain $h$ in Theorem~\\ref{thm:tame} to be of degree {\\em equal} to $n-m$.\n\\end{remark}\n\n\n\\subsection{Two ramified primes via small prime gaps}\n\\label{sec:twinprimes}\nIn this final subsection we present a different approach to produce extensions of group $S_n$ or $A_n$ with two ramified primes.\nFor this we will need the following result on primes with small gaps in $\\mathbb{F}_q[T]$, which follows directly from \\cite[Theorem 1.3]{CHLPT15}, which is a function field adaptation of the method of Maynard \\cite{May15}. \n\n\\begin{proposition}\\label{thm:twinprimes} Assume $q\\ge 107$ and fix $b\\in\\mathbb{F}_q^\\times$. For all sufficiently large $d$ (in terms of $q$) there exist polynomials $h\\in\\mathbb{F}_q[T]$ of degree $d$ (not necessarily monic) such that $h$ and $h-b$ are both irreducible.\n\\end{proposition}\n\n\\begin{proof}\nBy \\cite[Theorem 1.3]{CHLPT15} (applied with $m=2$ and $h_1,\\dots,h_k$ an enumeration of $\\mathbb{F}_q$) and the remark following it, as long as $q>k_0(2)=105$ (using the notation of the cited theorem) and $d\\ge d_0(q)$ is sufficiently large, one can find $f\\in\\mathbb{F}_q[T],\\deg f=d$ such that $f,f-a$ are irreducible for some $a\\in\\mathbb{F}_q^\\times$ (the statement of the cited theorem only asserts the existence of infinitely many such $f$, but the proof shows that such an $f$ exists with any sufficiently large degree). Then $h=\\frac ba f$ satisfies the assertion of the proposition.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:main2} \nLet $p$ be a prime number, $q$ a power of $p$, and $3\\leq n2$ (for both $A_n$ and $S_n$). In all other cases only elementary group theory (mainly Jordan's theorem on primitive permutation groups containing a prime cycle) is used.\\end{remark}\n\n\n\n\\newcommand{L_1^{\\mathrm{gt}}}{L_1^{\\mathrm{gt}}}\n\n\\subsection{$L_1^{\\mathrm{gt}}(q)$-realizable groups}\n\n\\begin{definition}\\label{def:quasip} \nA finite group $G$ is called \\emph{quasi}-$p$ if it is generated by its $p$-Sylow subgroups,\ni.e.~$G=p(G)$. \nA finite group $G$ is called \\emph{cyclic-by-quasi-$p$} if it is an extension of a cyclic group by a quasi-$p$ group,\ni.e.~$d(G\/p(G))\\leq1$.\n\\end{definition}\n\n\\begin{example} \\label{ex:cyclicbyquasip}\nThe groups $S_n$ and $A_n$ are cyclic-by-quasi-$p$ if $n\\ge p$.\n\\end{example}\n\n\\begin{definition} \nWe say that a finite group $G$ is \\emph{$L_1^{\\mathrm{gt}}(q)$-realizable} if there exists a geometric Galois extension $L\/\\mathbb{F}_q(T)$ with ${\\rm Gal}(L\/\\mathbb{F}_q(T))=G$ that is ramified over at most two primes of $\\mathbb{F}_q(T)$, both of degree one, and with wild ramification over at most one of them. \nAn extension $L$ with this property is called an $L_1^{\\mathrm{gt}}(q)$-\\emph{realization} of $G$.\\footnote{Abhyankar introduced the notation $L_1$ for the once-punctured affine line. The letters $\\mathrm{gt}$ remind us that (up to M\\\"obius transformations) we only consider geometric Galois $\\mathbb{F}_q$-covers of $L_1$ which are tamely ramified at $\\infty$.}\\end{definition}\nNote that an $L_1^{\\mathrm{gt}}(q)$-realizable group is automatically $L_1^{\\mathrm{gt}}(q^r)$-realizable for any $r\\ge 1$.\n\n\\begin{lemma}\\label{lemreal}\nLet $G$ be an $L_1^{\\mathrm{gt}}(q)$-realizable group.\nThen $G$ is cyclic-by-quasi-$p$ and satisfies $[G:p(G)]\\mid q-1$,\nand there exists an $L_1^{\\mathrm{gt}}(q)$-realization of $p(G)$ that is ramified over a single prime of degree one.\n\\end{lemma}\n\n\\begin{proof} \nLet $L\/\\mathbb{F}_q(T)$ be an $L_1^{\\mathrm{gt}}(q)$-realization of $G$.\nWe may assume that it is ramified only over $T=0,\\infty$, tamely ramified over $\\infty$. \nLet $H=p(G)\\unlhd G$ and denote $K=L^H$. \nWe have ${\\rm Gal}(K\/\\mathbb{F}_q(T))=C:=G\/H$. Since $(|C|,p)=1$ the extension $K\/\\mathbb{F}_q(T)$ is tamely ramified and by assumption it is\nramified at most over $T=0,\\infty$,\nso it must be of the form $K=\\mathbb{F}_q(T^{1\/n})$ with $n=|C|$ and $n|q-1$\n(e.g.~since $K(T^{1\/n})\/\\mathbb{F}_q(T^{1\/n})$ is geometric, and unramified by Lemma~\\ref{lem:abhyankar}, hence trivial).\nIn particular, $C={\\rm Gal}\\left(\\mathbb{F}_q(T^{1\/n})\/\\mathbb{F}_q(T)\\right)$ is cyclic, so $G$ is cyclic-by-quasi-$p$ and $[G:p(G)]=n\\mid q-1$.\n\nLet $U=T^{1\/n}$ and let $e$ be the ramification index of the prime $U=\\infty$ in $L\/\\mathbb{F}_q(U)$.\nNote that $(e,p)=1$\nand denote $K'=\\mathbb{F}_q(S)$, where $S^e=U$. \nThe extension $K'\\overline{\\F}_q\/\\overline{\\F}_q(U)$ is Galois with ${\\rm Gal}(K'\\overline{\\F}_q\/\\overline{\\F}_q(U))=\\mathbb{Z}\/e\\mathbb{Z}$ and since $L\/\\mathbb{F}_q(U)$ is geometric we have \nthat ${\\rm Gal}(L\\overline{\\F}_q\/\\overline{\\F}_q(U))=p(G)$. \nSince $p(G)$ is quasi-$p$, the groups $p(G)$ and $\\mathbb{Z}\/e\\mathbb{Z}$ have no nontrivial common quotients and therefore the extensions \n$K'\\overline{\\F}_q,L\\overline{\\F}_q$ of $\\overline{\\F}_q(U)$ are linearly disjoint. \nConsequently the geometric extensions $K',L$ of $\\mathbb{F}_q(U)$ are also linearly disjoint and we have ${\\rm Gal}(L\\mathbb{F}_q(S)\/\\mathbb{F}_q(S))={\\rm Gal}(L\/\\mathbb{F}_q(U))=p(G)$. \nBy Lemma~\\ref{lem:eliminateinfty}, $L\\mathbb{F}_q(S)\/\\mathbb{F}_q(S)$ is ramified only over the prime $S$.\n\\end{proof}\n\n\\begin{question}\\label{ques:Abhyankar}\nIs a cyclic-by-quasi-$p$ group $G$ with $[G:p(G)]\\mid q-1$ always $L_1^{\\mathrm{gt}}(q)$-realizable?\n\\end{question}\n\nThis question is a variant of the arithmetic Abhyankar conjecture (Conjecture \\ref{conj:abhyankar}) and is suggested by Abhyankar's general philosophy regarding ramified covers in characteristic $p$. \nWe remark that Harbater and van der Put \\cite[Theorem 5.5]{HavdP02} have given a non-obvious necessary condition for a group to be the Galois group of an extension of $\\mathbb{F}_q(T)$ ramified over two geometric points, but this condition is automatically satisfied for cyclic-by-quasi-$p$ groups, \nas can easily be shown using Frattini's argument. \nThe next theorem answers Question \\ref{ques:Abhyankar} affirmatively for the symmetric and alternating groups (with some mild restrictions on $n,p$).\n\n\\begin{theorem}\\label{thmlist} Let $p$ be a prime, $q$ a power of $p$, and $n\\ge p$. The following groups are $L_1^{\\mathrm{gt}}(q)$-realizable:\n\\begin{enumerate} \n\\item $S_n$ if $n\\neq p+1$ or $p=2$.\n\\item $A_n$ if $p>2$ and either $n\\neq p+1$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$ or $p=3$.\n\\item $A_n$ if $p=2$ and either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $10\\neq n\\ge 8$ and $n\\equiv 0,1,2,6,7\\pmod 8$.\n\\end{enumerate}\n\\end{theorem}\n\nTheorem~\\ref{thmlist} will be proved in Section \\ref{secthmlist}. \nNote that in the case of $A_n,n\\ge p>2$ Theorem~\\ref{thmlist} combined with Lemma~\\ref{lemreal}\nand Example \\ref{ex:cyclicbyquasip} immediately imply Theorem~\\ref{thm:abhyankar}.\n\n\\subsection{Application to the minimal ramification problem}\n\\label{sec:thmprod}\n\nWe note that $L_1^{\\mathrm{gt}}(q)$-re\\-al\\-iz\\-abil\\-ity implies\na positive answer to the minimal ramification problem:\n\n\n\\begin{lemma} \\label{removeinf} \\label{lemquasip} \nLet $G$ be an $L_1^{\\mathrm{gt}}(q)$-realizable nontrivial finite group. \nThen for infinitely many primes $h\\in\\mathbb{F}_q[T]$ there exists a geometric Galois extension $K\/\\mathbb{F}_q(T)$ with ${\\rm Gal}(K\/\\mathbb{F}_q(T))=G$ which is ramified only over $h$. \nIn particular, $r_{\\mathbb{F}_q(T)}(G)=1$ and $G$ satisfies Conjecture~\\ref{bm}.\n\\end{lemma}\n\n\\begin{proof} \nLet $K\/\\mathbb{F}_q(U)$ be a geometric realization of $G$ ramified over $U=0$ and (at most) tamely ramified over $U=\\infty$. \nDenote by $e$ the ramification index over $\\infty$,\nwhich by assumption is not divisible by $p$.\nLet $h\\in\\mathbb{F}_q[T]$ irreducible of degree divisible by $e$\nand identify $U=h(T)$.\nBy Lemma~\\ref{lem:disjoint}, $K$ and $\\mathbb{F}_q(T)$ are linearly disjoint over $\\mathbb{F}_q(U)$,\nand $K\\mathbb{F}_q(T)\/\\mathbb{F}_q(T)$ is ramified only over $h$ by Lemma~\\ref{lem:eliminateinfty}.\nTherefore, $K\\mathbb{F}_q(T)\/\\mathbb{F}_q(T)$ is geometric Galois of Galois group $G$ and ramified only over the single prime $h$,\nhence $r_{\\mathbb{F}_q(T)}(G)=1$.\n\\end{proof}\n\nIn this subsection we \nextend this to (certain) products of $L_1^{\\mathrm{gt}}(q)$-realizable groups,\nwhich we then combine with Theorem~\\ref{thmlist} to prove Theorem~\\ref{thm:main1}:\n\n\\begin{proposition}\\label{thmprod}\n Let $G=G_1\\times\\ldots\\times G_m$ be a product with each $G_i$ being $L_1^{\\mathrm{gt}}(q)$-realizable. \n Assume that there is a prime number $\\ell$ such that for each $1\\le i\\le m$ either $G_i$ is quasi-$p$ or $\\ell\\mid [G_i:p(G_i)]$. Then Conjecture \\ref{bm} holds for $G$ over $\\mathbb{F}_q(T)$.\n\\end{proposition}\n\nBy a $k$-{\\em variety} (for a field $k$) we mean a reduced $k$-scheme of finite type.\nIn what follows all maps are morphisms of $\\mathbb{F}_q$-varieties,\nand $\\mathbb{A}^n$ (resp. $\\P^n$) always denotes the affine (resp. projective) $n$-space over $\\mathbb{F}_q$ considered as an $\\mathbb{F}_q$-variety.\nThe function field of an irreducible $\\mathbb{F}_q$-variety $X$ is denoted by $\\mathbb{F}_q(X)$. \nWe will also use the notion of a Galois cover of $\\mathbb{F}_q$-varieties and its Galois group, see \\cite[\\S I.5]{Mil80} for the definition and basic properties. \nIn particular, if $w\\colon Y\\rightarrow X$ is a Galois cover of $\\mathbb{F}_q$-varieties, \nthen $Y$ and $X$ are connected and $w$ is \\'etale, and we denote its Galois group by ${\\rm Gal}(Y\/X)$. \nWe call $w$ a \\emph{geometric} Galois cover if $Y\\times\\mathrm{Spec}\\,\\overline{\\F}_q\\to X\\times\\mathrm{Spec}\\,\\overline{\\F}_q$ is also a Galois cover (if $X$ is geometrically irreducible this is equivalent to $Y$ being geometrically irreducible).\n\n\n\\begin{proposition}\\label{hit}\nLet $U\\subseteq\\mathbb{A}^m$ be an open $\\mathbb{F}_q$-subvariety and $Y\\to U$ a geometric Galois cover of $\\mathbb{F}_q$-varieties with ${\\rm Gal}(Y\/U)=G$.\nThen \nthere exists a morphism \n$\\psi\\colon\\mathbb{A}^1\\to\\mathbb{A}^m$ \n such that $\\psi^{-1}(U)\\neq\\emptyset$ and $Y\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is a geometric Galois cover with Galois group $G$.\\end{proposition}\n\n\\begin{proof} \nDenote $\\nu=|G|$ and $Y'=Y\\times_{\\mathbb{F}_q}\\mathrm{Spec}\\,\\mathbb{F}_{q^\\nu},U'=U\\times_{\\mathbb{F}_q}\\mathrm{Spec}\\,\\mathbb{F}_{q^\\nu}$. \nSince $Y\\rightarrow U$ is geometric, $Y'$ is irreducible and the cover $Y'=Y\\times_UU'\\to U$ is Galois with ${\\rm Gal}(Y'\/U)=G\\times {\\rm Gal}(\\mathbb{F}_{q^\\nu}\/\\mathbb{F}_q)$. (Generally, if $Z\\rightarrow X$ and $W\\rightarrow X$ are Galois covers then $Z\\times_X W$ is a union of isomorphic Galois covers with Galois group a subgroup of ${\\rm Gal}(Z\/X)\\times{\\rm Gal}(W\/X)$,\n and if $Z\\times_X W$ is irreducible we have ${\\rm Gal}(Z\\times_X W\/X)\\cong {\\rm Gal}(Z\/X)\\times{\\rm Gal}(W\/X)$. \nThis fact can be deduced from \\cite[Theorem 5.3]{Mil80} or by comparing with the Galois groups of the corresponding function field extensions.)\nWe will show that \none can choose $h_1,\\dots,h_m\\in\\mathbb{F}_q[T]$ such that the corresponding morphism $\\psi\\colon\\mathbb{A}^1\\rightarrow\\mathbb{A}^m$\nsatisfies $\\psi^{-1}(U)\\neq\\emptyset$ and\n$Y'\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is a Galois cover with Galois group \n${\\rm Gal}(Y'\/U)=G\\times{\\rm Gal}(\\mathbb{F}_{q^\\nu}\/\\mathbb{F}_q)$,\nwhich will imply that\n$Y\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is a {\\em geometric} Galois cover with Galois group $G$,\nsince any extension of the field of constants occurring in this cover must be of degree dividing $\\nu=|G|$.\n\nFirst note that \n$Y'\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is again \\'etale\nfor any $\\psi$,\nso it suffices to show that \n$h_1,\\dots,h_m$ can be chosen so that\nit is a Galois cover with Galois group ${\\rm Gal}(Y'\/U)$,\nfor which we are allowed to replace $U$ and $Y'$ by dense open subvarieties.\nSo since an \\'etale morphism is locally standard \\'etale (see e.g.~\\cite[Proposition I.3.19]{Mil80}),\nwe can assume that $U=\\{g(T_1,\\ldots,T_m)\\neq 0\\}$ with $0\\neq g\\in\\mathbb{F}_q[T_1,\\ldots,T_m]$ and\n$$\n Y'=\\{f(T_1,\\ldots,T_m,X)=0,g(T_1,\\ldots,T_m)\\neq 0\\}\\subseteq\\mathbb{A}^{m+1}\n$$ \nwith $f\\in\\mathbb{F}_q[T_1,\\ldots,T_m,X]$,\nand $Y'\\to U$ is the projection to the first $m$ coordinates.\nBy \\cite[Theorem 13.3.5 and Proposition 16.1.5]{FJ} there exist\n$h_1,\\ldots,h_m\\in\\mathbb{F}_q[T]$ with\n$g(h_1,\\ldots,h_m)\\neq 0$, $f(h_1,\\dots,h_m,X)\\in\\mathbb{F}_q(T)[X]$ is irreducible and \n$$\\label{eq:galproduct}\n {\\rm Gal}(f(h_1,\\ldots,h_m,X)\/\\mathbb{F}_q(T))={\\rm Gal}(f(T_1,\\ldots,T_m,X)\/\\mathbb{F}_q(T_1,\\ldots,T_m))={\\rm Gal}(Y'\/U).\n$$ \nThe corresponding $\\psi$ satisfies\n$\\psi^{-1}(U)\\neq\\emptyset$, $Y'\\times_U\\psi^{-1}(U)$ is irreducible, and\n$$\n {\\rm Gal}(Y'\\times_U\\psi^{-1}(U)\/\\psi^{-1}(U))={\\rm Gal}(f(h_1,\\ldots,h_m,X)\/\\mathbb{F}_q(T))={\\rm Gal}(Y'\/U),\n$$ \nconcluding the proof.\n\\end{proof}\n\n\nThe next proposition combined with Lemma $\\ref{lemquasip}$ implies Proposition~\\ref{thmprod} in the special case when all $G_i$ are quasi-$p$.\n\n\\begin{proposition}\\label{propprod} \nLet $G_1,G_2$ be $L_1^{\\mathrm{gt}}(q)$-realizable groups with $G_2$ quasi-$p$. Then $G_1\\times G_2$ is $L_1^{\\mathrm{gt}}(q)$-realizable.\n\\end{proposition}\n\n\\begin{proof} \nFirst assume that both $G_1,G_2$ are quasi-$p$. \nLet $T_1,T_2$ be independent variables and let $L_i\/\\mathbb{F}_q(T_i)$ be an $L_1^{\\mathrm{gt}}(q)$-realization of $G_i$ ramified only over $\\infty$ \n(we may assume this by Lemma~\\ref{lemreal}). \nIt corresponds to a geometric Galois cover $Y_i\\to\\mathbb{A}^1=\\mathrm{Spec}(\\mathbb{F}_q[T_i])$ with ${\\rm Gal}(Y_i\/\\mathbb{A}^1)=G_i$. \nTaking the product of these covers we obtain a geometric Galois cover $Y=Y_1\\times Y_2\\to \\mathbb{A}^2$ with ${\\rm Gal}(Y\/\\mathbb{A}^2)=G_1\\times G_2$.\nBy Proposition~\\ref{hit} there exists a morphism $h\\colon\\mathbb{A}^1\\to \\mathbb{A}^2$ such that $Z=Y\\times_{\\mathbb{A}^2}\\mathbb{A}^1\\to\\mathbb{A}^1$ \nis a geometric Galois cover with ${\\rm Gal}(Z\/\\mathbb{A}^1)={\\rm Gal}(Y\/\\mathbb{A}^2)=G$. \nThe extension $L=\\mathbb{F}_q(Z)$ of $\\mathbb{F}_q(\\mathbb{A}^1)$ now gives a geometric realization of $G$ ramified only over $\\infty$.\n\nNow let $G_1,G_2$ be $L_1^{\\mathrm{gt}}(q)$-realizable with only $G_2$ assumed quasi-$p$. Iterating the above claim we see that an arbitrary product of $L_1^{\\mathrm{gt}}(q)$-realizable quasi-$p$ groups is again $L_1^{\\mathrm{gt}}(q)$-realizable (and of course quasi-$p$ as well). \nIn particular since $G_2$ is quasi-$p$ and $L_1^{\\mathrm{gt}}(q)$-realizable, for every $m$ we can find a geometric Galois extension $L\/\\mathbb{F}_q(T)$ with Galois group $(G_2)^m$ ramified only over $\\infty$, and it contains linearly disjoint subextensions $L_1,\\ldots,L_m$ such that ${\\rm Gal}(L_i\/\\mathbb{F}_q(T))=G_2$. \nLet $L\/\\mathbb{F}_q(T)$ be a geometric realization of $G_1$ ramified over $\\infty$, tamely ramified over 0 and unramified elsewhere. \nFor $m$ sufficiently large, one of the $L_i$ is linearly disjoint from $L$ over $\\mathbb{F}_q(T)$, and then \n$K=LL_i$ is a geometric Galois extension of $\\mathbb{F}_q(T)$ with ${\\rm Gal}(K\/\\mathbb{F}_q(T))=G_1\\times G_2$, ramified only over $0,\\infty$, tamely over $0$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{thmprod}] \nLet $G=G_1\\times\\dots\\times G_m$ be a product of $L_1^{\\mathrm{gt}}(q)$-realizable groups.\nIf $G$ is quasi-$p$, then by Proposition~\\ref{propprod} the group $G$ is $L_1^{\\mathrm{gt}}(q)$-realizable and by Lemma~\\ref{lemquasip} it satisfies Conjecture \\ref{bm}.\n\nOtherwise we can use Proposition~\\ref{propprod} to absorb all the quasi-$p$ factors into one of the non-quasi-$p$ factors and reduce to the case when none of the $G_i$ are quasi-$p$. Assuming none of the $G_i$ are quasi-$p$ we have $d\\left((G\/p(G))^\\mathrm{ab}\\right)=m$,\nsince by Lemma~\\ref{lemreal} the $G_i$ are cyclic-by-quasi-$p$ and the assumptions of the proposition imply that each $G_i$ has a quotient isomorphic to $\\mathbb{Z}\/\\ell\\mathbb{Z}$. \nBy Lemma~\\ref{removeinf}, for each $G_i$ there exists a geometric Galois extension $L_i\/\\mathbb{F}_q(T)$ with Galois group $G_i$ branched only over a finite prime $h_i\\in\\mathbb{F}_q[T]$ and we may take the $h_i,i=1,\\ldots,m$ to be pairwise distinct. \nThen the extensions $L_i\/\\mathbb{F}_q(T)$ are pairwise linearly disjoint, since their branch loci are disjoint and $\\mathbb{F}_q(T)$ has no unramified geometric extensions. \nTaking $L=L_1\\cdots L_m$ we see that ${\\rm Gal}(L\/\\mathbb{F}_q(T))=G$ and $L\/\\mathbb{F}_q(T)$ is ramified exactly over $h_1,\\ldots,h_m$, so $r_{\\mathbb{F}_q(T)}(G)\\le m$.\nSince by \nCorollary \\ref{cor:lowerbound}\nwe also have $r_{\\mathbb{F}_q(T)}(G)\\geq m$,\nwe conclude that $r_{\\mathbb{F}_q(T)}(G)=m$,\nwhich is what Conjecture \\ref{conj} predicts.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main1}] \nLet $G=G_1\\times\\ldots\\times G_m$ with each $G_i$ a symmetric or alternating group satisfying the assumptions of Theorem~\\ref{thm:main1}. \nSince Theorem~\\ref{thmlist} has the same assumptions, each $G_i$ is $L_1^{\\mathrm{gt}}(q)$-realizable. \nIf $p>2$ we may take $\\ell=2$ and then each $G_i$ is either an alternating group and thus quasi-$p$ or is a symmetric group and then $\\ell\\mid 2=[G_i:p(G_i)]$. \nThus the conditions of Proposition~\\ref{thmprod} are satisfied and the conclusion follows. \nIf $p=2$ then we take $\\ell=3$ and $G_i$ is quasi-2 unless $G_i=A_3,A_4$ in which case $\\ell\\mid 3=[G_i:p(G_i)]$ and again Proposition~\\ref{thmprod} applies and the conclusion follows.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thmlist}}\\label{secthmlist}\n\nLet $n\\ge p$.\nIn the present subsection we prove for $G\\in\\{S_n,A_n\\}$\nand each of the pairs $(n,q)$ from Theorem~\\ref{thmlist}\nthat the cyclic-by-quasi-$p$ group $G$ is $L_1^{\\mathrm{gt}}(q)$-realizable. \nWe construct realizations for these groups by writing down explicit equations. \nOur constructions are inspired by the work of Abhyankar \\cite{Abh92},\n and in the case $p=2$ we directly use the results of Abhyankar, Ou, Sathaye and Yie \\cite{AOS94,AbYi94}. Some of the Galois group calculations rely on the classification of multiply transitive groups, which in turn relies on the Classification of Finite Simple Groups (CFSG). We will indicate when CFSG and its applications are used. \nIn all cases we will write an explicit polynomial $f\\in\\mathbb{F}_q(T)[X]$ of degree $n$ and show that ${\\rm Gal}(f\/\\mathbb{F}_q(T))=G$ and that the splitting field of $f$ over $\\mathbb{F}_q(T)$ is geometric \nand is ramified over a single prime of degree one if $G$ is quasi-$p$ and over two primes of degree one with tame ramification over at least one of them if $G$ is cyclic-by-quasi-$p$ but not quasi-$p$. Note that we always have the inclusions \n${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant{\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant S_n$,\nso for $G=S_n$, proving that ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ shows both that ${\\rm Gal}(f\/\\mathbb{F}_q(T))=G$ and that the splitting field of $f$ is geometric,\nand for $G=A_n$, proving that ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ and ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$ shows\nthat ${\\rm Gal}(f\/\\mathbb{F}_q(T))=G$ and that the splitting field of $f$ is geometric.\nRecall from Lemma~\\ref{lem:discgalois} that if $p>2$ then ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$\nif and only if $\\Delta(f)$ is a square in $\\mathbb{F}_q(T)$\n(where we always take the discriminant with respect to $X$).\nWe distinguish four main cases according to $G=S_n$ or $A_n$ and $p=2$ or $\\neq 2$,\nand several subcases according to $n$ and $q$.\n\n\n\n\n\\maincase{$G=S_n,p>2$}\\label{sec:sn}\nIn this case $G$ is cyclic-by-quasi-$p$ but not quasi-$p$, so we are looking for a polynomial $f\\in\\mathbb{F}_q(T)[X],\\deg f=n$ with \n${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ with splitting field ramified exactly over $T=0,\\infty$, tamely over one of these points. \n\n\\case{$n$ odd, $n>p, p\\nmid n+1$} \nLet $g=X^{n+1}-(1+T)X^p+T\\in\\mathbb{F}_p[T,X]$ and\n\\begin{equation}\\label{eq:snodd} \n f=\\frac{g}{X-1}=X^p\\frac{X^{n+1-p}-1}{X-1}-(X-1)^{p-1}T\\in\\mathbb{F}_p[T,X].\n\\end{equation}\nNote that $f$ is a monic polynomial of degree $n$ in $X$. It is irreducible over $\\overline{\\F}_q(T)$ because it is linear in the variable $T$ with coprime coefficients $X^p\\frac{X^{n+1-p}-1}{X-1}$ and $-(X-1)^{p-1}$ (here we use the assumption $p\\nmid n+1$).\nSince $g'=(n+1)X^n$, using (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}) we have \n$$\n \\disc(f)=\\disc(g)\/\\mathrm{Res}(X-1,f)^2=\\pm (n+1)^{n+1}T^n\/f(1)^2=\\pm (n+1)^{n-1}T^n.\n$$\nLet $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. \nSince $\\disc(f)=\\pm (n+1)^{n-1}T^n$, the extension $\\overline{\\F}_q(T,\\alpha)\/\\overline{\\F}_q(T)$ (and therefore its normal closure) is ramified at most over $0,\\infty$. We now compute the ramification indices of this extension over 0. To this end observe that\nby (\\ref{eq:snodd}) we have \n\\begin{equation}\\label{eq:snoddt} \n T=\\frac{\\alpha^p(\\frac{\\alpha^{n+1-p}-1}{\\alpha-1})}{(\\alpha-1)^{p-1}}\n\\end{equation} \nand we see that the divisor of zeros of $T$ over $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}_q(\\alpha)$ is composed of one prime of multiplicity $p$ and other primes of multiplicity 1. Therefore the ramification indices over $T=0$ are $p,1,\\ldots,1$, so by Lemma~\\ref{lemcycle}(ii) $G={\\rm Gal}(f\/\\overline{\\F}_q(T))$ contains a cycle of length $p$. Note also that by (\\ref{eq:snoddt}), the ramification indices over $T=\\infty$ are $p-1,n+1-p$,\nin particular the ramification over $T=\\infty$ is tame.\nTherefore the action of ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ on the roots of $f$ is primitive by Lemma~\\ref{lem:primitive},\nso since it\ncontains a cycle of length $p$ we can apply \nTheorem~\\ref{thm:jones} with $l=p$. \n\nIf $n>p+2$ then Jordan's Theorem (see Remark \n\\ref{remark:jordan}) implies that ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ and since $\\disc(f)$ is not a square \nwe have ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ and the polynomial $f\n$ satisfies all the required properties. \n\nIf $n=p+2$ then we use Theorem~\\ref{thm:jones}(ii) to \nconclude that either ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ (and then we argue as before) or $p=2^k-1$ is a \nMersenne prime and ${PGL}_2(2^k)\\leqslant {\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant{P\\Gamma L}_2(2^k)$ with its action \non the roots being the standard action of such a group on $\\P^1(\\mathbb{F}_{2^k})$. \nHowever,\nthe ramification indices over $\\infty$ are $3,p-1$ and since $p>3$ (as $p\\nmid n+1=p+3$),\n by Lemma \\ref{lemcycle}(i), ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ contains a permutation with cycle structure $3,p-1$. The group \n$P\\Gamma L_2(2^k)$ with its standard action on $\\P^1(\\mathbb{F}_{2^k})$ has no such element (see \n\\cite[Corollary 4.6]{GMPS16}), a contradiction.\n\n\\begin{remark}\nNote that in treating the case $n=p+2$ we made use of Theorem~\\ref{thm:jones}(ii), which makes use of the CFSG.\n\\end{remark}\n\n\\case{$n=p$} Let \n$$\n f=X^p+X^2-T\\in\\mathbb{F}_p[T,X].\n$$ \nWe have $f'=2X$ and therefore by (\\ref{eq:disc2}) we have $\\disc(f)=aT,a\\in\\mathbb{F}_p^\\times$, hence the splitting field of $f$ over $\\overline{\\F}_q(T)$ is ramified at most over 0 and $\\infty$. \nThe polynomial $f$ is irreducible over $\\overline{\\F}_q(T)$ since it is monic (up to sign) and linear in the variable $T$.\n\nLet $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. Then $T=\\alpha^2(\\alpha^{p-2}+1)\n$ and we get (reasoning as in the previous case) that in the extension $\\overline{\\F}_q(\\alpha)=\\overline{\\F}_q(T,\\alpha)\/\n\\overline{\\F}_q(T)$ the prime $T=0$ splits into $p-2$ unramified primes and the prime $\\alpha=0$ with \nramification index 2. \nIn particular, the splitting field of $f$ is tamely ramified over $T=0$\nand ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ contains a transposition by Lemma~\\ref{lemcycle}. \nOn the other hand $T=\\infty$ is totally ramified with ramification index $p$. \nBy Lemma~\\ref{lem:primitive}, ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ is primitive,\nso by Lemma~\\ref{lem:primtrans} we \nhave ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ and $f$ is as desired.\n\n\\case{$n$ odd, $p|n+1$, $n\\neq 2p-1$}\nNote that these conditions imply $n\\ge 4p-1$. \nChoose $h\\in\\mathbb{F}_p[X],\\deg h=3$ monic irreducible.\nThere exists $u\\in\\mathbb{F}_p[X],\\deg u=3$ with $X^{n+3-p}\\equiv u^p\\pmod h$,\nsince $a\\mapsto a^p$ \nis an automorphism of $\\mathbb{F}_p[X]\/h$ and we can\nassume without loss of generality that $X\\nmid u$ by adding $\\pm h$ if necessary.\nLet $g=X^{n+3}-u(X)^pX^p-h(X)^pT$ and\n\\begin{equation}\\label{eq:snodd1} \n f=\\frac{g}{h}=\\frac{X^{n+3-p}-u(X)^p}{h(X)}X^p-h(X)^{p-1}T\\in\\mathbb{F}_p[T,X].\n\\end{equation}\nThe polynomial $f$ is monic in $X$ with $\\deg_X f=n$.\nLet $v=X^{n+3-p}-u(X)^p$ and note that $v'=(n+3)X^{n+2-p}\\neq 0$, \nin particular $v$ is separable (as $X\\nmid u$),\nwhich also implies that $h^2\\nmid v$ and so $f$ \nis linear in $T$ with coprime coefficients, hence irreducible.\nUsing (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}) we compute\n$$\n \\disc(f)=\\disc(g)\/(\\mathrm{Res}(h,f)^2\\disc(h))=aT^{n+2}\n$$ for some $a\\in\\mathbb{F}_p^\\times$,\nas $\\mathrm{Res}(h,f)$ is independent of $T$ and non-zero since $h^2\\nmid v$.\nConsequently the splitting field of $f$\nis ramified only over $T=0,\\infty$. \nSince $n$ is odd, $\\disc(f)$ is not a square in $\\overline{\\F}_q(T)$ and so ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\not\\leqslant A_n$. \n\nLet $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. By (\\ref{eq:snodd1}) we have\n\\begin{equation}\\label{eq:snodd1t} \n T=\\frac{\\left(\\frac{\\alpha^{n+3-p}-u(\\alpha)^p}{h(\\alpha)}\\right)\\alpha^p}{h(\\alpha)^{p-1}}\n \\end{equation}\nand therefore $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}_q(\\alpha)$. Now since $n\\ge 4p-1$ we see from (\\ref{eq:snodd1t}) that the \nprimes of $\\overline{\\F}_q(\\alpha)$ over $T=0$ are $\\alpha=0$ with ramification index $p$ and the roots of $v\/h$ with ramification index 1,\nso by Lemma~\\ref{lemcycle} the group ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ contains a cycle of length $p$.\nSimilarly, the primes of $\\overline{\\F}_q(\\alpha)$ over $T=\\infty$ are \nthe 3 roots of $h$ with multiplicity $p-1$ each, and $\\alpha=\\infty$ with multiplicity $n-3(p-1)\\equiv 2\\pmod p$,\nin particular the splitting field of $f$ is tamely ramified over $T=\\infty$.\nBy Lemma~\\ref{lem:primitive},\n ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ acts primitively on the roots of $f$. \n By Jordan's Theorem (Theorem~\\ref{thm:jones}(i) in the elementary case $l=p\\le n-3$ prime) we have ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ and therefore ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$.\n\n\\case{\\bf $n=2p-1$} \nLet \n$$\n f=\\frac{X^{2p}-TX^p-X^2+T}{X-1}=\\frac{X^{2p-2}-1}{X-1}X^2-T(X-1)^{p-1}\\in\\mathbb{F}_p[T,X].\n$$ \nThe polynomial $f$ is monic in $X$ and irreducible over $\\overline{\\F}_q(T)$, since it is linear in $T$ with coprime coefficients. \nUsing (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}) we compute\n$$\n \\disc(f)=\\disc(X^{2p}-TX^p-X^2+T)\/\\mathrm{Res}(X-1,f)^2=aT\n$$ \nfor some $a\\in\\mathbb{F}_p^\\times$. We see that the splitting field of $f$ over $\\overline{\\F}_q(T)$ is ramified only over $0,\\infty$. Let $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. We have\n$$T=\\frac{\\left(\\frac{\\alpha^{2p-2}-1}{\\alpha-1}\\right)\\alpha^2}{(\\alpha-1)^{p-1}},$$\nfrom which we see that in the extension $\\overline{\\F}_q(\\alpha)=\\overline{\\F}_q(T,\\alpha)$ of $\\overline{\\F}_q(T)$ there are $2p-2$ primes lying over $T=0$ \nwith one of them having ramification index $2$ and the other unramified, \nwhile over $T=\\infty$ we have the primes $\\alpha=\\infty$ with ramification index $p$, and $\\alpha=1$ with ramification index $p-1$.\nTherefore, \nby Lemma~\\ref{lemcycle},\n${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ \ncontains both a transposition and cycle of length $p$.\nThe latter implies that it is primitive,\nand so ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ by Lemma~\\ref{lem:primtrans}. \n\n\\case{$n$ even, $p\\nmid n$, $n>p+1$} \nLet \n$$\n f=X^n+X^p-T\\in\\mathbb{F}_p[T,X].\n$$ \nSince $f$ is monic (up to sign) and linear in $T$, it is irreducible over $\\overline{\\F}_q(T)$. \nSince $f'=nX^{n-1}$, by (\\ref{eq:disc2}) we have $\\disc(f)=aT^{n-1},a\\in\\mathbb{F}_p^\\times$. \nTherefore the splitting field of $f$ is only ramified over $T=0$ and $T=\\infty$, and $\\disc(f)$ is not a square in $\\overline{\\F}_q(T)$ (since $n$ is even) and so ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\not\\leqslant A_n$. \nDenoting by $\\alpha$ a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$, we see that $T=\\alpha^p(\\alpha^{n-p}+1)$. \nThe polynomial $X^{n-p}+1$ is separable by the assumption $p\\nmid n$, and so the ramification indices of $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}(\\alpha)$ over $T=0$ are $p,1,\\ldots,1$, and $T=\\infty$ is totally ramified with ramification index $n$ (in particular tame). \nTherefore, ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ is primitive by Lemma~\\ref{lem:primitive} and contains a cycle of length $p$ by Lemma~\\ref{lemcycle},\nso by Jordan's Theorem (i.e. Theorem~\\ref{thm:jones}(i) in the case $l=p\\le n-3$ prime)\nwe get that ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$.\n\n\n\\case{$n$ even, $p|n$} \nBy our assumptions, $n\\ge 2p$.\nChoose $h(X)\\in\\mathbb{F}_p[X]$ monic irreducible with $\\deg h=2$ and $h'(0)\\neq 0$ (which always exists).\nSince $a\\mapsto a^p$ is an automorphism of $\\mathbb{F}_p[X]\/h$, there exists $u(X)\\in\\mathbb{F}_p[X]$ with $\\deg u2p$ we can add $\\pm h$ to $u$ if necessary, \nwhile if $n=2p$ and we take the unique $u$ with $\\deg u\\le 1$ such that $u^p\\equiv X^{p+2} \\pmod h$ then \nautomatically $u(0)\\neq 0$, since otherwise $u=cX,c\\in\\mathbb{F}_p^\\times$ and then $h=X^2-c$, contradicting our assumption $h'(0)\\neq 0$.\nLet $g=X^{n+2}-u(X)^pX^p-Th(X)^p\\in\\mathbb{F}_p[T,X]$ and\n\\begin{equation}\\label{eq:sneven}\nf=\\frac{g}{h}=\\frac{X^{n+2-p}-u(X)^p}{h(X)}X^p-Th(X)^{p-1}\\in\\mathbb{F}_p[T,X].\n\\end{equation}\nThe polynomial $f$ is monic in $X$ of degree $n$. \nLet $v=X^{n+2-p}-u(X)^p$. \nAs $v'=2X^{n+1-p}$ and $u(0)\\neq 0$, we have \nthat $v$ is separable, in particular $h^2\\nmid v$.\nSo $f$ is linear in $T$ with coprime coefficients, hence irreducible over $\\overline{\\F}_q(T)$. \nUsing (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}), we compute\n$$\n \\disc(f)=\\disc(g)\/(\\mathrm{Res}(h(X),f)^2\\Delta(h))=aT^{n+1},a\\in\\mathbb{F}_p^\\times\n$$ \nand conclude that the splitting field of $f$ over $\\overline{\\F}_q(T)$ is ramified at most over $T=0,\\infty$ and that ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\not\\leqslant A_n$ (since $n$ is even).\n\nDenoting by $\\alpha$ a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$ we have (by (\\ref{eq:sneven}))\n$$\n T=\\frac{\\alpha^p\\left(\\frac{\\alpha^{n+2-p}-u(\\alpha)^p}{h(\\alpha)}\\right)}{h(\\alpha)^{p-1}}.\n$$\nThe polynomial $v$ is separable, hence the primes of $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}_q(\\alpha)$ over $T=0$ are $\\alpha=0$ with ramification index $p$ and $n-p$ primes with ramification index 1. \nThe primes over $T=\\infty$ are the two roots of $h$ with ramification index $p-1$ and $\\alpha=\\infty$ with ramification index $n-2(p-1)\\equiv 2\\pmod p$,\nin particular $T=\\infty$ is tamely ramified.\nTherefore again, ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ is primitive by Lemma~\\ref{lem:primitive} and contains a cycle of length $p$ by Lemma~\\ref{lemcycle},\nso by Jordan's Theorem (i.e. Theorem~\\ref{thm:jones}(i) in the case $l=p\\le n-p\\leq n-3$ prime)\nwe get that ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$.\n\n\\maincase{$G=S_n,p=2$}\nIn this case $G$ is quasi-$p$. \n\n\\case{$n$ odd} \nWe have $n\\ge 3$. \nLet\n$$\n f(X)=X^n+TX^{n-2}+1\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[\\S 11.I.5]{Abh92} (with $t=n-2$, $a=1$, $s=1$), \nwe have ${\\rm Gal}(f\/\\overline{\\F}_2(T))=S_n$ and the splitting field of $f$ is ramified only over $T=\\infty$. \n\n\\case{$n$ even} \nLet\n$$\n f(X)=\\left((X+1)^{n-1}+X^{n-1}\\right)(X+1)^2+T^{n-1}X^{n-1}\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[\\S 12.IV.4]{Abh92} (with $t=n-1$, $s=n-1$, $a=1$, $b=1$) \nwe have ${\\rm Gal}(f\/\\overline{\\F}_2(T))=S_n$ and the splitting field of $f$ is ramified only over $T=0$. \n\n\\maincase{$G=A_n, p>2$}~\\\\\n\n\n\\case{$n\\neq p+1$}\nIn this case we have shown that $S_n$ is $L_1^{\\mathrm{gt}}(q)$-realizable, and since $p>2$ we have $p(S_n)=A_n$. Therefore by Lemma~\\ref{lemreal}, the group $A_n=p(S_n)$ is also $L_1^{\\mathrm{gt}}(q)$-realizable.\n\n\\case{$p=3,n=4$} \nLet\n$$\n f=X^4-TX^3+1\\in\\mathbb{F}_3[T,X].\n$$ \nAs $f'=X^3$ and $f(0)=1$ we compute that $\\Delta(f)=1$,\nand thus the splitting field of $f$ is ramified only over $T=\\infty$\nand ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_4$.\nThe group ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant A_4$ is transitive on the roots (since $f$ is irreducible, being linear in the variable $T$) and has order divisible by $3$ (otherwise the splitting field of $f$ over $\\overline{\\F}_q(T)$ would be tamely ramified and ramified only over infinity, which implies it is trivial). \nThus ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=A_4$.\n\n\\case{$n=p+1,p>3, \\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$}\\label{sec34} \nLet $s\\ge 1$ and $2\\le a\\le p-1$ be integers and let\n$$\n f=(X+1)\\left(X+\\frac{a}{a-1}\\right)^p-T^sX^a\\in\\mathbb{F}_p[T,X].\n$$ \nBy \\cite[\\S 22]{Abh92} (with $\\tau=a$, $Y=T^s$, $b=\\frac{a}{a-1}$, but note the nonstandard sign convention in the definition of the discriminant),\n\\begin{equation}\\label{eq:disc_mrt}\n \\disc(f)=(-1)^{(p+1)\/2}\\frac{a^{2a-1}}{(a-1)^{2a-3}}T^{s(p+1)}\\in\\mathbb{F}_p[T].\n\\end{equation}\nIt is shown in \\cite[\\S 12.IV.3]{Abh92} (with $t=a$, $b=\\frac{a}{a-1}$) that if $p>5$, $2\\le a\\le \\frac{p-1}2,(a,p+1)=1$ and $a(p+1-a)|s$, then ${\\rm Gal}(f\/\\overline{\\F}_q(T))=A_n$ and the splitting field of $f$ is ramified only over $T=0$. \nIf additionally $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$, then $\\disc(f)$ is a square in $\\mathbb{F}_q[T]$ and we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))=A_{n}$.\nNote that for all $p>5$, an $a$ with $(a,p+1)=1$ and $a\\not\\equiv\\pm 1\\pmod{p+1}$ can be found \n(as $\\phi(x)>2$ for $x>6$),\nand after replacing $a$ with $p+1-a$ if necessary to assume that $2\\leq a\\leq\\frac{p-1}{2}$,\nwe can set $s=a(p+1-a)$.\n\nIf $p=5,n=6,\\mathbb{F}_q\\supseteq\\mathbb{F}_{25}$ we take the polynomial\n$$\n f=(X+1)(X+2)^5-T^4X^2\\in\\mathbb{F}_p[T,X].\n$$\nThe discriminant of $f$ is computed by (\\ref{eq:disc_mrt}) with $a=2,s=4$ and equals $\\Delta(f)=2T^{24}$, which is a square in $\\mathbb{F}_{25}(T)$ and hence ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_5$. \nOn the other hand, by \\cite[\\S 12.IV.2]{Abh92} \n(with $t=2,b=2$), the splitting field of $f$ over $\\overline{\\F}_5(T)$ is ramified only over $T=0$, and its Galois group is $A_5$, \nhence ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=A_5$.\n\n\\begin{remark} If we want to drop the assumption $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$ in the case $p>5$ we would need to find an $a$ with $2\\le a\\le \\frac{p-1}2,(a,p+1)=1$ such that $(-1)^{(p+1)\/2}a(a-1)$ is a square modulo $p$. It is not hard to show (using a P\\'olya-Vinogradov-type inequality and an elementary sieve argument) that this is possible for all $p$ sufficiently large, and we conjecture that such an $a$ exists for all $p>13$. While proving this should be doable by means of a careful analysis and sufficiently large computer search, we did not pursue this.\\end{remark}\n\n\\maincase{$G=A_n,p=2$}\nThe group $A_n$ is always cyclic-by-quasi-2 and it is quasi-2 iff $n\\neq 3,4$. Most of the required realizations were constructed by Abhyankar, Ou, Sathaye and Yie \\cite{Abh92, Abh93, AOS94,AbYi94}.\n\n\\case{$n=3,4$, $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nFor $A_3\\cong\\mathbb{Z}\/3\\mathbb{Z}$ we can take the extension $K=\\mathbb{F}_q(s)\/F_q(T)$ with $s^3=T$ which is Galois with group $A_3$ if $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$\nand ramified only over $T=0,\\infty$. \nDenote by $\\zeta$ any element of $\\mathbb{F}_4\\setminus\\mathbb{F}_2$, and consider the splitting field $L$ of the polynomial \n$$\n f=(X^2+X+s)(X^2+X+\\zeta s)(X^2+X+\\zeta^2 s)\\in\\mathbb{F}_2(s)[X]\n$$\nover $K$. \nThe extension $L\/\\mathbb{F}_q(T)$ is Galois (since $s,\\zeta s,\\zeta^2 s$ are a Galois orbit over $\\mathbb{F}_q(T)$) and by Artin-Schreier theory ${\\rm Gal}(L\/K)=\\mathbb{Z}\/2\\mathbb{Z}\\times\\mathbb{Z}\/2\\mathbb{Z}$ with ${\\rm Gal}(L\/\\mathbb{F}_q(T))$ acting nontrivially on the order 2 subgroups of ${\\rm Gal}(L\/K)$ by conjugation, in particular ${\\rm Gal}(L\/\\mathbb{F}_q(T))$ is non-abelian.\nThe only non-abelian extension of $\\mathbb{Z}\/3\\mathbb{Z}$ by $\\mathbb{Z}\/2\\mathbb{Z}\\times\\mathbb{Z}\/2\\mathbb{Z}$ is $A_4$. \nFinally observe that $L\/\\mathbb{F}_q(T)$ is ramified only over $0,\\infty$ and is therefore an $L_1^{\\mathrm{gt}}(q)$-realization of $A_4$.\n\n\\begin{remark}\nNote that by Lemma~\\ref{lemreal}, both for $n=3$ and for $n=4$ the condition $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ is necessary for $A_n$ to be $L_1^{\\mathrm{gt}}(q)$-realizable since $A_3,A_4$ have cyclic quotients of order 3. \n\\end{remark}\n\n\\case{$n=5$, $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nLet\n$$\n f=X^5+TX+1\\in\\mathbb{F}_2[T,X].\n$$ \nIt follows from \\cite[\\S 11.III.1]{Abh92} (with $q=4$, $t=s=1$, $a=-1$, in the notation used there) that ${\\rm Gal}(f\/\\overline{\\F}_2(T))\\cong PSL_2(4)\\cong A_5$ and the splitting field of $f$ is ramified only over $T=\\infty$. \nBy \\cite[2.23]{AOS94} (with $K=\\mathbb{F}_q(T)$, $d=4$, $e=1$, $\\bar{b}_d=T$, $\\bar{b}_n=1$),\n$\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ implies that ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_5$.\nThus ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=A_5$.\n\n\n\\case{$n=6,7$, $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nConsider the polynomials\n\\begin{eqnarray*}\nf_6&=&X^6+T^{27}X^5+T^{54}X^4+(T^{18}+T^{36})X^3+T^{108}X^2+(T^{90}+T^{135})X+T^{162},\\\\\nf_7&=&X^7+TX^4+X^2+1.\n\\end{eqnarray*}\n\nThe polynomials $f_6,f_7$ were found by Abhyankar and Yie \\cite[Theorems 2.10 and 2.11]{AbYi94}, who showed that the splitting fields of $f_n,n=6,7$ are ramified only over $\\infty$ and,\nunder the assumption $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$,\nthat ${\\rm Gal}(f_n\/\\mathbb{F}_q(T))={\\rm Gal}(f_n\/\\overline{\\F}_q(T))=A_n$. \n\n\n\n\\case{$n\\ge 9$ odd, $n\\equiv 1,7\\pmod 8$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nLet\n$$\n f=X^n+TX^{n-4}+1\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[Theorem 2]{Abh93} (with $t=n-4$, $q=4$ in the notation of the cited paper), \nthe splitting field of $f$ is ramified only over $\\infty$ and we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\geqslant{\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$. \nConversely, by \\cite[(2.27)]{AOS94} (with $t=n-4$, $b_{n-t}^*=T$, $b_n^*=1$) we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$ if either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $n\\equiv 1,7\\pmod 8$.\nIn these cases we conclude that ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_2(T))=A_n$.\n\n\\case{$n\\ge 8$ even, $10\\neq n\\equiv 0,2,6\\pmod 8$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nLet $1\\le t\\le n$ with $(t,n)=1$, and\n$$\n f=X^n+X^t+T^t\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[\\S 11.II.5]{Abh92} (with $s=t$, $a=1$), if $2\\le t\\le n-4$ then ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\geqslant{\\rm Gal}(f\/\\overline{\\F}_2(T))\\geqslant A_n$ and the splitting field of $f$ is ramified only over $\\infty$. \nConversely, by \\cite[Theorem 2.27]{AOS94} (with $b_{n-t}^*=1$, $b_n^*=T^t$) we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$ if either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $n\\equiv 0\\pmod 8$ or $n\\equiv 2,6\\pmod 8,2t\\equiv n\\pmod 8$. \nIn each of these cases we can choose a suitable $t$:\nFor\n$n\\geq 8$\nthere exists $2\\leq t\\leq n-4$ with $(t,n)=1$\nsince $\\phi(n)>2$,\nand for $n\\equiv 2,6\\pmod 8,n>10$,\nthe choice $t=\\frac n2-4$ satisfies $2\\le t\\le n-4$, $(t,n)=1$ and $2t\\equiv n\\pmod 8$.\n\n\n\n\\section{Summary and application to the minimal ramification problem over $\\mathbb{Q}$}\n\\label{sec:Q}\n\\label{sec:summary}\n\n\\subsection{Summary}\n\\label{subsec:summary}\nWe first summarize some of the results for the groups $S_n$ and $A_n$:\n\n\\begin{theorem}[Main results for $S_n$]\nLet $n\\geq 2$ and $q=p^\\nu$ a prime power.\nThen $r_{\\mathbb{F}_q(T)}(S_n)\\leq 2$,\nand $r_{\\mathbb{F}_q(T)}(S_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $p(2n-3)^2$ \n\\item The function field analogue of Schinzel's hypothesis H (Conjecture \\ref{conj:SchinzelFF}) holds for $\\mathbb{F}_q(T)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe case $n=2$ is very easy and follows for example from Theorem~\\ref{thm:abelian}, so assume from now on that $n\\geq 3$.\nIf $pn$.\n\nFor the rest of this proof, we call an $S_n$-extension of $\\mathbb{F}_q(U)$\na $(q,n,m)$-realization\nif it is the splitting field of $f-Uc$ with $f,c\\in\\mathbb{F}_q[X]$, ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m(2n-7)^{2\/3}$ for all $n\\geq4$).\nFor $n=3,4,6$, Proposition~\\ref{prop:Morse2} gives a $(q,n,0)$-realization.\n\nWe now prove $r_{\\mathbb{F}_q(T)}(S_n)=1$ in cases (2)-(4),\nfor which it suffices by Lemmas~\\ref{lem:disjoint} and \\ref{lem:eliminateinfty} to exhibit a $(q,n,m)$-realization such that $(p,n-m)=1$ and\n$H(q,n+m-1,n-m)$ holds:\n\nIf (4) holds, then $H(q,d,e)$ always holds by Proposition~\\ref{lem:Hqdm}(1),\nso we can take the realizations from above.\n\nIf (3) holds\nand $n\\neq 4$,\nTheorem~\\ref{thm:tame2} with $m=n-2$ gives a $(q,n,n-2)$-realization (as $q>(2n-3)^2=(2m+1)^2$),\nand $H(q,2n-3,2)$ always holds by Proposition~\\ref{lem:Hqdm}(3) (as $q$ is odd).\nIf (3) holds and $n=4$, \nwe take the $(q,4,0)$-realization from above,\nand Proposition~\\ref{lem:Hqdm}(5) gives $H(q,3,4)$ for every $q\\geq (2^1\\cdot 3-1)^2=25=(2n-3)^2$. \n\nIf (2) holds,\nthen $H(q,d,4)$ holds for every odd $d$ by\nProposition~\\ref{lem:Hqdm}(3);\nif $n=5$ or $n\\geq 7$,\nwe obtained a $(q,n,n-4)$-realization above,\nand $H(q,2n-5,4)$ holds;\nif $n=4$ we take the $(q,4,0)$-realization from above,\nand $H(q,3,4)$ holds;\nif $n=3$ or $n=6$ we are in case (3) as soon as $q> 9$ resp.~$q>81$; \nfor $n=6$ and $q\\equiv 1\\mbox{ mod }3$,\n$H(q,5,6)$ holds by Proposition~\\ref{lem:Hqdm}(3)\nso we can take the $(q,6,0)$-realization from above.\nIn each of the remaining cases, the following table provides \n$f,c\\in\\mathbb{F}_p[X]$ with $\\deg f=n$, $\\deg c=m{$}l<{$}}\n\\begin{center}\n\\begin{tabular}{|ll|l|l|l|}\n\\hline\n $n$ & $p$ & $f$ & $c$ & $h$ \\\\\n\\hline\n$3$ &$5$ & $X^3 + 1$ &$X+2$ &$T^2$ \\\\\n$6 $&$5 $&$X^6 + 1 $&$X^2+X $&$T^4+1$ \\\\\n$6 $&$17 $& $X^6 + X^2 + X$ & $1 $ & $T^6+T+2$ \\\\\n$6 $&$29 $&$X^6 + X^2 + X $&$1 $&$T^6+T+6$ \\\\\n$6 $&$41 $&$X^6+X $&$1 $&$T^6+T+1$ \\\\\n$6 $&$53 $&$X^6+X^2+13X $&$1 $&$T^6+T$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{proof}\n\n\\begin{remark}\nNote that instead of $q\\equiv 1\\mbox{ mod }4$ we could also treat other arithmetic progressions:\nFor any prime number $\\ell$, if $q\\equiv 1\\mbox{ mod }2\\ell$, $n\\geq 2\\ell+3$, \nand $2n\\not\\equiv 1\\mbox{ mod }\\ell$,\nlet $m=n-2\\ell$. \nThen, in the language of the previous proof,\nTheorem~\\ref{thm:tame2}\ngives a $(q,n,m)$-realization,\nand $H(q,n+m-1,n-m)=H(q,2n-2\\ell-1,2\\ell)$ holds by Proposition~\\ref{lem:Hqdm}(3).\n\\end{remark}\n\n\\begin{theorem}[Main results for $A_n$]\nLet $n\\geq 3$ and $q=p^\\nu$ a prime power.\nIf $p>2$ \nor \n$\\mathbb{F}_q\\supseteq\\mathbb{F}_{4}$, then $r_{\\mathbb{F}_q(T)}(A_n)\\leq 2$,\nand $r_{\\mathbb{F}_q(T)}(A_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $23$.\n\nIf $p\\le n,p\\neq n-1$ or $p=n-1,\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$, all assertions follow from Theorem~\\ref{thm:main1}. \n\nIf $p=n-1$ and $n\\ge 14$ we may use Theorem~\\ref{thm:tame2} with $m=3$ to obtain $r_{\\mathbb{F}_q(T)}(A_n)\\le 2$ in this case. \nAssuming Conjecture $\\ref{conj:SchinzelFF}$ allows us to apply Theorem~\\ref{thm:tame} combined with Proposition~\\ref{lem:Hqdm}(1) (once again with $m=3$) to conclude $r_{\\mathbb{F}_q(T)}(A_n)=1$ in this case.\n\nIf $p=n-1$ and $n\\in\\{6,8,12\\}$,\nthe following table provides irreducible $f\\in\\mathbb{F}_p[T,X]$ \nwith $\\deg f=n$ and $\\disc_X(f)$ a square in $\\mathbb{F}_p(T)$ with only one prime divisor $T$.\nComputer verification shows that ${\\rm Gal}(f\/\\mathbb{F}_p(T))$\ncontains a $3$-cycle and an $(n-1)$-cycle, hence by Theorem~\\ref{thm:jones} contains $A_n$.\nThus the splitting field of $f$ over $\\mathbb{F}_q(T)$ is ramified at most over $T$ and the infinite prime,\nand ${\\rm Gal}(f\/\\mathbb{F}_q(T))=A_n$ as $A_n$ is simple.\n\\begin{center}\n\\begin{tabular}{|ll|l|l|}\n\\hline\n$n$ &$p$ &$f$ & $\\disc_X(f)$ \\\\\n\\hline\n$6$ &$5$ & $X^6+X^5T - 2X^3T^3 + XT + T^2$ & $4T^{18}$ \\\\\n$8$ & $7$ & $X^8 + 3X^2 + XT - 2$ & $4T^2$ \\\\\n$12$ & $11$ & $X^{12} + 5XT^3 - 5X^2 - 2$ & $4T^6$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nNow assume that $p>n$.\nIf $n\\ge 13$ or $n=11$ we apply Theorem~\\ref{thm:tame2} \nwith $m=2$ or $m=3$ chosen so that $m\\not\\equiv n\\pmod 2$\nand thus obtain $r_{\\mathbb{F}_q(T)}(A_n)\\le 2$ in this case. \nIf $n=10$ we apply Theorem~\\ref{thm:tame2} with $m=1$ (note that $\\left(\\frac{q}{n-m}\\right)=\\left(\\frac{q}{3}\\right)^2=1$).\nAssuming Conjecture $\\ref{conj:SchinzelFF}$ allows us to apply Theorem~\\ref{thm:tame} combined with Proposition~\\ref{lem:Hqdm}(1) (with $m$ as above) to conclude $r_{\\mathbb{F}_q(T)}(A_n)=1$ in these cases.\nIn all other cases, Theorem~\\ref{thm:main2} gives that \n$r_{\\mathbb{F}_q(T)}(A_n)\\le 2$.\n\\end{proof}\n\n\n\\subsection{Application to the minimal ramification problem over $\\mathbb{Q}$}\nFinally, we apply our results for $S_n$ over $\\mathbb{F}_q(T)$\nto give a conditional proof of Conjecture~\\ref{conj:BM} for $S_n$ over $\\mathbb{Q}$:\n\n\\begin{lemma}\\label{lem:lift_real}\nLet $n\\in\\mathbb{N}$, $h_0\\in\\mathbb{Z}$, and $S$ a finite set of primes numbers,\nand for each $p\\in S$ let $f_p\\in\\mathbb{Z}[X]$ be monic of degree $n$.\nThere exist $f\\in\\mathbb{Z}[X]$ monic of degree $n$ and $c\\in\\mathbb{Z}$ such that $f\\equiv f_p\\pmod p$ and $c\\equiv 1\\pmod p$\nfor every $p\\in S$,\nand $f-c^nh_0$ has $n$ roots in $\\mathbb{R}$.\n\\end{lemma}\n\n\\begin{proof}\nBy the Chinese Remainder Theorem there exists a monic $f_0\\in\\mathbb{Z}[X]$\nwith $f_0\\equiv f_p\\pmod p$ for every finite $p\\in S$.\nLet $\\pi=\\prod_{p\\in S}p$,\nwrite $f_0=\\sum_{i=0}^na_iX^i$ and \nlet $f_\\infty\\in\\mathbb{Z}[X]$ be any monic polynomial of degree $n$\nwith $n$ roots in $\\mathbb{R}$.\nSince fractions of the form $\\frac{x\\pi}{y\\pi+1}$ with $x,y\\in\\mathbb{Z}$ are dense in $\\mathbb{R}$, \nwe can choose\n$x_i,y_i\\in\\mathbb{Z}$ such that\n$\\tilde{f}:=X^n+\\sum_{i=0}^{n-1}(a_i+\\frac{x_i\\pi}{y_i\\pi+1})X^i$\nis arbitrarily close to $f_\\infty+h_0$, \nin particular so close that also $\\tilde{f}-h_0$ has $n$ roots in $\\mathbb{R}$.\nThen with $c=\\prod_{i=0}^{n-1}(y_i\\pi+1)$,\nthe polynomial\n$f(X):=c^n\\tilde{f}(c^{-1}X)\\in\\mathbb{Z}[X]$ satisfies the claim.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm:S_n_over_Q}\nSchinzel's hypothesis H (Conjecture \\ref{conj:Schinzel}) implies that\n$r_\\mathbb{Q}(S_n)=1$ for every $n\\geq2$, i.e.~Conjecture \\ref{conj:BM} holds for symmetric groups.\n\\end{theorem}\n\n\\begin{proof}\nIn case $n=2$, for example $L=\\mathbb{Q}(\\sqrt{2})$ does the job,\nso assume that $n\\geq 3$.\nLet\n$S_0$ denote the set of prime numbers $\\ell d$,\nso for every such $\\ell$ there exists $a\\in\\mathbb{Z}$ such that\n$g(X,a)\\mbox{ mod }\\ell$ is separable, \nand thus the same argument shows that $\\ell$ is unramified in\n$A^\\phi_{[a:1]}$.\n\nThus in the language of \\cite[Definition 1.10]{BSS},\n$G=S_n$ has a $(U;\\mathbf{d})$-realization with $U=\\emptyset$ and $\\mathbf{d}=(d)$.\nTherefore, \\cite[Proposition 1.11]{BSS} gives a realization of $G$\nover $\\mathbb{Q}$\nwith at most $B(\\mathbf{d})+\\#U$ many ramified primes,\nand under Schinzel's hypothesis H, $B(\\mathbf{d})\\leq 1$, cf.~\\cite[(13) on p.~923]{BSS}.\n\\end{proof}\n\n\n\\begin{remark}\\label{remark:plans}\nPlans \\cite[Remark 3.10]{Plans} sketches how to use Schinzel's hypothesis H (Conjecture \\ref{conj:Schinzel}) to construct $S_n$-extensions $L$ of $\\mathbb{Q}$ ramified in only one finite prime. However, he constructs $L$ as the splitting field of an irreducible trinomial $f=X^n+aX^i+b$, and therefore $L$ is not totally real if $n\\geq5$: \nIndeed, replacing $X$ by $X^{-1}$ if necessary we may assume that $i\\geq\\frac{n}{2}$, \nso since the derivative $f'=X^{i-1}(nX^{n-i}+ia)$ has at most $n-i+1$ distinct roots, in particular at most $n-i+1$ distinct roots in $\\mathbb{R}$,\nRolle's theorem shows that $f$ has at most $n-i+2\\leq\\frac{n}{2}+2\\!0$, they have different set of mathematical equations in the (singular) limit $\\varepsilon=0$ (detailed below). This highlights the fact that these systems are \\textit{singularly} and not \\textit{regularly} perturbed. Resolving these singular limits and reconciling these dual mathematical equations has been the basis of several mathematical and numerical developments within, e.g., \\textit{Catastrophe theory}, \\textit{Singularity theory}, \\textit{Bifurcation theory}, \\textit{Geometrical Singular perturbation theory (Fenichel theory)}, \\textit{geometric desingularization or blow-up method}. In short, these two different limits give rise to the so-called \\textit{slow subsystem} and \\textit{fast subsystem}, respectively. The slow subsystem is a differential-algebraic problem, where the slow variables remain explicitly dynamic through the differential equation $\\dot{y}=g(x,y)$, while the fast variables are enslaved to the slow ones through the algebraic equation $f(x,y)=0$. On the other hand, the fast subsystem is a family of dynamical systems on the fast variables $x$, where the slow variables $y$ have lost their dynamics and have become parameters. The set $\\{f=0\\}$ is referred to as the \\textit{critical manifold} of the system, and it plays a central role in both subsystems. Specifically, it is the phase space of the slow subsystem and it is the set of equilibria of the fast subsystem. Hence the locus of the limiting slow motion corresponds to a subset of the bifurcation diagram of the fast subsystem obtained when varying one or more slow variables. Moreover, varying other system parameters can induce the limiting slow motion to have non trivial trajectories, such as \\textit{canard solutions} that emerge via \\textit{dynamic bifurcations} and that visit both the stable and unstable regions of the critical manifold (see e.g.~\\cite{desroches12} for details). \n\\begin{figure}[!t]\n\\centering\n\\includegraphics{BurstingData.pdf}\n\\caption{Example of electrophysiological recordings of bursting oscillations in four types of neurons: (a) parabolic-type bursting from the CeN neuron from the melibe (a sea slug)~cite{newcomb08}; (b) square-wave-type bursting from a human $\\beta$-cell~\\cite{riz14}; (c) elliptic-type bursting from a dorsal-root-ganglia (DRG) neuron of a rat~\\cite{jian04}; (d) Pseudo-plateau-type bursting from a pituitary cell of a rat~\\cite{tabak11}.}\n\\label{fig.bursting_data}\n\\end{figure}\nThe knowledge of this dissection between the slow and fast subsystems, together with the knowledge of how the trajectory of the full system (i.e. for small $\\varepsilon\\!>\\!0$) evolves along these subsystems is at the basis of the various classification systems for complex slow-fast oscillations that we will subsequently review. Specifically, we will focus on the state-of-the-art mathematical classification systems for \\textit{bursting dynamics}, their limitations and finally propose a novel classification framework. \\red{Our novel classification fundamentally relies upon a certain type of canard configuration, referred to as \\textit{folded node}. Noteworthy, canards are indirectly included in the previous classification frameworks as boundaries between the spiking and the bursting regimes via so-called \\textit{spike-adding transitions} or \\textit{torus canards}, which we will review, thus highlighting the importance of canards in the classification process. We will define the notion of bursting oscillations in the context of neuronal systems} (see examples in Figure~\\ref{fig.bursting_data}) since for historical reasons the notion of bursting emerged within the neuroscientific literature. In particular, \\mdsr{bursting models} appeared in the context of classical single neuron electrophysiological measurements, where \\mdsr{the neuron's voltage time-series \\red{displays} a bursting oscillation either in response to a brief input stimulus or, in absence of any stimuli, in an endogenous manner}. These oscillations are defined as having a periodic succession (sometimes irregular) of two distinct epochs of activity. One epoch features slow and low-amplitude activity, and it is typically referred to as the \\textit{quiescent} (or \\textit{silent}) phase. The other epoch features fast and high-amplitude activations (i.e. several action potentials or spikes) is classically denoted \\textit{active} or \\textit{burst} phase as shown by several examples in Figure~\\ref{fig.bursting_data}. Although a great deal of our discussions will be in the context of neuronal dynamics, the mathematical framework intends to capture complex slow-fast oscillations beyond the scope of neuroscientific applications (e.g. in chemical reactions, genetic switches, material transitions, etc.). Hence some of the mathematical model constructs that we will present here display bursting oscillations not necessarily observed in neural data. Indeed, our \\mdsr{idealized} models will not have direct biophysical interpretation as we aim to be as general as possible in describing the fundamental mathematical mechanisms, which can then be applied to explain complex bursting oscillations in multiple contexts. Moreover, we will focus on the minimal deterministic mathematical setting for bursting oscillations. Specifically, we will cover the case of two-timescale systems with explicit timescale separation (dictated by a single small parameter $0\\!<\\!\\varepsilon\\!\\ll\\!1$), with two fast variables that will enable the description of the active phase of a bursting oscillation (i.e. $x\\in\\mathbb{R}^2$), and one or two slow variables describing the quiescent phase of the bursting dynamics (i.e. $y\\in\\mathbb{R}$ or $y\\in\\mathbb{R}^2$). This minimal setting will inform more complex scenarios involving multi-dimensional systems with multiple timescales.\n\n\\mdsr{This paper is organized as follows. In Section~\\ref{sec:review}, we will review existing classification frameworks for bursting oscillations. \\red{Subsequently}, in Section~\\ref{sec:beyond}, we first introduce the \\red{key} idea of our novel bursting classification based upon the concept of folded-node bursting dynamics. \\red{This is followed by showcasing} several \\red{new} examples of folded-node burster idealized models, first in the case of classical folded node and then in the case of cyclic folded node. Finally, in the conclusion section, we review our findings and propose a number of perspectives and future directions to explore. We \\red{complete the manuscript} by proposing, in Appendix~\\ref{sec:newfast}, few \\red{novel} additional bursting scenarios within the Rinzel-Izhikevich's classification. \\red{These include cases} with transcritical and pitchfork bifurcations of limit cycles, \\textit{isola bursting} and a two-slow-variable bursting scenario with a family of transcritical bifurcations of equilibria.}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{RinzelClassification.pdf}\n\\caption{Rinzel classification of bursting patterns: square-wave bursting, here in the Hindmarsh-Rose model~\\cite{hindmarsh84} (panels (a1)-(b1)); elliptic bursting, here in the FitzHugh-Rinzel model~\\cite{rinzel86,rinzel87} (panels (a2)-(b2)); parabolic bursting, here in Plant's model~\\cite{plant81} (panels (a3)-(b3)).}\n\\label{fig:rinzel}\n\\end{figure}\n\n\\section{Review of the state-of-the-art classification of bursting patterns}\n\\label{sec:review}\n\\subsection{Rinzel's classification (mid 1980s)} Historically, John Rinzel opened the door towards mathematically understanding bursting oscillations. His seminal work on a mathematical analysis and classification of bursting oscillatory patterns, were first published within two companion manuscripts~\\cite{rinzel86,rinzel87}. The fundamental insight behind Rinzel's classification is based on slow-fast dissection and in particular describing the bifurcation structure of the fast subsystem where the slow variables are frozen. Subsequently, the time trajectory of the full system (i.e. for small $\\varepsilon\\!>\\!0$) is superimposed on top of the bifurcation structure of the fast subsystem. This reveals that the quiescent phase of the bursting cycle correspond to trajectory segments where the solution slowly tracks families of stable equilibria, or low-amplitude (subthreshold) limit cycles, of the fast subsystem. Conversely, the burst phase of the full system's cycle correspond to trajectory segments where the solution slowly tracks families of limit cycles of the fast subsystem. The transitions between these two main phases of bursting cycles occur near bifurcation points of the fast subsystem. With this approach, Rinzel proposed three classes of bursting dynamics based on both the bifurcation structure of the fast subsystem and the salient features of the main fast variable's time profile (in the neuronal context this is typically the neuronal membrane potential). These features include spike frequency during the burst, dynamics during the silent phase (oscillatory or not), shape of the burst (on a plateau compared to the silent phase or on the contrary with undershoots). These three features led Rinzel to name three classes as \\textit{square-wave}, \\textit{elliptic} and \\textit{parabolic} bursting. We show an example of each class in Figure~\\ref{fig:rinzel}.\n\n\\subsection{Izhikevich's classification (ca. 2000)} Eugene Izhikevich generalised Rinzel's approach by considering that a bursting pattern is entirely characterised by a pair of bifurcations ($\\mathbf{b_1}$, $\\mathbf{b_2}$) of the fast subsystem. One bifurcation, say $\\mathbf{b_1}$, explains the transition from quiescence to burst, and the other, $\\mathbf{b_2}$, marks the inverse transition, from burst to quiescence. Due to the well established bifurcation theory and indeed knowledge of classes of bifurcation, this led to a systematic identification of at least 120 bursting patterns~\\cite{izhikevich00}. An example of a bursting model that is not within the Rinzel classification is depicted in Fig~\\ref{fig:homhom}. In this example the bursting pattern has a transition from quiescence to burst via a homoclinic bifurcation (involving a small homoclinic connection) and equally, the transition from burst to quiescence is via homoclinic bifurcation (involving a large homoclinic connection). In many ways, Izhikevich's work serves as a key source of reference for classification of complex slow-fast oscillations. This is particularly the case in Neuroscience since some of the assembled examples were motivated by existing conductance-based neuronal models and demonstrated how complex neuronal oscillations could be achieved by adding one slow equation to a spiking system. Indeed, a dedicated book towards Neuroscience was later published, where the derived models where also put into context with neurophysiological processes~\\cite{izhikevich07}. The result of this deeply insightful work is a quasi-complete classification of bursting patterns in terms of pairs of codimension-one bifurcations of the fast subsystem.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics{homhom.pdf}\n\\caption{Small homoclinic \/ big homoclinic bursting, corresponding to Fig.88 of~\\cite{izhikevich00}. Panel (a) shows the slow-fast dissection in the $(u,V)$ phase plane and panel (b) shows the $V$-time series of this bursting solution.}\n\\label{fig:homhom}\n\\end{figure}\n\\subsection{Bertram et al.'s \/ Golubitsky et al.'s classification (mid 1990)} An alternative approach to classification was proposed by Bertram and colleagues in 1995~\\cite{bertram95} and extended mathematically by Golubitsky and collaborators in 2001~\\cite{golubitsky01} using a singularity theory viewpoint. The fundamental idea consists in identifying a codimension-$k$ bifurcation point ($k \\geq 2$) in the fast subsystem and subsequently consider the slow variables of the bursting system as the unfolding parameters of this high codimension bifurcation point. The bursting is then obtained via a slow path made by the slow variables in the unfolding of that point (i.e. within a multidimensional parameter space). The minimum codimension, whose unfolding allows to create a given bursting pattern, defines the class of the associated bursting patterns provided a notion of path equivalence is properly defined. Specifically, two paths are equivalent if one can pass from one to the other via diffeomorphism and a re-parameterization. Recently, a review and a show-case demonstrating the construction of bursting oscillations via this approach, including cases for higher codimension bifurcation points was published in~\\cite{saggio17}. It is worth noting that the Rinzel-Izhikevich approach and the Bertram-Golubitsky approach both focus on the fast subsystem only. Moreover, a way to see a link between the two approaches is that the two bifurcation points ($\\mathbf{b_1}$, $\\mathbf{b_2}$) of the fast subsystem (as characterised by Izhikevich's approach) belong to bifurcation curves in a two-parameter plane, which coalesce at a codimension-two bifurcation point that characterises this particular bursting pattern from the singularity theory viewpoint. This implies that in principle the Rinzel-Izhikevich and the Bertram-Golubitsky approaches both lead to the same number of bursting oscillation cases.\n\n\\mdsr{The bursting patterns covered by these three existing classification schemes have not been exhausted yet, even though a large number (way above one hundred) have already been reported and analysed in previous works. However, we propose a few more cases which we believe have not been considered before and which will be presented with associated idealized models in Appendix~\\ref{sec:newfast}. In particular, we will show bursting scenarios where the burst phase ends due to a transcritical or a pitchfork bifurcation of limit cycles of the fast subsystem. We also propose the concept of \\textit{isola bursting}, where the burst starts and ends through saddle-node bifurcations of limit cycles of the fast subsystem which happen to lie on an isola of limit cycles. Finally, we propose one example (amongst many) of bursting pattern with two slow variables where the burst initiates through a family of transcritical bifurcation of equilibria.}\n\n\n\\section{\\mdsr{Towards a new classification of bursting patterns}}\n\\label{sec:beyond}\n\n\\subsection{\\mdsr{Main idea to go beyond the state-of-the-art}}\nIt is compelling to ask if there are other bursting oscillations beyond the Rinzel-Izhikevich and Bertram-Golubitsky classification approaches, as summarised in Figure~\\ref{fig:sketch} (top panel), and which cannot be explained invoking these state-of-the-art results. If so, could there be an improved classification system that captures a larger class of bursting dynamics beyond the ca. 120 cases captured by the state-of-the-art? Noteworthy, it is reasonable to contemplate that there could be possible extensions for classifying bursting patterns in systems with more than two timescales. In this context, the mathematical analysis would have to deal with nested fast subsystems, which has not yet been achieved (except in very particular cases) and therefore there is substantial work to do in order to extend the state-of-the-art approaches. However, still within the two-timescale framework the question of bursting classification extension remains. This question gains further support since there are electrophysiological recordings of bursting dynamics, which resist the state-of-the-art classification system. A case in point is depicted in Figure~\\ref{fig:MMBO_exp}, where the bursting oscillations has two phases, but the quiescent phase has the peculiarity that it appears to periodically rise close to a threshold, however the neuron does not have a transition to the active phase and instead descends back to its baseline activity and only on the second run the active phase emerges. \n\\begin{figure}[!t]\n\\centering\n\\includegraphics{MMBOexp.pdf}\n\\caption{Electrophysiological recordings of the lateral thalamic nuclei neuron in cat from~\\cite{roy84} show complex bursting oscillations.}\n\\label{fig:MMBO_exp}\n\\end{figure}\nThese observations suggest that there is an underlying complex mechanism for the quiescent phase of the oscillations and therefore point towards a bursting classification framework that has to also incorporate the analysis of the slow subsystem, which is in stark contrast to state-of-the-art approaches. Further motivating this view is our earlier study, which constructed the first example (to the best of our knowledge) of a slow-fast bursting model whereby the initiation of the bursting oscillations could not be explained by the fast subsystem of the underlying slow-fast model~\\cite{desroches13a}. However, therein we did not attempt to derive an improved bursting classification framework. Thus, we herein propose an extension of the state-of-the-art classification system that enforces the importance of considering complex dynamical mechanisms within the slow subsystem as the underlying cause of the initiation or termination of bursting oscillations, which can not be explained by the fast subsystem's analysis solely. To substantiate this novelty we will also show in subsequent sections how to construct a variety of these new cases of bursting oscillations. However, to better guide the reader throughout this manuscript we first replicate the results of our previous work in Figure~\\ref{fig:sketch} (panels (a1)-(c3)) and further summarise the key insights of the proposed novel classification framework. We first considered the minimal setting of systems with two-timescales and that possess at least two slow variables. We then constructed a bursting model whose quiescent phase displays small-amplitude near-threshold oscillations. Mathematically, it turns out that these observations are best explained by the so-called \\textit{folded-node singularities} defined in the slow subsystem ($\\varepsilon=0$) and associated \\textit{canard solutions}, which persist for small enough $\\varepsilon>0$; see Figure~\\ref{fig:sketch} panels (a1)-(c1). Noteworthy, these are not \\emph{per se} bifurcations of the fast subsystem but lie on saddle-node bifurcation curves of the fast subsystem (see Figure~\\ref{fig:sketch} panel (a1)). \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=14cm]{fnburstRationale.pdf}\n\\caption{(Top part) Rationale behind bursting classifications: that of Rinzel, Izhikevich, Golubitsky et al., respectively, as well as the folded-node bursting approach proposed in this paper. (Main part (ai)-(ci), i=1,2,3) An exemplary folded-node bursting scenario, namely folded-node-homoclinic bursting, is shown in (a2)-(c2) in different 3D phase-space projections; (b2) is a zoomed view of (a2) where the critical manifold $S^0$ has been replaced by attracting $S^a_{\\varepsilon}$ and repelling $S^r_{\\varepsilon}$ slow manifolds in the vicinity of the folded node (dot). The classical analysis using the approaches of Bertram et al. \/ Golubitsky et al. (a3), Rinzel and Izhikevich (b3)-(c3) is based upon the fast subsystem's bifurcation structure. The complementary key approach to fully characterize this bursting pattern, presented in panels (a1)-(c1), uses the slow subsystem and its flow which reveals the presence of the folded node \\textbf{fn}. \\mdsr{Note that the fold curve $\\mathcal{F}^-$ of the critical manifold $S^0$ (panel (a2)) is also a curve of saddle-node bifurcation points (labelled LP in (a1)) of the fast subsystem when both slow variables are considered as parameters, the same curve used in the singularity theory approach on panel (a3).} (Side part) Full system's equations as well as the various subsystems, fast and slow, obtained for $\\varepsilon=0$.}\n\\label{fig:sketch}\n\\end{figure}\nSuch unexpected and non-trivial emergent mathematical objects allow trajectories of the slow subsystem to visit both the attracting ($S^0_a$) and repelling ($S^0_r$) parts of the critical manifold. In the full system (for small $\\varepsilon\\!>\\!0$) the perturbed versions of these manifolds --attracting $S^{\\varepsilon}_a$ and repelling $S^{\\varepsilon}_r$ slow manifolds-- twist and intersect multiple times (see Figure~\\ref{fig:sketch} panels (a2)-(b2)) thereby causing trajectories to non-trivially and robustly oscillate during the quiescent phase. \\mdsr{In essence, if the slow dynamics change direction along a fold then this can create a folded node scenario.} The transition from quiescent to active phase is caused by a repulsion of the trajectory away from the unstable sheet of the critical manifold; this phenomenon is mediated by canard solutions due to the presence of the folded node in the slow subsystem. In this particular example, the fast oscillations of the active phase are due to a nearby family of Hopf bifurcations in the fast subsystem (Figure~\\ref{fig:sketch} panel (c2)). The return back to quiescence is then caused by a family of homoclinic bifurcations of the fast subsystem. The key insight is that the fast subsystem is blind to what is causing these small-amplitude oscillations during the quiescent phase, and thus it is unable to classify the initiation of these oscillations based upon the bifurcations of fast subsystem only. This point is illustrated by the Rinzel-Izhikevich slow-fast dissection and projection of the trajectory of the full system onto the bifurcation diagram of the fast subsystem (see Figure~\\ref{fig:sketch} panels (b3)-(c3)). Note that by employing the Rinzel-Izhikevich classification system, the bursting dynamics would be explained by two bifurcations of the fast subsystem, namely the fold bifurcation LP$_2$ and the homoclinic bifurcation Ho$_2$. In particular, a fold bifurcation (LP$_2$) does not explain an oscillation. Moreover, a similar argument applies to the Golubitsky approach (see Figure~\\ref{fig:sketch} panel (a3)). This panel displays curves of codimension-one bifurcation points of the fast subsystem, which meet at codimension-two e.g. a Bogdanov-Takens BT (within a two-dimensional parameter space). It can then be shown that it is impossible to construct a path for the slow dynamics (within this two-dimensional parameter space), in particular along the homoclinic and saddle-node curves (since these characterise the bursting in the fast subsystem), which could explain folded-node-initiated quiescent phase oscillations. It turns out amongst all possible folded singularities, only folded nodes (and in limiting cases, so-called \\textit{folded saddle-nodes}) can generate such robust small-amplitude oscillations in the full system, and this is due to the twisting of nearby attracting and repelling slow manifolds. The key message is that these folded-node singularities only appear in the slow limit of the underlying two-slow-two-fast bursting systems and are invisible in the fast limit. Therefore one must consider both fast and slow subsystems in order to fully characterise the novel bursting scenarios associated with folded-node singularities, which leads us to a novel bursting classification system (see Figure~\\ref{fig:sketch} top panel in blue for the new framework). We believe these insights will fuel subsequent developments in higher dimensional multi-scale systems.\n\nAs hinted \\mdsr{above}, there are emergent slow dynamical mechanisms (captured by slow variables) that are blind to the state-of-the-art classification and thus this suggests the need to extend the classification of bursting oscillations. An example of such emergent slow dynamics is the so-called \\textit{folded node} and, in a minimal setting, it emerges due to non-trivial interactions between two slow variables. Indeed, herein we focus on new classes of bursting oscillations modelled via slow-fast systems with (at least) two slow variables and (at least) two fast variables and for which some epochs of the oscillatory time-series is explained by folded-node dynamics. This underlying folded-node signature leads us to name the resulting new classes of bursting models, \\textit{folded-node bursters}. Three fundamental cases are envisaged. The first case are bursters characterised by small-amplitude oscillations that occur during the quiescence phase, in which case we will refer to the \\textit{classical folded-node} bursting scenario. The second case involves slow-amplitude modulation of the burst, which we will denote as the \\textit{cyclic folded-node} bursting scenario. The third case, combines \\textit{classical folded-node} and \\textit{cyclic folded-node}. These classes of bursting patterns involve both the fast subsystem and the slow subsystem of the model, unlike traditional bursters. A second key aspect of these new classes is the central role played by \\textit{canards}, namely, \\textit{spike-adding canard} cycles involved in the classical folded-node bursting case, and \\textit{torus canards} in the cyclic folded-node bursting case. In the following subsections, we describe in details these two scenarios.\n\n\\subsection{Classical folded-node case}\n\\label{sec:classicalFN}\nHere we propose several bursting oscillations mediated by a classical folded node and the modelling steps of underlying \\mdsr{idealized} models are given. To guide the reader towards a modelling strategy of these systems, we first recall key concepts and mechanisms.\n\n\\subsubsection{A necessary preliminary step: spike-adding canard explosion}\n\\label{sec:spikeadd}\nFirst, recall that \\textit{canards} are non-trivial trajectories that emerge due timescale separation and unexpectedly, these trajectories contain segments that follow both an attracting and a repelling slow manifold. This phenomenon has been thoroughly studied in planar systems (i.e. with 1 slow variable and 1 fast variable)~\\cite{benoit81,dumortier96,eckhaus83,krupa01,mishchenko94}, as well as, in 3D systems with two slow variables~\\cite{benoit90,desroches12,wechselberger05}. In applications, canards can be associated to complex (bio)physical mechanisms, for example in neuroscience it provides the best approximation to the excitability threshold in certain single-neuron models. This observation was first made by Izhikevich~\\cite{izhikevich07}, who showed that canards organise the transition to the spiking regime of type II neurons. This was later analysed in more details in~\\cite{demaesschalck15,desroches13b,wechselberger13}. Yet another important mechanism is the so-called \\textit{spike-adding canard explosion}. This canard phenomenon arises in bursting oscillations and can be described as a sequence of canard explosions which organise the transition from subthreshold oscillations to bursting solutions with more and more spikes per bursts. This phenomenon was first described and analysed (in the case of chaotic dynamics regime) in~\\cite{terman91} in the context of square-wave bursting. This was revisited more recently in~\\cite{guckenheimer09} from the computational standpoint of saddle-type slow manifolds and further described in~\\cite{nowacki12} in a modeling context to explain transient spikes. These analyses were later refined (from a canard standpoint) in~\\cite{desroches13a} and the canard-mediated spike-adding dynamics was fully analysed in~\\cite{desroches16} in the context of parabolic bursters (with two slow variables), revealing the central role of folded-saddle canards. Noteworthy, bursting oscillations that possess spike-adding mechanism is a limiting (border line) case that already hints to the importance of possibly including the analysis of the slow flow in a bursting classification framework. That is, spike-adding requires a turning point of the slow flow, a so-called \\textit{canard point}, whereby each new added spike (within the bursting phase) is born via a slow (delayed) passage through this turning point. Crucially, the fast subsystem is blind to the underlying canard trajectories occurring near the turning point (well-defined as such only in the slow flow) and instead only sees a fold bifurcation. Therefore the state-of-the-art bursting classification systems does not capture this aspect. Nevertheless, we refrain from declaring this phenomenon as a new bursting mechanism because a spike-adding canard explosion gives rise to canard cycles that exist only within exponentially thin parameter regions. Hence, the robust dynamics is the fold-initiated bursting dynamics, and the fast subsystem analysis still prevail in order to classify it. In contrast, if we consider a fold-initiated bursting scenario undergoing spike-adding canard explosion and if we further add a slow dynamics for the parameter that displays the spike-adding canard explosion (i.e. a second slow variable in the extended model) then we obtain a folded-node bursting system. This has a similar effect to the case in classical (van der Pol type) systems where the canard phenomenon becomes robust if one adds a second slow variable, which has the effect of creating a folded singularity in the resulting two-dimensional slow flow and allows for multiple canard trajectories to exist. The idea here is similar, but with two fast variables, allowing for bursting dynamics in conjunction with folded-node dynamics. A first example of this scenario was termed \\textit{mixed-mode bursting oscillations} in~\\cite{desroches13a} but we prefer to denote it more generally folded-node bursting. Indeed, folded-node bursting is a new form of bursting pattern with two slow variables where the silent phase contains small-amplitude (subthreshold) oscillations due to the presence of a folded node in the slow subsystem. This folded node is responsible for the presence of a funnel region in the full system and trajectories entering this funnel make a number of rotations (which can be controlled by adjusting parameters) before they leave it and start to burst. Hence, the passage through the folded-node funnel organises the transition from quiescence to burst and it can only be understood by suitably analysing the slow subsystem. We subsequently describe a strategy for constructing folded-node bursting systems.\n\\subsubsection{Construction of minimal folded-node bursting systems}\nAs a staring point, we consider the prototypical fold-initiated burster of Hindmarsh-Rose type. By this we mean a three-dimensional slow-fast system with two fast variables and one slow and a cubic-shaped family of equilibria in the fast subsystem, namely the critical manifold $S^0$. We can write the following set of differential equations (using the fast time $\\tau$) to describe the dynamics of such a system\n\\begin{equation}\\label{eq:protoburster}\n\\begin{split}\nx' &= y - f(x) + az,\\\\\ny' &= g(x)-y,\\\\\nz' &= \\varepsilon(\\alpha x + \\gamma\\beta - \\delta z),\n\\end{split}\n\\end{equation}\nwhere $f$ is a cubic polynomial function, $g$ is (at least) quadratic; moreover, $0\\!<\\varepsilon\\!\\ll1$ is a small parameter and $(a,\\alpha,\\beta,\\gamma,\\delta)$ are potential bifurcation parameters; why we use a product of two parameters in the $z$ equation will become clear in the next section. As we shall see in the example section to follow, one can also obtain all fold-initiated scenarios by using an unfolding of a codimension-3 degenerate Bogdanov-Takens (BT) bifurcation; see~\\cite{dumortier91} for details.\n\nA few assumptions are required in order for the system~\\eqref{eq:protoburster} to display fold-initiated bursting. First of all, we assume that $f$ and $g$ are adequately chosen so that the fast subsystem has a cubic-shaped family of equilibria that depends on $z$ as a parameter (for the fast subsystem). This family can be written as a cubic function: \n$$z=\\left(f(x)-g(x)\\right)\/a.$$ \nTherefore, the corresponding bifurcation diagram (of the fast subsystem) in $z$ is S-shaped and will have fold points. \nThe critical manifold is then given by \n\\begin{eqnarray}\\label{eq:critman}\nS^0:=\\left\\{(x,y,z)\\in\\mathbb{R}^3\\;\\big\/\\;\\;y=g(x)\\;,\\;z=\\left(f(x)-y\\right)\/a\\right\\}.\n\\end{eqnarray}\nWe also require bistability in the fast subsystem between equilibria and limit cycles, in an interval of $z$-values. One bound of this interval correspond to a fold bifurcation and, geometrically, to one fold point of the cubic family of equilibria. The other boundary of the region of bistability of the fast subsystem will be a bifurcation of limit cycles and we shall consider three main cases, namely, saddle-homoclinic bifurcation (see Fig.~\\ref{fig:fnhom}), Hopf bifurcation (see Fig.~\\ref{fig:fnhopf}) and fold bifurcation of cycles (see Fig.~\\ref{fig:fnsnp}), but the list is not exhaustive. Now, considering the linear slow dynamics of system~\\eqref{eq:protoburster} for the slow variable $z$, we assume that a variation of one of the two parameters $\\alpha$ and $\\beta$ in the full system induces the linear $z$-nullsurface to cut through the fold point of the critical manifold $S^0$ for a certain value of this parameter. One can show that this creates a Hopf bifurcation in the full system which induces limit cycles to appear. Provided this transversal cut of the $z$-nullsurface with the critical manifold takes place, then a spike-adding canard explosion will emerge, whereby bursting solutions appear from subthreshold (spikeless) periodic solutions along branch of limit cycles undergoing multiple canard explosions; see~\\cite{desroches13a} for an example of this phenomenon in the context of square-wave bursting. As explained in the previous section, one salient feature of the spike-adding canard explosion is the presence of a turning point (a canard point) in the slow flow of system~\\eqref{eq:protoburster}. To compute the slow-flow, we first rescale time in~\\eqref{eq:protoburster} by a factor $\\varepsilon$. That is, we rescale the fast time $\\tau$ (with $x'=dx\/d\\tau$) into the slow time $t$ defined by $t=\\varepsilon\\tau$. This bring the system to the slow-time parametrisation \n\\begin{equation}\\label{eq:protobursterslowtime}\n\\begin{split}\n\\varepsilon\\dot{x} &= y - f(x) + z,\\\\\n\\varepsilon\\dot{y} &= g(x)-y,\\\\\n~~\\dot{z} &= (\\alpha x + \\gamma\\beta - \\delta z),\n\\end{split}\n\\end{equation}\nwhose $\\varepsilon=0$ limit corresponds to the slow subsystem. The slow subsystem is a differential-algebraic equation (DAE), where the dynamics of $z$ is explicitly preserved while $x$ and $y$ are slaved to $z$ by the algebraic constraints that corresponds to the equation of the (here one-dimensional) critical manifold $S^0$. The dynamics of $x$ and $y$ can be revealed by differentiating the algebraic constraint with respect to the slow time, which gives after rearranging the following one-dimensional dynamical system defined on $S^0$\n\\begin{equation}\\label{eq:protobursterslowflow}\n\\begin{split}\n\\dot{x} &= a\\frac{\\alpha x + \\gamma\\beta - \\delta z}{f'(x)-g'(x)}.\n\\end{split}\n\\end{equation}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{FNHom.pdf}\n\\caption{Folded-node\/Homoclinic bursting. We take $f(x)=x^3-3x^2$ and \\bluebis{$G(x,y,z)=g(x)-y$ with $g(x)=1-5x^2$. The parameter values are: $a=1$, $c=1$, $\\alpha=0.3$, $\\gamma=1$, $\\delta=1.2$, $\\varepsilon=0.002$, $\\mu=0.033$, $\\gamma_y=0.0005$ and $\\gamma_{\\beta}=-0.008$}. Panels (a-b) show the spike-adding transition in system~\\eqref{eq:fnburster}: (a) in the $(z,x)$ plane; (b) associated bifurcation diagram with respect to parameter $\\beta$. Panels (c-d) show a folded-node\/homoclinic bursting orbit: (c) in the $(\\beta,z,x)$ space; (d) $x$-time series. The bottom panels show a comparison between this folded-node bursting orbit from~\\eqref{eq:fnburster} and experimental data from~\\cite{roy84}.}\n\\label{fig:fnhom}\n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{FNHopf.pdf}\n\\caption{Folded-node\/Hopf bursting. We take $f(x)=x^3-3x^2$ and \\bluebis{$G(x,y,z)=g(x)-y$ with $g(x)=1-5x^2$. The parameter values are: $a=1$, $c=2$, $\\alpha=0.3$, $\\gamma=1$, $\\delta=1$, $\\varepsilon=0.004$, $\\mu=0.0104$, $\\gamma_y=0.0003$ and $\\gamma_{\\beta}=-0.05$}. Panels (a-b) show the spike-adding transition in system~\\eqref{eq:fnburster}: (a) in the $(z,x)$ plane; (b) associated bifurcation diagram with respect to parameter $\\beta$. Panels (c-d) show a folded-node\/Hopf bursting orbit: (c) in the $(\\beta,z,x)$ space; (d) $x$-time series.}\n\\label{fig:fnhopf}\n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{FNSNP.pdf}\n\\caption{Folded-node\/Fold of cycles bursting. We take $f(x)=0$ and \\bluebis{$G(x,y,z)=-x^3+A_1(z)x+A_2(z)-y(A_3(z)-x+x^2)$}, where \\bluebis{$A_1(z)=0.1201z+0.1871$, $A_2(z)=0.0906z-0.0251$, $A_3(z)=0.105z-0.3526$}. \\bluebis{The parameter values are: $a=0$, $c=1$, $\\alpha=0$, $\\gamma=-1$, $\\delta=1$, $\\varepsilon=0.01$, $\\mu=-0.00012$, $\\gamma_y=-0.003$, $\\gamma_{\\beta}=0.0001$}. Panels (a-b) show the spike-adding transition in system~\\eqref{eq:fnburster}: (a) in the $(z,x)$ plane; (b) associated bifurcation diagram with respect to parameter $\\beta$. Panels (c-d) show a folded-node\/fold of cycles bursting orbit: (c) in the $(\\beta,z,x)$ space; (d) $x$-time series.}\n\\label{fig:fnsnp}\n\\end{figure}\nAs is \\red{typical} in slow-fast systems with folded critical manifolds, note that the denominator of the right-hand side of~\\eqref{eq:protobursterslowflow} vanishes at fold points of $S^0$, which makes \\mdsr{generically} the dynamics of $x$ explode \\mdsr{and the corresponding fold point is referred to as a \\textit{jump point}. However, if} the numerator has a zero of the same order \\mdsr{as the denominator, then there can be a cancellation and the dynamics of $x$ does not explode; in this case, the fold point is referred to as a \\textit{canard point} or a \\textit{turning point}. The condition for a canard point to occur \\red{in this system is}} \\mdsr{given} by the following condition: \n\\begin{equation}\\label{eq:protoburstercanardpoint}\n\\begin{split}\nz_f &= (\\alpha x_f + \\gamma\\beta)\/\\delta,\n\\end{split}\n\\end{equation}\nwhere $(x_f,z_f)$ is a fold point of $S^0$. This indeed gives a transversal crossing of the slow nullsurface with the critical manifold at one of its fold points. Even though~\\eqref{eq:protoburstercanardpoint} depends on several parameters, it is a codimension-1 condition, therefore by fixing all parameters but one, then the condition can be satisfied by adjusting the last parameter. We arbitrarily choose to vary $\\beta$, which will become a second slow variable in the full 4D folded-node bursting system that we will construct below. Therefore, the spike-adding transitions leading to bursting in system~\\eqref{eq:protoburster} are obtained as the result of the slow nullsurface moving though one fold point of the critical manifold upon variation of $\\beta$. The same dynamics would be obtained by varying a parameter affecting the critical manifold while maintaining the slow nullsurface fixed, in particular if we were to append an additive parameter $I$ to the $x$ equation of the system. This would mimick the effect of an applied (external) current in a neuron-type model such as the Hindmarsh-Rose model~\\cite{hindmarsh84} or the Morris-Lecar model~\\cite{morris81,terman91}. However, from the pure dynamical viewpoint, varying a parameter in the slow equation results in the same effect and this is the scenario that we chose in order to construct fold-initiated spike-adding transitions in the original 3D burster and folded-node bursting in the extended 4D model.\n\nStarting from a fold-initiated bursting scenario with spike-adding canard explosion (controlled via a static variation of parameter $\\beta$) then a folded-node bursting is obtained by prescribing the dynamics on $\\beta$ by a slow differential equation. That is, we consider the following extended bursting system\n\\bluebis{\n\\begin{equation}\\label{eq:fnburster}\n\\begin{aligned}\nx' &= (y - f(x) + az)\/c,\\\\\ny' &= G(x,y,z),\\\\\nz' &= \\varepsilon(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\beta' &= \\varepsilon\\big(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\big).\n\\end{aligned}\n\\end{equation}\n}\n\\bluebis{Note that we will consider prototype systems where either $G$ is directly given as a graph over $x$, described as, $G(x,y,z)=g(x)-y$ (i.e. Folded-node\/homoclinic and folded-node\/Hopf cases), or the level set $\\{G(x,y,z)=0\\}$ is a graph over $(x,z)$, expressed as, $\\{y=g(x,z)\\}$ (i.e. Folded-node\/fold of cycles case). We claim that all folded-node initiated bursting scenarios can be obtained in either of these two ways. In the latter case, our minimal model is inspired by the codimension-3 degenerate Bogdanov-Takens unfolding introduced in~\\cite{dumortier91} and further applied in the context of bursting in~\\cite{saggio17}. In practice (for simulation purposes), $\\mu$, $\\gamma_y$ and $\\gamma_z$ will be taken $O(\\varepsilon)$.} Therefore, system~\\eqref{eq:fnburster} is effectively a three-timescale dynamical systems with dynamics evolving on $O(1)$, $O(\\varepsilon)$ and $O(\\varepsilon^2)$ timescales. For convenience and to ease the folded-node analysis, we will keep the equations written as in~\\eqref{eq:fnburster} with only $\\varepsilon$ has an apparent timescale separation parameter. System~\\eqref{eq:fnburster} is parametrised by the fast time $\\tau$, meaning that the $\\varepsilon=0$ limit of that parametrisation of the system gives the fast subsystem where the slow variables $z$ and $I$ are frozen and become parameters. Introducing the slow time $t=\\varepsilon\\tau$ brings the system into the different parametrisation \n\\bluebis{\n\\begin{equation}\\label{eq:fnburstertau}\n\\begin{aligned}\n\\varepsilon\\dot{x} &= (y - f(x) + az)\/c,\\\\\n\\varepsilon\\dot{y} &= G(x,y,z),\\\\\n\\dot{z} &= \\alpha z + \\gamma\\beta - \\delta x,\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\n}\nwhose $\\varepsilon\\!=\\!0$ limit corresponds to the slow subsystem. We will show that, all other parameters being fixed, the slow subsystem of~\\eqref{eq:fnburstertau} possesses a folded-node singularity, which creates transient subthreshold oscillations that initiate the burst when $0<\\varepsilon\\ll1$, regardless of the values of other parameters. However, simulations require that $\\mu$\\, $\\gamma_y$ and $\\gamma_I$ be $O(\\varepsilon)$ in order for these small subthreshold oscillations to be recurrent, hence entering into a robust periodic bursting attractor which we name folded-node bursting. We provide numerical evidence of this point, based on the strength of the global return mechanism, even though we do not provide a rigorous proof of it.\n\n\\bluebis{Applying the same strategy as in the three-dimensional (bursting) case, and projecting onto the $(x,\\beta)$-plane (the dimension of the slow flow corresponds to the number of slow variables), we obtain the following system for the reduced system (or slow subsystem)\n\\begin{equation}\\label{eq:fnbursterslowflow}\n\\begin{aligned}\n\\dot{x} &= \\frac{(g_z(x,z)+a)(\\alpha z + \\gamma\\beta - \\delta x)}{f'(x)-g_x(x,z)},\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y\\left(g(x,z)-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\nafter substituting for $g(x,z)$ for $y$ from the critical manifold condition. Note that in two of the three examples that we will consider, $g(x,z)=g(x)$ depends only on $x$ and hence $g_z(x,z)=0$. The critical manifold of system~\\eqref{eq:fnburster} is not normally hyperbolic everywhere and, hence, the system possesses a (1D here) fold set defined by $$\\mathcal{F}:=\\{(x,y,z)\\in S^0;\\;f'(x)=g_x(x,z)\\}.$$\nThis implies that the slow flow~\\eqref{eq:fnbursterslowflow} of system~\\eqref{eq:fnburster} is not defined along $\\mathcal{F}$. The slow flow can be extended along $\\mathcal{F}$ by preforming an $x$-dependent time rescaling which amounts to multiply the right-hand side of~\\eqref{eq:fnburster} by a factor $f'(x)-g_x(x,z)$, hence yielding the so-called \\textit{desingularised reduced system (DRS)}\n\\begin{equation}\\label{eq:fnbursterDRS}\n\\begin{aligned}\n\\dot{x} &= \\left(g_z(x,z)+a\\right)(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\dot{\\beta} &= \\left(f'(x)-g_x(x,z)\\right)\\left(\\mu-\\gamma_y\\left(g(x,z)-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right),\n\\end{aligned}\n\\end{equation}\nwith $z=z(x)$ defined by $S^0$, that is, $g(x,z)-f(x)+\\alpha z=0$. In all cases we shall consider (including the general codimension-3 unfolding of a degenerate BT bifurcation from~\\cite{dumortier91}), $z$ can be written as a function of $x$ on $S^0$. \nAs a consequence of this $x$-dependent time rescaling, the DRS~\\eqref{eq:fnbursterDRS} is regular everywhere in $\\mathbb{R}^2$ including on $\\mathcal{F}$, along which it has the possibility for equilibria simply by appearance of the factor $f'(x)-g_x(x,z)$ in the $\\beta$-equation. The equilibrium condition is then that $\\dot{x}=0$ in~\\eqref{eq:fnbursterDRS} together with $f'(x)-g_x(x,z)$, which conveys the idea already seen in the 3D (bursting) case. That is, a singularity of the reduced system at a point on $\\mathcal{F}$ can be resolved if and only if the numerator of the right-hand side of $\\dot{x}$ in that system vanishes at this point and the zeros of the two algebraic expressions to be of the same order. Such points are called \\textit{folded singularities} (or \\textit{folded equilibria}) and they are the equivalent of canard points in the cases with (at least) two slow variables. Folded equilibria are equilibria of the DRS~\\eqref{eq:fnbursterDRS} and, according to their topological type as equilibria of the DRS, one can generically define \\textit{folded nodes}, \\textit{folded saddles} and \\textit{folded foci}. However they are not equilibria of the reduced system~\\eqref{eq:fnbursterslowflow} due to the singular time rescaling performed to pass from one to the other. Indeed, this time rescaling is chosen so that trajectories of the DRS have reversed orientation on the repelling sheet of $S^0$ compared to trajectories of the reduced system (both have the same orientation along the attracting sheet). Hence, in the case of folded nodes and folded saddles, trajectories starting on the attracting sheet of $S^0$ may cross the folded singularity in finite time and with finite speed.\\newline\nThe Jacobian matrix of~\\eqref{eq:fnbursterDRS} evaluated at a folded equilibrium has the form\n\\begin{equation}\\label{eq:JacDRS}\n\\begin{aligned}\n\\mathrm{J}=\\begin{pmatrix}\n(-\\delta+\\alpha z'(x))(g_z(x,z)+a) & \\gamma(g_z(x,z)+a)\\\\\nK_2 & 0\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwhere $$K_2=(f''(x)-\\partial_xg_x(x,z))\\left(\\mu-\\gamma_y\\left(g(x,z)-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right).$$ From~\\eqref{eq:JacDRS}, one can easily write down conditions that enable the emergence of a folded-node singularity ($\\mathrm{tr}(J)<0$, $\\det{J}>0$, $\\mathrm{tr}(J)^2-4\\det{J}>0$) or a folded-saddle singularity ($\\det{J}<0$) in the reduced system. As we will explain below, even though only the folded-node case gives rise to robust bursting patterns, the folded-saddle case is still interesting in the study of 4D bursters with two slow variables. One also can easily verify that our minimal example systems all give rise to a folded-node case. Indeed, in the folded-node\/homoclinic (Figure~\\ref{fig:fnhom}) and folded-node\/Hopf (Figure~\\ref{fig:fnhopf}) bursting cases, system~\\eqref{eq:fnburster} has the form\n\\begin{equation}\\label{eq:fnhomburster}\n\\begin{aligned}\nx' &= (y - x^3 + 3x^2 + z)\/c,\\\\\ny' &= 1 - 5x^2 - y,\\\\\nz' &= \\varepsilon(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\beta' &= \\varepsilon\\left(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right),\n\\end{aligned}\n\\end{equation}\nwhich hence gives the following DRS's Jacobian matrix\n\\begin{equation}\\label{eq:JachomDRS}\n\\begin{aligned}\n\\mathrm{J_{1,2}}=\\begin{pmatrix}\n-\\delta & \\gamma\\\\\nK_2 & 0\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwith: $K_2=(-6x_{\\mathrm{fs}}-4)\\left(\\mu-\\gamma_y\\left(1-5x_{\\mathrm{fs}}^2-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right)$, and $x_{\\mathrm{fs}}=-4\/3$. Given the chosen parameter values corresponding to Figures.~\\ref{fig:fnhom} and~\\ref{fig:fnhopf}, then we immediately conclude that we have indeed a folded node. Likewise, in the folded-node\/fold of cycles case illustrated in Figure~\\ref{fig:fnsnp}, the slow-fast system corresponding to~\\eqref{eq:fnburster} is \n\\begin{equation}\\label{eq:fnsnpburster}\n\\begin{aligned}\nx' &= y,\\\\\ny' &= -x^3+A_1(z)x+A_2(z)-y(A_3(z)-x+x^2),\\\\\nz' &= \\varepsilon(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\beta' &= \\varepsilon\\left(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right),\n\\end{aligned}\n\\end{equation}\nwhere $A_i=a_i z+b_i$ ($i=1,2,3$) are linear functions of $z$. Therefore, we obtain the associated DRS's Jacobian matrix\n\\begin{equation}\\label{eq:JacsnpDRS}\n\\begin{aligned}\n\\mathrm{J_{1,2}}=\\begin{pmatrix}\n\\left(-\\delta+\\alpha\\frac{3x_{\\mathrm{fs}}^2-b_1}{a_1+a_2}\\right)(a_1x_{\\mathrm{fs}}+a_2) & \\gamma(a_1x_{\\mathrm{fs}}+a_2)\\\\\nK_2 & 0\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwith: $K_2=(6x_{\\mathrm{fs}}-a_1)\\left(\\mu-\\gamma_yy_{\\mathrm{fold}}^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right)$ and $x_{\\mathrm{fs}}$ solution to $$-3x_{\\mathrm{fs}}^2+a\\frac{x_{\\mathrm{fs}}^3-b_1x_{\\mathrm{fs}}-b_2}{a_1+a_2}+b_1=0.$$ Substituting the parameter values for their chosen numerical value mentioned in the caption of Figure~\\ref{fig:fnsnp} allows to conclude that we are indeed dealing with a folded node.}\\newline\nOne can obtain the general DRS~\\eqref{eq:fnbursterDRS} by applying implicit differentiation to one algebraic equation only (the right-hand side of the $\\dot{x}$ equation in the original system) and substituting $g(x,z)$ for $y$ (coming from the second algebraic equation). This gives the same result as the DRS obtained from both algebraic constraint together. Indeed, in all generality, applying implicit differentiation to the two algebraic equations of the slow subsystem gives \n\\begin{equation}\\label{eq:RS2fast}\n\\begin{aligned}\n\\begin{pmatrix}\n-f'(x) & 1 \\\\\n-g_x(x,z) & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n\\dot{x} \\\\\n\\dot{y}\n\\end{pmatrix}\n&=\n\\begin{pmatrix}\n-a \\\\\ng_z(x,z)\n\\end{pmatrix}\n(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{z} &= \\alpha z + \\gamma\\beta - \\delta x,\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\nwhich by Kramer's rule is equivalent, after posing $$\\mathrm{J}=\n\\begin{pmatrix}\n-f'(x) & 1 \\\\\n-g_x(x,z) & 1\n\\end{pmatrix}\n,$$ (Jacobian matrix of the original vector field with respect to the fast variables at $\\varepsilon=0$) to \n\\begin{equation}\\label{eq:RS2fast2}\n\\begin{aligned}\n\\det(\\mathrm{J})\n\\begin{pmatrix}\n\\dot{x} \\\\\n\\dot{y}\n\\end{pmatrix}\n&= \\mathrm{Adj(J)}\\begin{pmatrix}\na \\\\\n-g_z(x,z)\n\\end{pmatrix}\n(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{z} &= \\alpha z + \\gamma\\beta - \\delta x,\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\nwhere $\\det(\\mathrm{J})=g_x(x,z)-f'(x)$ and \n$$\\mathrm{Adj}(\\mathrm{J})=\n\\begin{pmatrix}\n1 & -1\\\\\ng_x(x,z) & -f'(x)\n\\end{pmatrix},$$ denote the determinant and the adjugate matrix of $\\mathrm{J}$, respectively. The previous system is singular when $\\det(\\mathrm{J})$ vanishes, which happens on the fold set. It can be desingularized by rescaling time by a factor $\\det(\\mathrm{J})$, which brings the DRS in its most general form, namely\n\\begin{equation}\\label{eq:DRS2fast}\n\\begin{aligned}\n\\begin{pmatrix}\n\\dot{x} \\\\\n\\dot{y}\n\\end{pmatrix}\n&= \\mathrm{Adj(J)}\\begin{pmatrix}\na \\\\\n-g_z(x,z)\n\\end{pmatrix}\n(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{z} &= \\det(\\mathrm{J})(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{\\beta} &= \\det(\\mathrm{J})\\big(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2\\big).\n\\end{aligned}\n\\end{equation}\nAfter being projected onto the $(x,\\beta)$-space, system~\\eqref{eq:DRS2fast} then takes the form\n\\begin{equation}\\label{eq:DRS2fast2}\n\\begin{aligned}\n\\dot{x} &=(a+g_z(x,z))(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{\\beta} &= (f'(x)-g_x(x,z))\\big(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2\\big),\n\\end{aligned}\n\\end{equation}\nwhich indeed agrees with~\\eqref{eq:fnbursterDRS}.\n\nWith the above analysis, we can construct in principle any folded-node burster of our liking. We showcase three examples. First, a folded-node homoclinic burster is presented in Fig.~\\ref{fig:fnhom} and we also show in the bottom panels a comparison with data from~\\cite{roy84} (also displayed in Fig.~\\ref{fig:MMBO_exp}). Note that our \\mdsr{idealized} model was not initially designed to explain these data, yet the time profiles match remarkably well. The strong similarity between our \\mdsr{idealized} model and these data suggest that folded-node bursting constructions could potentially inform the design of biophysical models. Then, a folded-node Hopf burster is presented in Fig.~\\ref{fig:fnhopf}. Last, a folded-node fold-cycle burster is shown in Fig.~\\ref{fig:fnsnp}. In each of these three figures, we show in panel (a) the classical Rinzel dissection between the bifurcation diagram of the 2D fast subsystem of the underlying 3D fold-initiated bursting model, and several limit cycles of this 3D bursting model undergoing a spike-adding canard explosion. In panel (b), we show the bifurcation diagram of the 3D bursting system upon variation of the parameter which will become the second slow variable of the folded-node burster, and this diagram displays spike-adding canard explosion. Note that a full analysis of these spike-adding canard explosion scenarios is beyond the scope of the present work as it is by-and-large an open research question. In panels (c), we show a folded-node bursting cycle in a 3D phase-space projection together with the critical manifold $S^0$. In panel (d), we plot the time series of this cycle for the fast variable $x$. More examples of folded-node bursting scenarios can be constructed by following the procedure highlighted above and by choosing a different bifurcation of the fast subsystem ending the burst.\n\nFinally, we quickly reflect on why folded-saddle bursting is not robust. The folded-saddle case is simply a different parameter regime in the slow subsystem, however the resulting dynamics is substantially different than that generated by a folded node. In neuron models with (at least) two slow variables, folded saddles and their associated canard solutions play the role of firing threshold. In particular, in the context of bursting system, they have recently been shown to organise the spike-adding transition in parabolic bursters~\\cite{desroches16,desroches18}. Counterintuitively, small-amplitude oscillations can also emerge in the vicinity of a folded saddle; see~\\cite{mitry17} for a rigorous analysis of this phenomenon and also~\\cite{desroches16b,desroches18} for further related work. However, there is no funnel near a folded saddle and the canard dynamics is hence not robust, which applies no matter how many fast variables the system possesses, so in particular in the context of bursting. This is why, in systems with (at least) two fast and two slow variables, only the folded-node case gives rise to a new class of bursting oscillations.\n\n\\subsubsection{Existence of folded-node bursting solutions}\nMixed-mode oscillations have been the subject of intense research in the past few decades, in particular (more recently) in the context of multiple-timescale systems; see~\\cite{desroches12}. However, there are very few results guaranteeing the existence of MMOs. One example is due to Br{\\o}ns, Krupa and Wechselberger in~\\cite{brons06}, where they prove the existence of ``simple'' patterns of MMOs ---with one large-amplitude oscillations and $s$ small-amplitude oscillations, hence denoted a $1^s$ MMO--- using a perturbation argument (in $\\varepsilon$) from a singular orbit that they construct using both slow and fast limits of the original system. Their strategy can be adapted to prove the existence of a folded-node bursting periodic orbit by $\\varepsilon$-perturbation of a singular orbit. The segment of that singular orbit constructed using the slow flow is basically the same as in the MMO case, that is, it lies in the singular funnel or on the strong canard of the folded node. However, in the case of folded-node bursting the fast subsystem is of dimension 2 and displays limit cycles in the fast part of the cycle. Therefore one needs to concatenate the slow-flow segment with a segment along the \\textit{averaged slow flow} of the system; see e.g.~\\cite{roberts15} for details. The averaged slow flow is defined using the fast subsystem in its oscillatory regime and averaging out the fast variables $x$ and $y$ along one cycle to define a slow motion for the slow variables $z$ and $\\beta$; its equations are hence given by:\n\\begin{equation}\\label{eq:avslowflow}\n\\begin{aligned}\nx' &= (y - f(x) + az)\/c,\\\\\ny' &= G(x,y,z),\\\\\nz' &= \\alpha z + \\gamma\\beta - \\delta \\langle x \\rangle,\\\\\n\\beta' &= \\mu-\\frac{\\gamma_y}{T_z}\\int_0^{T(z)}(y(s)-y_{\\mathrm{fold}})^2 -\\gamma_{\\beta}(\\beta(s)-\\beta_{\\mathrm{fold}})^2 ds.\n\\end{aligned}\n\\end{equation}\n\nHence the singular orbit is formed by the following segments (see Figure~\\ref{fig:singorb}):\n\\begin{enumerate}\n\\item A critical fiber connecting the folded node to the landing-up point $\\mathrm{p_u}$;\n\\item A trajectory of the averaged slow flow ending along the line of bifurcation points of the fast subsystem ending the burst;\n\\item A fast fiber connecting that point to landing-down point $\\mathrm{p_d}$;\n\\item A segment of the slow flow connecting $\\mathrm{p_d}$ to the folded node.\n\\end{enumerate}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{SingPerorb.pdf}\n\\caption{Singular periodic orbit from which an MMBO perturbs for small enough $\\varepsilon>0$. Together with the attracting $S^0_a$ and repelling $S^0_r$ sheets of the critical manifold, the lower fold curve $\\mathcal{F}^-$, and the folded node fn, also shown on this figure are the average slow nullsurface $\\mathcal{A}$, the singular strong $\\gamma_s^0$ and weak $\\gamma_w^0$ canards, the landing-up point $\\mathrm{p_u}$, the landing-down point $\\mathrm{p_d}$ and the singular periodic orbit $\\Gamma_{\\mathrm{sing}}$.}\n\\label{fig:singorb}\n\\end{figure}\n\n\\subsection{Cyclic folded-node case}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{fnodecyc.pdf}\n\\caption{Cyclic folded-node bursting cases. We use a polar-coordinates formulation in order to construct idealized models. The top panels show the slow-fast dissection for the amplitude $r$ of the underlying bursting model, with three different torus canard scenarios (a), (b) and (c). Adding a slow dynamics on parameter $a$ yields associated cyclic folded-node bursting scenarios for which we show both the slow-fast dissection in the $(a,r)$ plane and the $x$ time series : (a) initiated by a subcritical Hopf bifurcation; (b) terminated by a fold of cycles; (c) initiated by a fold of cycles.}\n\\label{fig:fncy}\n\\end{figure}\n\\mdsr{In the same spirit as in Section~\\ref{sec:classicalFN}, one can construct interesting bursting rhythms where the slow oscillations occur on the envelope of the burst and this is due to what we will \\red{denote} \\textit{cyclic folded node}. Parallel to the construction of a folded-node burster system, one can construct a cyclic-folded-node burster system by considering a three-dimensional slow-fast system which possesses \\textit{torus canard} solutions. Loosely speaking, this corresponds to a canard phenomenon with a fast rotation. Already mentioned by Izhikevich in~\\cite{izhikevich01} in a canonical model, it \\red{was later} found in a biophysical model of Cerebellar Purkinje cell exhibiting fold\/fold cycle bursting~\\cite{kramer08}, and subsequently analysed with more mathematical details in, e.g.,~\\cite{benes11,burke12}. Even though to date not all elements of torus canard transitions have been \\red{mathematically} unravelled, one can summarise this phenomenon by \\red{emphasising that its key feature} corresponds to a canard explosion within a fast oscillatory motion. Instead of following slowly a family of equilibria past a fold bifurcation, the fast-oscillating system follows slowly a family of equilibria past a cyclic fold bifurcation. \\red{Moreover, one can draw} a parallel between classical canards and torus canards in their role of transitional regime in neuronal dynamics: classical canards can explain the rapid transition from rest to the spiking regime, likewise torus canards can explain the rapid transition from the spiking to the bursting regime. \\red{Furthermore}, torus canards are also not robust and only exist within exponentially thin parameter regions. \\red{Thus, the} very same idea that leads from canard point to folded singularities, can lead from torus canard to cyclic folded-node canards, when adding a second slow variable. \\red{In this way}, a cyclic folded-node can be robust even if the torus canards are not robust. This has been proposed very recently by Vo and collaborators~\\cite{vo17,vo16} \\red{via a specific example that links} the resulting dynamics to the amplitude-modulated bursting already mentioned in~\\cite{izhikevich01,kramer08}; see also~\\cite{han18} for other examples of amplitude-modulated bursting. \\red{In summary, we herein} propose a taxonomy of cyclic folded-node bursting patterns, with several numerical examples, which completes our extension of the previous bursting classifications. \\red{We complement this} with few examples of idealized models displaying cyclic-folded-node bursting. \\red{This is achieved by considering systems expressed} in polar form, in which case the condition for cyclic folded node and then for cyclic folded-node bursting reduce to folded-node conditions on $r$; see Figure~\\ref{fig:fncy}. In general, \\red{it is possible to reduce} the system locally near the cyclic fold bifurcation of the fast subsystem \\red{enabling the computation of} normal form coefficients (see~\\cite{roberts15,roberts17,vo16,vo17}) \\red{that effectively characterise} the cyclic folded-node. However, the bursting conditions have not been established in general. \\red{Finally, for sake of completeness, we construct a limiting case of a non-trivial system that displays both classical folded-node bursting and cyclic-folded-node bursting, as depicted in} Figure~\\ref{fig:fnfncy}.}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{fnfncbursting.pdf}\n\\caption{An example of a folded-node and cyclic folded-node bursting case. The $x$-times series of the folded-node \/ cyclic folded-node bursting solution is shown; panel (b) is a zoom of panel (a) near the folded node.}\n\\label{fig:fnfncy}\n\\end{figure}\n\n\\section{Conclusion and perspectives}\n\\label{sec:conclusion}\n\\mdsr{The research topic of mathematical classification of bursting patterns was initiated with seminal papers published in the mid 1980s with three proposed classes of bursting oscillations~\\cite{rinzel85,rinzel86,rinzel87}.} The key idea of comparing the fast subsystem's bifurcation diagram and the full systems' dynamics may seem natural with \\red{hindsight, but in fact it was a genuine breakthrough, which shaped the way bursting oscillations have been modelled and dissected ever since}. The present manuscript follows \\mdsr{these} footsteps, as well as those of the subsequent contributors on this topic~\\cite{bertram95,golubitsky01,izhikevich00}\\mdsr{, hence it was important to review in detail \\red{these results since they form one pilar of Mathematical Neuroscience but also have impact in other fields. We then take a step forward by proposing an extension of the classification scheme, which allows to cover more types of burster systems, in particular those with two slow variables, namely folded-node bursters}. What we propose is} a conceptual framework which does not pretend to full mathematical rigour \\red{(i.e. theorem derivations since its beyond the focus of the paper)} but rather it aims to \\red{provide key insights that will enable} further theoretical and modelling work on the vast question of bursting. \\mdsr{The extended bursting classification crucially focuses on the dynamics \\red{during the} silent phase where the termination \\red{of the trajectory profile} is not just a simple rise over the fold of the critical manifold but can involve subthreshold oscillations. We emphasize the importance of the slow flow in slow-fast systems with (at least) two slow variables, which was somehow previously overlooked in the context of bursting. In such two-slow-variable bursting systems, the silent phase termination is due to the presence of folded node. This scenario is known to give rise to canard solutions that organise, upon parameter variation but also, transiently, upon change of initial conditions, the number of subthreshold oscillations. This slow cycle-adding phenomenon is indeed entirely due to canards \\red{and it} controls the profile of the underlying bursting oscillations. \\red{Importantly,} it does so in a robust manner in the sense that such bursting patterns with subthreshold oscillations exist over order one ranges of parameter values. Therefore, we have added to the state-of-the-art classifications of bursting patterns the cases of classical and cycle folded-node bursters, which we can summarise in~\\ref{tab:fnburst} below. We also propose a few more cases to the existing classifications (refer to Appendix~\\ref{sec:newfast}), which to the best of our knowledge, have not been reported before.}\n\\begin{table}[!h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\\hline\n\\diaghead{\\theadfont Diag ColumnmnHead II}%\n {Initiation\\\\of the burst}{Termination\\\\of the burst}&\n\\thead{codim. 1 bif.\\\\of cycles}&\\thead{\\blue{cyclic \\textbf{fn}}}&\\thead{\\blue{classical \\textbf{fn}}}\\\\ \\hline\n\\blue{classical \\textbf{fn}} & \\blue{\\ding{51}},~\\cite{desroches13a} & \\blue{\\ding{51}} & \\cite{vbw}+1 slow var.\\\\ \\hline\ncodim. 1 bif. equilibria & \\cite{izhikevich00,rinzel85} & \\blue{\\ding{51}},~\\cite{vo16} & \\cite{vbw} \\\\ \\hline\ncodim. 1 bif. cycles & \\cite{izhikevich00,rinzel85} & \\blue{\\ding{51}} & \\cite{vbw} \\\\ \\hline\n\\blue{cyclic \\textbf{fn}} & \\blue{\\ding{51}} & \\texttt{modify Fig.~\\ref{fig:fncy} (b)} & \\blue{\\ding{51}} \\\\ \\hline\n\\end{tabular}\n\\vspace*{0.2cm}\n\\caption{Extended classification of bursting patterns.}\n\\label{tab:fnburst}\n\\end{center}\n\\end{table}\n\n\\mdsr{Where do we go from here? Following this initial framework for folded-node bursting, it will be important to develop this approach in the context of biophysical excitable cell models with more than one slow processes. To this extent, a very interesting question for follow-up work is to rethink \\red{about} folded-node bursting dynamics from a \\red{biophysical} modelling viewpoint. In all our \\mdsr{idealized} models of folded-node bursting, we have added feedback terms in the second slow differential equation with both positive and negative coefficients, which tends to indicate that both positive and negative feedback loops are useful to produce the desired output behaviour. \\red{In this context, we hight two interesting aspects associated with the experimental time-series that we attempted to model with our idealized model (folded-node bursting) reproduced in} Figs.~\\ref{fig:MMBO_exp} and~\\ref{fig:fnhom}. First, the subthreshold oscillations \\red{appear to be following} the excitability threshold, which may be harder to obtain in a three-dimensional model, even though some elliptic bursting models --e.g. FitzHugh-Rinzel, Morris-Lecar as well as some MMO models-- could potentially reproduce this aspect. However, \\red{these elliptic bursting models cannot} capture the second aspect. Note that our example of folded-node bursting has 3 time scales; this was done for convenience in the construction and may not be absolutely necessary. Second, the burst phase is located on a plateau (in terms of \\red{neuronal} membrane potential values) compared to the quiescent phase, which is reminiscent of a square-wave type bursting. Indeed our \\mdsr{idealized} folded-node bursting model \\red{reproduces quite well these data and in fact it can effectively be designated as a} folded-node homoclinic bursting model. Three-dimensional elliptic bursting models, or MMO models, would not be able to capture this aspect. \\red{One interesting possibility to find biophysical models with folded-node bursting dynamics is perhaps via existing models} of thalamic bursting, or \\red{alternatively to extend these models to explain the observational data published in}~\\cite{roy84}. In terms of application to neural dynamics, it is legitimate to ask about neural coding and the implications of folded-node dynamics within a bursting regime. There, one would want to compare spike-adding to folded-node cycle-adding. The cycle adding can quantize the slow phase duration which might have significant effect on silent phase (and therefore on active phase) durations. On the other hand, spike-adding has less impact on macroscopic timing and less impact if a spike is added to a burst of several, say, 6 or more, spikes. A single spike added in a 2-4 spike burst might have coding contributions (synaptic transmission) but less so if there are already more than 6 spikes in a burst. These questions go beyond the scope of the present paper but are certainly of direct interest for follow-up work. \\red{On the theoretical side, as aforementioned} we do not claim to have reached mathematical rigour but rather to have introduced a new framework for analysing bursting oscillations with two slow variables. There are clearly several open avenues for \\red{rigorous theorem-driven directions} to be explored. \\red{For instance,} we have \\red{elucidated} how to construct a singular orbit that will perturb to a folded-node bursting orbit, but, proving this perturbation result is not immediate. In general, proving rigorously the existence of canards in this 4D context and how both 3D parts (subthreshold and superthreshold) combine to organise the global dynamics require \\red{non-trivial} mathematical analysis.}\n\n\\mdsr{The question of noise is also a natural one to consider. If small to moderate noise is added to a folded-node bursting systems, \\red{it is likely that noise will not affect significantly the burst phase. However, it is expected that the phase} of spiking oscillations during the burst will be affected, but not the qualitative dynamics. Folded-node dynamics is known to be robust to noise, its time course is parametrically robust and noise-tolerant. The canard phenomenon accounts for subtle dynamic features like cycle-adding however the subthreshold oscillations near a folded node are robust. The noise will affect these subthreshold oscillations by modifying the rotation sector in which the trajectory falls into from one passage to the the next, however the oscillations will remain. To quantify this variability of the \\red{sector of a} folded-node burster with noise, one could use results by Berglund et al.~\\cite{berglund12}. However, here as well the qualitative dynamics and the key role of the slow subsystem and its folded node will remain. A rigorous understanding of the impact of noise on a folded-node burster model is certainly an interesting question that goes beyond the scope of the present work.}\n\n\\mdsr{Finally, the question of bursting dynamics with at least two slow variables and more than two timescales is also of interest and related to the present work. As \\red{aforementioned}, in the limit of folded-saddle-node singularities, small subthreshold oscillations will remain and increase in number and shape. In the context of slow-fast systems with two slow variables, this scenario is well-known to be akin to three-timescale dynamics~\\cite{krupa10}. The associated bifurcation structure is already involved in the three-dimensional setup, with involvement of adding organizing centers such as singular Hopf bifurcation points~\\cite{guckenheimer08}. \\red{Thus,} it is to be expected that the folded-saddle-node bursting profiles will be more rich and \\red{complex} to fully describe than the folded-node bursting cases presented \\red{herein}. Yet, the underlying robust mechanism that gives a bursting pattern and requires the analysis of both slow and fast subsystem will be similar as the one proposed in the present work. A full analysis of this limiting case is a very interesting and natural question for future work. Besides, bursting systems with more than two timescales have recently gained further interest in link with canard solutions~\\cite{desroches18,krupa12,letson17,nan15}, where the additional timescales bring more structure to the system and allow for further geometric singular perturbation analysis. Such approaches would certainly shed further light onto folded-node bursting dynamics as presented here and we regard it as a natural and interesting question for future work.} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\nThe standard setting of \\emph{prediction with expert advice} \\citep{littlestone1994weighted,freund1997boosting,vovk1998mixability,cesabianchi2006plg} aims to provide sound strategies for sequential prediction that combine the forecasts from different sources.\nMore precisely, in the so-called \\emph{Hedge problem} \\citep{freund1997boosting}, at each round the learner has to output a probability distribution on a finite set of \\emph{experts} $\\{1, \\ldots, M\\}$; the losses of the experts are then revealed, and the learner incurs the expected loss from its chosen probability distribution. \nThe goal is then to control the \\emph{regret}, defined as the difference between the cumulative loss of the learner and that of the best expert (with smallest loss).\nThis online prediction problem is typically considered in the \\emph{individual sequences} framework, where the losses may be arbitrary and in fact set by an adversary that seeks to maximize the regret.\nThis leads to regret bounds that hold under virtually no assumption~\\citep{cesabianchi2006plg}.\n\nIn this setting, arguably the simplest and most standard strategy is the \\emph{Hedge algorithm} \\citep{freund1997boosting}, also called the \\emph{exponentially weighted averaged forecaster} \\citep{cesabianchi2006plg}.\nThis algorithm depends on a time-varying parameter $\\eta_t$ called the \\emph{learning rate}, which quantifies by how much the algorithm departs from its initial probability distribution to put more weight on the currently leading experts.\nGiven a known finite time horizon $T$, the standard tuning of the learning rate is fixed and given by $\\eta_t = \\eta \\propto \\sqrt{\\log(M) \/ T}$, which guarantees an optimal worst-case regret of order $O( \\sqrt{T \\log M} )$.\nAlternatively, when $T$ is unknown, one can set $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$ at round $t$, which leads to an \\emph{anytime} $O(\\sqrt{T \\log M})$ regret bound valid for all $T \\geq 1$.\n\n\nWhile worst-case regret bounds are robust and always valid, they turn out to be overly pessimistic in some situations.\nA recent line of research \\citep{cesabianchi2007secondorder,derooij2014followtheleader,gaillard2014secondorder,koolen2014learning,sani2014exploiting,koolen2015squint,luo2015adanormalhedge} designs algorithms that combine $O(\\sqrt{T \\log M})$ worst-case regret guarantees with an improved regret on easier instances of the problem.\nAn interesting example of such an easier instance is the\nstochastic problem, where it is assumed that the losses are stochastic and that at each round the expected loss of a ``best'' expert is smaller than those of the other experts by some gap $\\Delta$.\nSuch algorithms rely either on a more careful, data-dependent tuning of the learning rate $\\eta_t$ \\citep{cesabianchi2007secondorder,derooij2014followtheleader,koolen2014learning,gaillard2014secondorder}, or on more sophisticated strategies \\citep{koolen2015squint,luo2015adanormalhedge}.\nAs shown by~\\citet{gaillard2014secondorder} (see also \\citealt{koolen2016combining}), one particular type of adaptive regret bounds (so-called \\emph{second-order bounds}) \nimplies at the same time a $O (\\sqrt{T\\log M})$ worst-case bound and a better \\emph{constant} $O(\\log (M) \/ \\Delta)$ bound in the stochastic problem with gap $\\Delta$.\nArguably starting with the early work on {second-order bounds} \\citep{cesabianchi2007secondorder}, the design of online learning algorithms that combine robust worst-case guarantees with improved performance on easier instances has been an active research goal in recent years \\citep{derooij2014followtheleader,gaillard2014secondorder,koolen2014learning,sani2014exploiting}.\nHowever, to the best of our knowledge, existing work on the Hedge problem has focused on developing new adaptive algorithms rather than on analyzing the behavior of ``conservative'' algorithms in favorable scenarios. \nOwing to the fact that the standard Hedge algorithm is designed for --- and analyzed in --- the adversarial setting \\citep{littlestone1994weighted,freund1997boosting,cesabianchi2006plg}, and that its parameters are not tuned adaptively to obtain better bounds in easier instances, it may be considered as overly conservative and not adapted to stochastic environments.\n\n\n\\paragraph{Our contribution.}\n\nThis paper fills a gap in the existing literature by providing an analysis of the standard Hedge algorithm in the stochastic setting.\nWe show that the anytime Hedge algorithm with default learning rate $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$ actually \\emph{adapts} to the stochastic setting, in which it achieves an optimal \\emph{constant} $O(\\log (M) \/ \\Delta)$ regret bound \\emph{without any dedicated tuning} for the easier instance, which might be surprising at first sight.\nThis contrasts with previous works, which require the construction of new adaptive (and more involved) algorithms.\nRemarkably, this property is \\emph{not} shared by the variant of Hedge for a known fixed-horizon $T$ with constant learning rate $\\eta \\propto \\sqrt{\\log(M) \/ T}$, since it suffers a $\\Theta (\\sqrt{T \\log M})$ regret even in easier instances.\nThis exhibits a strong difference between the performances of the anytime and the fixed-horizon variants of the Hedge algorithm.\n\nGiven the aforementioned adaptivity of Decreasing Hedge, one may wonder whether there is in fact any benefit in using more sophisticated algorithms in the stochastic regime.\n We answer this question affirmatively, by considering a more refined measure of complexity of a stochastic instance than the gap $\\Delta$.\n Specifically, we show that Decreasing Hedge does not admit improved regret under Bernstein conditions, which are standard low-noise conditions from statistical learning \\citep{mammen1999margin,tsybakov2004aggregation,bartlett2006empirical}.\n By contrast, it was shown by \\citet{koolen2016combining} that algorithms which satisfy some adaptive adversarial regret bound achieve improved regret under Bernstein conditions.\n Finally, we characterize the behavior of Decreasing Hedge in the stochastic regime, by showing that its eventual regret on \\emph{any} stochastic instance is governed\n by the gap $\\Delta$.\n\n\\paragraph{Related work.}\n\nIn the bandit setting, where the feedback only consists of the loss of the selected action, there has also been some interest in ``best-of-both-worlds'' algorithms that combine optimal $O (\\sqrt{M T})$ worst-case regret in the adversarial regime with improved $O (M \\log T)$ regret (up to logarithmic factors) in the stochastic case \\citep{bubeck2012best2worlds,seldin2014practical,auer2016both}.\nIn particular, \\citet{seldin2014practical,seldin2017improved} showed that by augmenting the standard EXP3 algorithm for the adversarial regime (an analogue of Hedge with $\\Theta (1\/\\sqrt{t})$ learning rate)\nwith a special-purpose gap detection mechanism, one can achieve poly-logarithmic regret in the stochastic case.\nThis result is strengthened in some recent follow-up work \\citep{zimmert2019optimal,zimmert2019beating},\nthat appeared since the completion of the first version of the present paper,\nwhich obtains optimal regret in the stochastic and adversarial regimes through a variant of the Follow-The-Regularized-Leader (FTRL) algorithm with $\\Theta (1\/\\sqrt{t})$ learning rate and a proper regularizer choice.\nThis result can be seen as an analogue in the bandit case of our upper bound for Decreasing Hedge.\nNote that, in the bandit setting, the hardness of an instance is essentially characterized by the gap $\\Delta$ \\citep{bubeck2012regret}; in particular, the Bernstein condition, which depends on the correlations between the losses of the experts, cannot be exploited under bandit feedback, where one only observes one arm at each round.\nHence, it appears that the negative part of our results (on the limitations of Hedge) does not have an analogue in the bandit case.\n\nA similar adaptivity result for FTRL with decreasing $\\Theta (1\/\\sqrt{t})$ learning rate has been observed in a different context by \\citet{huang2017curved}. Specifically, it is shown that, in the case of online linear optimization on a Euclidean ball, FTRL with squared norm regularizer and learning rate $\\Theta (1\/\\sqrt{t})$ achieves $O (\\log T)$ regret when the loss vectors are i.i.d.\\@\\xspace\nThis result is an analogue of our upper bound for Hedge, since this algorithm corresponds to FTRL on the simplex with entropic regularizer \\citep{cesabianchi2006plg,hazan2016online}.\nOn the other hand, the simplex lacks the curvature of the Euclidean ball, which is important to achieve small regret; here, the improved regret is ensured by a condition on the distribution, namely the existence of a gap $\\Delta$.\nOur lower bound for Hedge shows that this condition is necessary, thereby characterizing the long-term regret of FTRL on the simplex with entropic regularizer.\nIn the case of the Euclidean ball with squared norm regularizer, the norm of the expected loss vector appears to play a similar role, as shown by the upper bound from \\citet{huang2017curved}.\n\n\\paragraph{Outline.}\n\nWe define the setting of prediction with expert advice and the Hedge algorithm in Section~\\ref{sec:expert-problem-hedge}, and we recall herein its standard worst-case regret bound.\nIn Section~\\ref{sec:hedge-easy}, we consider the behavior of the Hedge algorithm on easier instances, namely the stochastic setting with a gap $\\Delta$ on the best expert. \nUnder an i.i.d assumption on the sequence of losses, we provide in Theorem~\\ref{thm:hedge-stochastic} an upper bound on the regret of order $(\\log M) \/ \\Delta$ for Decreasing Hedge. \nIn Proposition~\\ref{prop:lowerbound-gap}, we prove that the rate $(\\log M) \/ \\Delta$ cannot be improved in this setting.\nIn Theorem~\\ref{thm:adv-gap} and Corollary~\\ref{cor:hedge-martingale}, we extend the regret guarantees to the adversarial with a gap setting, where a leading expert linearly outperforms the others.\nThese results stand for any Hedge algorithm which is worst-case optimal and with any learning rate which is larger than the one of Decreasing Hedge, namely $O(\\sqrt{\\log M\/ t})$.\nIn Proposition~\\ref{prop:lower-bound-hedge-cst}, we prove the sub-optimality of the fixed-horizon Hedge algorithm, and of another version of Hedge based on the so-called ``doubling trick''.\nIn Section~\\ref{sec:advant-second-order}, we discuss the advantages of adaptive Hedge algorithms, and explain what the limitations of Decreasing Hedge are compared to such versions.\nWe include numerical illustrations of our theoretical findings in Section~\\ref{sec:experiments}, conclude in Section~\\ref{sec:conclusion} and provide the proofs in Section~\\ref{sec:proofs}.\n\n\n\\section{The expert problem and the Hedge algorithm}\n\\label{sec:expert-problem-hedge}\n\n\nIn the Hedge setting, also called \\emph{decision-theoretic online learning} \\citep{freund1997boosting}, the learner and its adversary (the Environment) sequentially compete on the following game: at each round $t\\geq 1$,\n\\begin{enumerate}\n\\item the Learner chooses a probability vector $\\bm v_t = (v_{i,t})_{1\\leq i \\leq M}$ on the $M$ experts $1, \\dots, M$;\n\\item the Environment picks a bounded loss vector $\\bm \\ell_t = (\\ell_{i,t})_{1\\leq i \\leq M} \n\\in [0, 1]^M$, where $\\ell_{i, t}$ is the loss of expert $i$ at round $t$, while the Learner suffers loss $\\widehat \\ell_t = \\bm v_t^\\top \\bm \\ell_t$.\n\\end{enumerate}\nThe goal of the Learner is to control its \\emph{regret}\n\\begin{equation}\n \\label{eq:regret}\n R_T = \\sum_{t=1}^T \\widehat \\ell_t - \\min_{1\\leq i \\leq M} \\sum_{t=1}^T \\ell_{i,t}\n\\end{equation}\nfor every $T \\geq 1$, irrespective of the sequence of loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ chosen by the Environment.\nOne of the most standard algorithms for this setting is the \\emph{Hedge} algorithm.\nThe Hedge algorithm, also called the exponentially weighted averaged forecaster, uses the vector of probabilities $\\bm v_t = (v_{i,t})_{1\\leq i \\leq M}$ given by\n\\begin{equation}\n \\label{eq:hedge-algorithm}\n v_{i, t} = \\frac{e^{-\\eta_t L_{i,t-1}}}{\\sum_{j=1}^M e^{-\\eta_t L_{j,t-1}}}\n\\end{equation}\nat each $t\\geq 1$, where $L_{i,T} = \\sum_{t=1}^T \\ell_{i,t}$ denotes the cumulative loss of \nexpert~$i$ for every $T\\geq 1$.\nLet us also denote $\\widehat L_T := \\sum_{t=1}^T \\widehat \\ell_t$ and $R_{i,T} = \\widehat L_T - L_{i,T}$ the regret with respect to expert $i$.\nWe consider in this paper the following variants of Hedge, where $c_0 > 0$ is a constant.\n\n\\medskip\\noindent\n\\textbf{Decreasing Hedge}~\\citep{auer2002adaptive}.\nThis is Hedge with the sequence of learning rates $\\eta_t = c_0 \\sqrt{\\log(M) \/ t}$.\n\n\\medskip\\noindent\n\\textbf{Constant Hedge}~\\citep{littlestone1994weighted}.\nGiven a finite time horizon $T \\geq 1$, this is Hedge with constant learning rate $\\eta_t = c_0 \\sqrt{\\log(M) \/ T}$.\n\n\\medskip\\noindent\n\\textbf{Hedge with doubling trick}~\\citep{cesabianchi1997doublingtrick, cesabianchi2006plg}.\nThis variant of Hedge uses a constant learning rate on geometrically increasing intervals, restarting the algorithm at the beginning of each interval. Namely, it uses\n\\begin{equation}\n \\label{eq:hedge-doubling-trick}\n v_{i,t} = \\frac{\\exp ( - \\eta_t \\sum_{s = T_k}^{t-1} \\ell_{i,s})}{\\sum_{j=1}^M \\exp ( - \\eta_t \\sum_{s = T_k}^{t-1} \\ell_{j,s})},\n\\end{equation}\nwith $T_l = 2^l$ for $l \\geq 0$, $k \\in \\mathbf N$ such that $T_k \\leq t < T_{k+1}$ and\n$\\eta_t = c_0 \\sqrt{\\log (M) \/ T_k}$.\n\n\\medskip\nLet us recall the following standard regret bound for the Hedge algorithm from \\citet{chernov2010prediction}.\n\\begin{proposition}\n \\label{prop:hedge-adversarial}\n Let $\\eta_1, \\eta_2, \\dots$ be a decreasing sequence of learning rates.\n The Hedge algorithm \\eqref{eq:hedge-algorithm} satisfies the following regret bound:\n \\begin{equation}\n \\label{eq:hedge-adversarial}\n R_T \\leq \\frac{1}{\\eta_T} \\log M + \\frac{1}{8} \\sum_{t=1}^T \\eta_t\n \\, .\n \\end{equation}\n In particular, the choice $\\eta_t = 2 \\sqrt{{\\log (M)}\/{t}}$ yields a regret bound of $\\sqrt{T \\log M}$ for every $T \\geq 1$.\n\\end{proposition}\n\nNote that the regret bound stated in Equation~\\eqref{eq:hedge-adversarial} holds for every sequence of losses $\\bm \\ell_1, \\bm \\ell_2, \\dots$, which makes it valid under no assumption (aside from the boundedness of the losses).\nThe worst-case regret bound in $O(\\sqrt{T \\log M})$ is achieved by Decreasing Hedge, Hedge with doubling trick and Constant Hedge (whenever $T$ is known in advance).\nThe $O(\\sqrt{T \\log M})$ rate cannot be improved either by Hedge or any other algorithm: it is known to be the minimax optimal regret \\citep{cesabianchi2006plg}.\nContrary to Constant Hedge, Decreasing Hedge is anytime, in the sense that it achieves the $O(\\sqrt{T \\log M})$ regret bound simultaneously for each $T \\geq 1$.\nWe note that this worst-case regret analysis fails to exhibit any difference between these three algorithms.\n\n\nIn many cases, this $\\sqrt{T}$ regret bound is pessimistic, and more ``aggressive'' strategies (such as the follow-the-leader algorithm, which plays at each round the uniform distribution on the experts with smallest loss, \\citealp{cesabianchi2006plg}) may achieve constant regret in easier instances, even though they lack regret guarantees in the adversarial regime.\nWe show in Section~\\ref{sec:hedge-easy} below that Decreasing Hedge is actually\nbetter than both Constant Hedge and Hedge with doubling trick in some easier instance of the problem (including in the stochastic setting).\nThis entails that Decreasing Hedge is actually able to adapt, without any modification, to the easiness of the problem considered.\n\n\n\\section{Regret of Hedge variants on easy instances}\n\\label{sec:hedge-easy}\n\n\nIn this section, we depart from the worst-case regret analysis and study the regret of the considered variants of the Hedge algorithm on easier instances of the prediction with expert advice problem.\n\n\\subsection{Optimal regret for Decreasing Hedge in the stochastic regime}\n\\label{sub:optimal-stoch}\n\nWe examine the behavior of Decreasing Hedge in the stochastic regime, where the losses are the realization of some (unknown) stochastic process.\nMore precisely, we consider the standard i.i.d.\\@\\xspace case, where the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace (independence holds over rounds, but not necessarily across experts).\nIn this setting, the regret can be much smaller than the worst-case $\\sqrt{T \\log M}$ regret, since the best expert (with smallest expected loss) will dominate the rest after some time.\nFollowing \\citet{gaillard2014secondorder,luo2015adanormalhedge}, the easiness parameter we consider in this case, which governs the time needed for the best expert to have the smallest cumulative loss and hence the incurred regret, is the sub-optimality gap $\\Delta = \\min_{i \\neq i^*} \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t} ]$, where $i^* = \\mathop{\\mathrm{argmin}}_{i} \\mathbb E [ \\ell_{i,t} ]$.\n\nWe show below that, despite the fact that Decreasing Hedge is designed for the worst-case setting described in Section~\\ref{sec:expert-problem-hedge}, it is able to {adapt} to the easier problem considered here, \nIndeed, Theorem~\\ref{thm:hedge-stochastic} shows that Decreasing Hedge achieves a \\emph{constant}, and in fact \\emph{optimal} (by Proposition~\\ref{prop:lowerbound-gap} below) regret bound in this setting, in spite of its ``conservative'' learning rate.\n\nWith the exception of the high-probability bound of Corollary~\\ref{cor:hedge-martingale}, the upper and lower bounds in the stochastic case are stated for the \\emph{pseudo-regret} $\\mathcal{R}_T = \\mathbb E [R_{i^*,T}]$ (similar bounds hold for the the expected regret $\\mathbb E [R_T]$, since $\\mathcal{R}_T \\leq \\mathbb E [R_T]$ and by Remark~\\ref{rem:pseudoregret} in Section~\\ref{sec:proof-theorem-1}).\n\n\\begin{theorem}\n \\label{thm:hedge-stochastic}\n Let $M \\geq 3$.\n Assume that the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace random variables, where $\\bm \\ell_t = (\\ell_{i,t})_{1\\leq i \\leq M}$.\n Also, assume that there exists $i^* \\in \\{1, \\dots, M\\}$ and $\\Delta > 0$ such that\n \\begin{equation}\n \\label{eq:gap-condition}\n \\mathbb E [ \\ell_{i,t} - \\ell_{i^*,t} ] \\geq \\Delta\n \\end{equation}\n for every $i \\neq i^*$.\n Then, the Decreasing Hedge algorithm with learning rate $\\eta_t = 2 \\sqrt{(\\log M)\/t}$\n achieves the following pseudo-regret bound\\textup: for every $T \\geq 1$\\textup,\n \\begin{equation}\n \\label{eq:regret-stochastic-exp}\n \\mathcal{R}_T \\leq \\frac{4 \\log M + 25}{\\Delta}\n \\, .\n \\end{equation}\n\\end{theorem}\n\n\nThe proof of Theorem~\\ref{thm:hedge-stochastic} is given in Section~\\ref{sec:proof-theorem-1}.\nTheorem~\\ref{thm:hedge-stochastic} proves that, in the stochastic setting with a gap $\\Delta$, the Decreasing Hedge algorithm achieves a regret $O(\\log (M) \/ \\Delta)$, without any prior knowledge of $\\Delta$.\nThis matches the guarantees of adaptive Hedge algorithms which are explicitly designed to adapt to easier instances \\citep{gaillard2014secondorder,luo2015adanormalhedge}.\nThis result may seem surprising at first: indeed, adaptive exponential weights algorithms\nthat combine optimal regret in the adversarial setting and constant regret in\neasier scenarios, such as Hedge with a second-order tuning \\citep{cesabianchi2007secondorder} or AdaHedge \\citep{derooij2014followtheleader}, typically use a data-dependent learning rate $\\eta_t$ that adapts to the properties of the losses.\nWhile the learning rate $\\eta_t$ chosen by these algorithms may be as low as the worst-case tuning $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$, in the stochastic case those algorithms will use larger, lower-bounded learning rates to ensure constant regret.\nAs Theorem~\\ref{thm:hedge-stochastic} above shows, it turns out that the data-independent, ``safe'' learning rates $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$ used by ``vanilla'' Decreasing Hedge are still large enough to adapt to the stochastic case.\n\n\\paragraph{Idea of the proof.} \n\nThe idea of the proof of Theorem~\\ref{thm:hedge-stochastic} is to divide time in two phases: a short initial phase $\\iint 1{t_1}$,\nwhere $t_1 = O (\\frac{\\log M}{\\Delta^2})$,\nand a second phase $\\iint{t_1}{T}$.\nThe initial phase is dominated by noise, and regret during this period is bounded\nthrough the worst-case regret bound of Proposition~\\ref{prop:hedge-adversarial}, which gives a regret of $O(\\sqrt{t_1 \\log M}) = O (\\frac{\\log M}{\\Delta})$.\nIn the second phase, the best expert dominates the rest, and the weights concentrate on this best expert fast enough that the total regret incurred is small.\nThe control of the regret in the second phase relies on the critical fact that, if $\\eta_t$ is at least as large as $\\sqrt{(\\log M)\/t}$, then the following two things occur simultaneously at $t_1 \\asymp \\frac{\\log M}{\\Delta^2}$, namely at the beginning of the late phase:\n\\begin{enumerate}\n\\item with high probability, the best expert $i^*$ dominates all the others linearly: for every $i \\neq i^*$ and $t \\geq t_1$, $L_{i,t} - L_{i^*,t} \\geq \\frac{\\Delta t}{2}$;\n\\item the total weight of all suboptimal experts is controlled: $\\sum_{i \\neq i^*} v_{i,t_1} \\leq \\frac{1}{2}$. If $\\eta_t \\geq \\sqrt{(\\log M)\/t}$ and the first condition holds, this amounts to $M \\exp (- \\frac{\\Delta}{2} \\sqrt{t \\log M}) \\leq \\frac{1}{2}$, namely $t_1 \\gtrsim \\frac{\\log M}{\\Delta^2}$.\n\\end{enumerate}\nIn other words, the learning rate $\\eta_t \\asymp \\sqrt{(\\log M)\/t}$ ensures that the total weight of suboptimal experts starts vanishing at about the same time as when the best expert starts to dominate the others with a large probability (and remarkably, this property holds for every value of the sub-optimality gap $\\Delta$).\nFinally, the upper bound on the regret in the second phase rests on the two conditions above, together with the bound $\\sum_{t \\geq 1} e^{-c \\sqrt{t}} = O (\\frac{1}{c^2})$ for $c > 0$.\n\n\\begin{remark}\n The fact that $\\sum_{t \\geq 1} e^{-c \\sqrt{t}} = O (1 \/ c^2)$ is also used in the analysis of the EXP3++ bandit algorithm~\\citep[Lemma 10]{seldin2014practical}.\n In the expert setting considered here, summing the contribution of all experts (which suffices in the bandit setting to obtain the correct order of regret) would yield a significantly suboptimal $O(M \/ \\Delta)$ regret bound, with a linear dependence on the number of experts $M$.\n In our case, the decomposition of the regret in two phases, which is explained above, removes the linear dependence on $M$ and allows to obtain the optimal rate $(\\log M) \/ \\Delta$.\n\\end{remark}\n\nWe complement Theorem~\\ref{thm:hedge-stochastic} by showing that the $O((\\log M) \/ \\Delta)$ regret under the gap condition cannot be improved, in the sense that its dependence on both $M$ and $\\Delta$ is optimal.\n\n\\begin{proposition}\n \\label{prop:lowerbound-gap}\n Let $\\Delta \\in (0, \\frac{1}{4})$, $M \\geq 4$ and $T \\geq (\\log M) \/ (16 \\Delta^2)$.\n Then, for any algorithm for the Hedge setting, there exists an i.i.d.\\@\\xspace distribution over the sequence of losses $(\\bm \\ell_{t})_{t \\geq 1}$ such that\\textup:\n \\begin{itemize}\n \\item there exists $i^* \\in \\{ 1, \\dots, M \\}$ such that, for any $i \\neq i^*$, $\\mathbb E [\\ell_{i,t} - \\ell_{i^*, t}] \\geq \\Delta$\\textup;\n \\item the pseudo-regret of the algorithm satisfies\\textup:\n \\begin{equation}\n \\label{eq:lowerbound-gap}\n \\mathcal{R}_T\n \\geq \\frac{\\log M}{256 \\Delta}\n \\, .\n \\end{equation}\n \\end{itemize}\n\\end{proposition}\n\nThe proof of Proposition~\\ref{prop:lowerbound-gap} is given in Section~\\ref{sec:proof-lowerbound-gap}.\nProposition~\\ref{prop:lowerbound-gap} generalizes the well-known minimax lower bound of $\\Theta (\\sqrt{T \\log M})$, which is recovered by taking $\\Delta \\asymp \\sqrt{(\\log M)\/T}$.\n\n\\subsection{Small regret for Decreasing Hedge in the adversarial with a gap problem}\n\\label{sub:adv-gap}\n\nIn this section, we extend the regret guarantee of Decreasing Hedge in the stochastic setting (Theorem~\\ref{thm:hedge-stochastic}), by showing that it holds for more general algorithms and under more general assumptions.\nSpecifically, we consider an ``adversarial with a gap'' regime, similar to the one introduced by \\citet{seldin2014practical} in the bandit case, where the leading expert linearly outperforms the others after some time.\nAs Theorem~\\ref{thm:adv-gap} shows, essentially the same regret guarantee can be obtained in this case, up to an additional $\\log (\\Delta^{-1}) \/ \\Delta$ term.\nTheorem~\\ref{thm:adv-gap} also applies to any Hedge algorithm whose (possibly data-dependent) learning rate $\\eta_t$ is at least as large as that of Decreasing Hedge, and which satisfies a $O(\\sqrt{T \\log M})$ worst-case regret bound;\nthis includes algorithms with \\emph{anytime} first and second-order tuning of the learning rate \\citep{auer2002adaptive,cesabianchi2007secondorder,derooij2014followtheleader}.\nIn what follows, we will assume $M \\geq 3$ for convenience; similar results holds for $M=2$.\n\\begin{theorem}\n \\label{thm:adv-gap}\n Let $M \\geq 3$.\n Assume that there exists $\\tau_0 \\geq 1$\\textup, $\\Delta \\in (0, 1)$ and $i^* \\in \\{1, \\dots, M \\}$ \n such that\\textup, for every $t \\geq \\tau_0$ and $i \\neq i^*$\\textup, one has\n \\begin{equation}\n \\label{eq:condition-gap-adv}\n L_{i, t} - L_{i^*, t} \\geq \\Delta t.\n \\end{equation}\n Consider any Hedge algorithm with \\textup(possibly data-dependent\\textup) learning rate $\\eta_t$ such that \n \\begin{itemize}\n \\item $\\eta_t \\geq c_0 \\sqrt{(\\log M) \/ t}$ for some constant $c_0 > 0$\\textup;\n \\item it admits the following worst-case regret bound: $R_T \\leq c_1 \\sqrt{T \\log M}$ for every $T \\geq 1$\\textup,\n for some $c_1 > 0$.\n \\end{itemize}\n Then, for every $T\\geq 1$, the regret of this algorithm is upper bounded as\n \\begin{equation}\n \\label{eq:regret-gap-adv}\n R_T \n \\leq c_1 \\sqrt{\\tau_0 \\log M} + \\frac{c_2 \\log M + c_3 \\log {\\Delta}^{-1} + c_4}{\\Delta} \n \\end{equation}\n where $c_2 = c_1 + \\frac{\\sqrt{8}}{c_0}$, $c_3 = \\frac{\\sqrt{8}}{c_0}$ and $c_4 = \\frac{16}{c_0^2}$.\n\\end{theorem}\n\nThe idea of the proof of Theorem~\\ref{thm:adv-gap} is the same as that of Theorem~\\ref{thm:hedge-stochastic}, the only difference being the slightly longer initial phase to account for the adversarial nature of the losses.\nAs a consequence of the general bound of Theorem~\\ref{thm:adv-gap}, we can recover the guarantee of Theorem~\\ref{thm:hedge-stochastic} (up to an additional $\\log (\\Delta^{-1}) \/\\Delta$ term), both in expectation and with high probability, under more general stochastic assumptions than i.i.d.\\@\\xspace over time.\nThe proofs of Theorem~\\ref{thm:adv-gap} and Corollary~\\ref{cor:hedge-martingale} are provided in Section~\\ref{sec:proof-thm-adv-gap}.\n\n\\begin{corollary}\n \\label{cor:hedge-martingale}\n Assume that the losses $(\\ell_{i,t})_{1\\leq i \\leq M, t\\geq 1}$ are random variables. \n Also, denoting $\\mathcal{F}_t = \\sigma \\big( (\\ell_{i,s})_{1\\leq i \\leq M, 1\\leq s \\leq t} \\big)$, assume that there exists $i^*$ and $\\Delta >0 $ such that\n \\begin{equation}\n \\label{eq:gap-condition-martingale}\n \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} \\right] \\geq \\Delta\n \\end{equation}\n for every $i\\neq i^*$ and every $t\\geq 1$.\n Then, for any Hedge algorithm satisfying the conditions of Theorem~\\ref{thm:adv-gap}, and every $T \\geq 1$\\textup:\n \\begin{equation}\n \\label{eq:regret-martingale-exp}\n \n \\mathcal{R}_T\n \\leq (5 c_1 + 2 c_2) \\frac{\\log M}{\\Delta} + 2 c_3 \\frac{\\log \\Delta^{-1}}{\\Delta} \n + \\frac{2 c_4}{\\Delta},\n \\end{equation}\n with $c_1, c_2, c_3, c_4$ as in Theorem~\\ref{thm:adv-gap}.\n In addition, for every $\\varepsilon \\in (0, 1)$, we have\n \\begin{equation}\n \\label{eq:regret-martingale-prob}\n R_T\n \\leq \\left( c_1 \\sqrt{8} + 2 c_2 \\right) \\frac{\\log M}{\\Delta} + c_1 \\frac{\\sqrt{8\\log M \\log \\varepsilon^{-1}}}{\\Delta} + 2 c_3 \\frac{\\log \\Delta^{-1}}{\\Delta} + \\frac{2 c_4}{\\Delta}\n \\end{equation}\n with probability at least $1 - \\varepsilon$. \n\\end{corollary}\n\n\n\\subsection{Constant Hedge and Hedge with the doubling trick do not adapt to the stochastic case}\n\\label{sec:negative-results}\n\nNow, we show that the adaptivity of Decreasing Hedge to gaps in the losses, established in Sections~\\ref{sub:optimal-stoch} and~\\ref{sub:adv-gap}, is not shared by the two closely related Constant Hedge and Hedge with the doubling trick, despite the fact that they both achieve the minimax optimal worst-case $O (\\sqrt{T \\log M})$ regret.\nProposition~\\ref{prop:lower-bound-hedge-cst} below shows that both algorithms fail to achieve a constant regret, and in fact to improve over their worst-case $\\Theta (\\sqrt{T \\log M})$ regret guarantee, even in the extreme case of experts with constant losses $0$ (for the leader), and $1$ for the rest (\\ie, $\\Delta = 1$).\n\n\\begin{proposition}\n \\label{prop:lower-bound-hedge-cst}\n Let $T \\geq 1$\\textup, $M \\geq 2$\\textup, and consider the experts $i=1, \\dots, M$ with losses $\\ell_{1, t} = 0$\\textup, \n $\\ell_{i, t} = 1$ $(1 \\leq t \\leq T, 2 \\leq i \\leq M)$.\n Then\\textup, the pseudo-regret of Constant Hedge with learning rate $\\eta_t = c_0 \\sqrt{\\log (M)\/T}$ \n \\textup(where $c_0 > 0$ is a numerical constant\\textup) is lower bounded as follows\\textup:\n \\begin{equation}\n \\label{eq:lower-bound-hedge-cst}\n \\mathcal{R}_T\n \\geq \\min \\Big( \\frac{\\sqrt{T \\log M}}{3 c_0}, \\frac{T}{3} \\Big)\n \\, .\n \\end{equation}\n In addition, Hedge with doubling trick~\\eqref{eq:hedge-doubling-trick} also suffers a pseudo-regret satisfying\n \\begin{equation}\n \\label{eq:lower-bound-hedge-doubling}\n \\mathcal{R}_T \\geq\n \\min \\Big ( \\frac{\\sqrt{T \\log M}}{6 c_0} , \\frac{T}{12} \\Big)\n \\, .\n \\end{equation}\n\\end{proposition}\n\n\nThe proof of Proposition~\\ref{prop:lower-bound-hedge-cst} is given in Section~\\ref{sec:proof-lower-bound-hedge-cst}.\nAlthough Hedge with a doubling trick is typically considered as overly conservative and only suitable for worst-case scenarios \\citealp{cesabianchi2006plg} (especially due to its periodic restarts, after which it discards past observations), to the best of our knowledge Proposition~\\ref{prop:lower-bound-hedge-cst} (together with Theorem~\\ref{thm:hedge-stochastic}) is the first to formally demonstrate the advantage of Decreasing Hedge over the doubling trick version.\nThis implies that Decreasing Hedge should not be seen as merely a substitute for Constant Hedge to achieve anytime regret bounds.\nIndeed, even when the horizon $T$ is fixed, Decreasing Hedge outperforms Constant Hedge in the stochastic setting.\n\n\\section{Limitations of Decreasing Hedge in the stochastic case}\n\\label{sec:advant-second-order}\n\nIn this section, we explore the limitations of the simple Decreasing Hedge algorithm in the stochastic regime, and exhibit situations where it performs worse than more sophisticated algorithms.\nThe starting observation is that the sub-optimality gap $\\Delta$ is a rather brittle measure of ``hardness'' of a stochastic instance, which does not fully reflect the achievable rates.\nWe therefore consider the following fast-rate condition from statistical learning, which refines the sub-optimality gap as a measure of complexity of a stochastic instance.\n\n\\begin{definition}[Bernstein condition]\n \\label{def:bernstein-condition}\n Assume that the losses $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are the realization of a stochastic process.\n Denote $\\mathcal{F}_{t} = \\sigma (\\bm \\ell_1, \\dots, \\bm \\ell_t)$ the $\\sigma$-algebra generated by $\\bm \\ell_1, \\dots, \\bm \\ell_t$.\n For $\\beta \\in [0,1]$ and $B >0$, the losses are said to satisfy the \\emph{$(\\beta,B)$-Bernstein condition} if there exists $i^*$ such that, for every $t \\geq 1$ and $i \\neq i^*$,\n \\begin{equation}\n \\label{eq:bernstein-condition}\n \\mathbb E [(\\ell_{i,t} - \\ell_{i^*,t})^2 \\,|\\, \\mathcal{F}_{t-1}]\n \\leq B \\mathbb E [ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} ]^\\beta\n \\, . \n \\end{equation}\n\\end{definition}\n\nThe Bernstein condition \\citep{bartlett2006empirical}, a generalization of the Tsybakov margin condition \\citep{tsybakov2004aggregation,mammen1999margin}, is a geometric property on the losses which enables to obtain fast rates (e.g., faster than $O(1\/\\sqrt{n})$ for parametric classes) in statistical learning; we refer to \\citet{vanerven2015fastrates} for a discussion of fast rates conditions.\n The Bernstein condition~\\eqref{eq:bernstein-condition}\n quantifies the \n ``easiness'' of a stochastic instance, and generalizes the gap condition considered in the previous section (see Example~\\ref{ex:bernstein-gap} below).\nRoughly speaking, it states that good experts (with near-optimal expected loss) are highly correlated with the best expert.\nIn the examples below, we assume that the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace\n\n\\begin{example}[Gap implies Bernstein]\n \\label{ex:bernstein-gap}\n If $\\Delta_i = \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}] \\geq \\Delta$ for $i \\neq i^*$, then the $(1, \\frac{1}{\\Delta})$-Bernstein condition holds \\citep[Lemma~4]{koolen2016combining}.\n Furthermore, letting $\\alpha = \\mathbb E [\\ell_{i^*,t}]$ denote the expected loss of the best expert, the $(1, 1 + \\frac{2 \\alpha}{\\Delta})$-Bernstein condition holds.\n Indeed, for any $i \\neq i^*$,\n denoting $\\mu_i := \\mathbb E [\\ell_{i,t}] = \\alpha + \\Delta_i$, we have (since $(u-v)^2 \\leq \\max(u^2,v^2) \\leq u^2+v^2 \\leq u+v$ for $u,v \\in [0, 1]$):\n \\begin{align*}\n \\mathbb E \\big[ (\\ell_{i,t} - \\ell_{i^*,t})^2 \\big]\n &\\leq \\mathbb E \\left[ \\ell_{i,t} + \\ell_{i^*,t} \\right]\n = \\frac{\\mu_i + \\alpha}{\\mu_i - \\alpha} \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\right]\n = \\Big( 1 + \\frac{2 \\alpha}{\\Delta_i} \\Big) \\, \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\right]\n \\, ,\n \\end{align*}\n which establishes the claim\n since $\\Delta_i \\geq \\Delta$.\n This provides an improvement when $\\alpha$ is small.\n\\end{example}\n\n\\begin{example}[Bernstein without a gap]\n \\label{ex:bernstein-without-gap}\n Let $P$ be a distribution on $\\mathcal{X} \\times \\{ 0, 1\\}$, where $\\mathcal{X}$ is some measurable space. Assume that $(X_1, Y_1), (X_2, Y_2) \\dots$ are i.i.d.\\@\\xspace samples from $P$, and that the experts $i \\in \\{ 1, \\dots, M\\}$ correspond to classifiers $f_i : \\mathcal{X} \\to \\{ 0, 1 \\}$: $\\ell_{i, t} = \\bm 1 ( f_i (X_t) \\neq Y_t ) $, and that expert $i^*$ is the Bayes classifier: $f_{i^*} (X) = \\bm 1 ( \\eta (X) \\geq 1\/2 ) $, where $\\eta(X) = \\P (Y=1 \\,|\\, X)$.\n Tsybakov's low noise condition \\citep{tsybakov2004aggregation}, namely $\\P ( | 2 \\eta (X) - 1 | \\leq t ) \\leq C t^{\\kappa}$ for some $C > 0$, $\\kappa \\geq 0$ and every $t > 0$, implies the $(\\frac{\\kappa}{\\kappa + 1}, B)$-Bernstein condition for some $B$ (see, e.g., \\citealp{boucheron2005survey}).\n In addition, under the Massart condition \\citep{massart2006risk} that\n $| \\eta (X) - 1\/2 | \\geq c > 0$, the $(1, 1\/(2c))$-Bernstein condition holds.\n Note that these conditions may hold even with an arbitrarily small sub-optimality gap $\\Delta$, since the $f_i$, $i \\neq i^*$, may be arbitrary.\n\\end{example}\n\nTheorem~\\ref{thm:hedge-no-bernstein} below shows that Decreasing Hedge fails to achieve improved rates under Bernstein conditions.\n\n\\begin{theorem}\n \\label{thm:hedge-no-bernstein}\n For every $T \\geq 1$, there exists a $(1, 1)$-Bernstein stochastic instance on which the pseudo-regret of the Decreasing Hedge algorithm with $\\eta_t = c_0 \\sqrt{(\\log M) \/ t}$ satisfies\n $\\mathcal{R}_{T} \\geq \\frac{1}{3} \\min( \\frac{1}{c_0} \\sqrt{T \\log M}, {T})$.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{thm:hedge-no-bernstein} is given in Section~\\ref{sec:proof-lowerbound-no-bernstein}.\nBy contrast, it was shown by \\citet{koolen2016combining}\n(and implicitly used by \\citealp{gaillard2014secondorder}) that\nsome adaptive algorithms with data-dependent regret bounds enjoy improved regret under the Bernstein condition.\nFor the sake of completeness, we state this fact in Proposition~\\ref{prop:second-order-bernstein} below, which corresponds to~\\citet[Theorem~2]{koolen2016combining}, but where the dependence on $B$ is made explicit. We also only provide a bound in expectation, which considerably simplifies the proof.\nThe proof of Proposition~\\ref{prop:second-order-bernstein}, which uses the same ideas as \\citet[Theorem~11]{gaillard2014secondorder}, is provided in Section~\\ref{sec:proof-second-order-bernstein}.\n\n\\begin{proposition}\n \\label{prop:second-order-bernstein}\n Consider an algorithm for the Hedge problem which satisfies the following regret bound: for \n every $i\\in \\{ 1, \\dots, M\\}$\\textup,\n \\begin{equation}\n \\label{eq:second-order-regret}\n R_{i,T} \\leq C_1 \\sqrt{(\\log M) \\sum_{t=1}^T (\\widehat \\ell_t - \\ell_{i,t})^2} + C_2 \\log M\n \\end{equation}\n where\n $C_1, C_2 >0$ are constants.\n Assume that the losses satisfy the $(\\beta, B)$-Bernstein condition.\n Then, the pseudo-regret of the algorithm satisfies\\textup:\n \\begin{equation}\n \\label{eq:regret-bernstein}\n \\mathcal{R}_T\n \\leq C_3 (B \\log M)^{\\frac{1}{2-\\beta}} T^{\\frac{1-\\beta}{2-\\beta}} + C_4 \\log M \n \\end{equation}\n where $C_3 = \\max (1, 4C_1^2)$ and $C_4 = 2 C_2$.\n\\end{proposition}\n\nThe data-dependent regret bound~\\eqref{eq:second-order-regret}, a ``second-order'' bound, is satisfied by adaptive algorithms such as Adapt-ML-Prod \\citep{gaillard2014secondorder} and Squint \\citep{koolen2015squint}.\nA slightly different variant of second-order regret bounds, which depends on some cumulative variance of the losses across experts, has been considered by \\citet{cesabianchi2007secondorder,derooij2014followtheleader}, and is achieved by Hedge algorithms with a data-dependent tuning of the learning rate. Second-order bounds refine so-called \\emph{first-order} bounds \\citep{cesabianchi1997doublingtrick,auer2002adaptive,cesabianchi2006plg}, which are adversarial regret bounds that scale as $O(\\sqrt{L_T^* \\log M} + \\log M)$, where $L_T^*$ denotes the cumulative loss of the best expert.\n While first-order bounds may still scale as the worst-case $O(\\sqrt{T \\log M})$ rate in a typical stochastic instance (where the best expert has a positive expected loss), second-order algorithms are known to achieve constant $O((\\log M) \/ \\Delta)$ regret in the stochastic case with gap $\\Delta$ \\citep{gaillard2014secondorder,koolen2015squint}.\n\nTheorem~\\ref{thm:hedge-no-bernstein}, in light of Proposition~\\ref{prop:second-order-bernstein}, clarifies where the advantage of second-order algorithms compared to Decreasing Hedge lies: unlike the latter, they can exploit Bernstein conditions on the losses.\nThe contrast is most apparent for Bernstein instances with $\\beta = 1$. \nBy Example~\\ref{ex:bernstein-gap}, the existence of a gap $\\Delta$ implies that the $(1,B)$-Bernstein condition holds with $B \\leq \\frac{1}{\\Delta}$.\nHowever, as shown by Example~\\ref{ex:bernstein-without-gap}, $B$ can in fact be much smaller than $\\Delta$, in which case the regret bound~\\eqref{eq:regret-bernstein} satisfied by second-order algorithms, namely $O (B \\log M)$, significantly improves over the upper bound of $O( (\\log M)\/\\Delta)$ of Decreasing Hedge from Theorem~\\ref{thm:hedge-stochastic}.\nTheorem~\\ref{thm:hedge-no-bernstein} provides an instance where the difference does occur, in the most pronounced case where $B =1$, so that second-order algorithms enjoy small $O (\\log M)$ regret, while Decreasing Hedge suffers $\\Theta (\\sqrt{T \\log M})$ regret.\n\n\\begin{remark}\n The advantage of larger learning rates on some stochastic instances may be understood intuitively as follows.\n Consider an instance with $B$ small but small gap $\\Delta$.\n The learning rate of Decreasing Hedge is large enough that it can rule out bad experts (with large enough gap $\\Delta_i$) at the optimal rate (\\ie, at time $(\\log M)\/\\Delta_i^2$).\n However, once these bad experts are ruled out, near-optimal experts (with small gap $\\Delta_i$) are ruled out late (after $(\\log M)\/\\Delta_i^2$ rounds).\n On the other hand, the Bernstein assumption entails that those experts are highly correlated with the best expert, the amount of noise on the relative losses of these near-optimal experts is small, so that a larger learning rate could be safely used and would enable to dismiss near-optimal experts sooner.\n\\end{remark}\n\nSetting the Bernstein condition aside, we conclude by investigating the intrinsic limitations of Decreasing Hedge in the stochastic setting.\nIndeed, it is natural to ask whether Decreasing Hedge can exploit some other regularity of a stochastic instance, apart from the gap $\\Delta$.\nTheorem~\\ref{thm:hedge-characterize-gap} shows that this is in fact not the case.\n\n\\begin{theorem}\n \\label{thm:hedge-characterize-gap} \n For every i.i.d.\\@\\xspace \\textup(over time\\textup) stochastic instance with a unique best expert \n \\begin{equation*}\n i^* = \\mathop{\\mathrm{argmin}}_{1 \\leq i \\leq M} \\mathbb E [\\ell_{i,t}],\n \\end{equation*}\n the pseudo-regret of Decreasing Hedge \\textup(with $c_0 \\geq 1$\\textup) satisfies\n \\begin{equation*}\n \\mathcal{R}_T \\geq \\frac{1}{450 c_0^4 (\\log M)^2 \\Delta}\n \\end{equation*}\n for $T \\geq \\frac{1}{4 \\Delta^2}$\\textup, where $\\Delta := \\inf_{i \\neq i^*} \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}]$.\n\\end{theorem}\n\nTheorem~\\ref{thm:hedge-characterize-gap} shows (together with the upper bound of Theorem~\\ref{thm:hedge-stochastic}) that the eventual regret of Decreasing Hedge on \\emph{any} stochastic instance is determined by the sub-optimality gap $\\Delta$, and scales (up to a $\\log^3 M$ factor, depending on the number of near-optimal experts) as $\\Theta (\\frac{1}{\\Delta})$.\nThis characterizes the behavior of Decreasing Hedge on any stochastic instance.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\nIn this section, we illustrate our theoretical results by numerical experiments that compare the behavior of various Hedge algorithms in the stochastic regime.\n\n\\paragraph{Algorithms.} We consider the following algorithms: \\texttt{hedge} is Decreasing Hedge with the default learning rates $\\eta_t = 2\\sqrt{\\log (M) \/ t}$, \\texttt{hedge\\_constant} is Constant Hedge with constant learning rate $\\eta_t = \\sqrt{8 \\log (M) \/ T}$, \\texttt{hedge\\_doubling} is Hedge with doubling trick with $c_0 = \\sqrt{8}$, \\texttt{adahedge} is the AdaHedge algorithm from \\citet{derooij2014followtheleader}, which is a variant of the Hedge algorithm with a data-dependent tuning of the learning rate $\\eta_t$ (based on $\\bm \\ell_1, \\dots, \\bm \\ell_{t-1}$).\nAs shown in the note \\citet{blog}, AdaHedge also benefits from Bernstein conditions.\nA related algorithm, namely Hedge with second-order tuning of the learning rate \\citep{cesabianchi2007secondorder}, performed similarly to AdaHedge on the examples considered below, and was therefore not included. \\texttt{FTL} is Follow-the-Leader \\citep{cesabianchi2006plg} which puts all mass on the expert with the smallest loss (breaking ties randomly).\nWhile FTL serves as a benchmark in the stochastic setting, unlike the other algorithms it lacks any guarantee in the adversarial regime, where its worst-case regret is \\emph{linear} in $T$.\n\n\\paragraph{Results.}\n\nWe report in Figure~\\ref{fig:experiments} the cumulative regrets of the considered algorithms in four examples.\nThe results for the stochastic instances (a), (b) and (c) described below are averaged over $50$ trials.\n\n\n\\medskip\n\\noindent\n\\emph{\\textup(a\\textup) Stochastic instance with a gap.}\nThis is the standard instance considered in this paper.\nThe losses are drawn independently from Bernoulli distributions (one of parameter $0.3$, $2$ of parameter $0.4$ and $7$ of parameter $0.5$, so that $M=10$ and $\\Delta = 0.1$).\nThe results of Figure~\\ref{fig:1a} confirm our theoretical results: Decreasing Hedge achieves a small, constant regret which is close to that of AdaHedge and FTL, while Constant Hedge and Hedge with doubling trick suffer a larger regret of order $\\sqrt{T}$ (note that, although the expected regret of Constant Hedge converges in this case, the value of this limit depends on its learning rate and hence on $T$).\n\n\\medskip\n\\noindent\n\\emph{\\textup(b\\textup) ``Hard'' stochastic instance.}\nThis example has a zero gap $\\Delta = 0$ between the two leading experts and $M=10$, which makes it ``hard'' from the standpoint of Theorem~\\ref{thm:hedge-stochastic} (which no longer applies in this limit case).\nThe losses are drawn from independent Bernoulli distributions, of parameters $0.5$ for the $2$ leading experts, and $0.7$ for the $8$ remaining ones.\nAlthough all algorithms suffer an unavoidable $\\Theta (\\sqrt{T})$ regret due to pure noise, Decreasing Hedge, AdaHedge and FTL achieve better regret than the two conservative Hedge variants (Figure~\\ref{fig:1b}).\nThis is due to the fact that for the former algorithms, the weights of suboptimal experts decrease quickly and only induce a constant regret.\n\n\\medskip\n\\noindent\n\\emph{\\textup(c\\textup) Small loss for the best expert.}\nIn this experiment, we illustrate one advantage of adaptive Hedge algorithms such as AdaHedge over Decreasing Hedge, namely the fact that they admit improved regret bounds when the leading expert has small loss. We considered in this experiment $M = 10$, $\\Delta = 0.04$ and the leading expert is $\\mathsf{Beta}(0.04,0.96)$, then $4$ $\\mathsf{Beta}(0.08,0.92)$, then $5$ $\\mathsf{Beta}(0.5, 0.5)$.\n\n\\medskip\n\\noindent\n\\emph{\\textup(d\\textup) Adversarial with a gap instance.}\nThis simple instance is not random, and satisfies the assumptions of Theorem~\\ref{thm:adv-gap}.\nIt is defined by $M=3$, $\\Delta = 0.04$, $\\ell_{3, t} = \\frac{3}{4}$ for $t \\geq 1$, $(\\ell_{1, t}, \\ell_{2,t}) = (\\frac{1}{2}, 0)$ if $t=1$, $(0, 1)$ if $t \\geq 80$ or if $t$ is even, and $(1, 0)$ otherwise.\nFTL suffers linear regret in the first phase, while Constant Hedge and Hedge with doubling trick suffer $\\Theta (\\sqrt{T})$ during the second phase.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.48\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{stoch_gap_2_legends.pdf}\n \\caption{\n \n }\n \\label{fig:1a}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.48\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth] {stoch_hard.pdf}\n \\caption{\n \n }\\label{fig:1b}\n\\end{subfigure} \\\\ %\n\\begin{subfigure}[b]{.48\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth] {small_loss_1000.pdf}\n \\caption{\n \n }\\label{fig:1c}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.48\\linewidth}\n \\includegraphics[width=\\linewidth] {adv_gap_simple.pdf} \n \\caption{\n \n }\\label{fig:1d}\n\\end{subfigure}\n\\caption{Cumulative regret of Hedge algorithms on four examples, see text for a precise description and discussion about the results. (a) Stochastic instance with a gap; (b) ``Hard'' stochastic instance; (c) Small loss for the best expert; (d) Adversarial with a gap instance.}\n\\label{fig:experiments}\n\\end{figure}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\n\nIn this article, we carried the regret analysis of the standard exponential weights (Hedge) algorithm in the stochastic expert setting, closing a gap in the existing literature.\nOur analysis reveals that, despite being tuned for the worst-case adversarial setting and lacking any adaptive tuning of the learning rate, Decreasing Hedge achieves optimal regret in the stochastic setting.\nThis property also enables one to distinguish it qualitatively from other variants including the one with fixed (horizon-dependent) learning rate or the one with doubling trick, which both fail to adapt to gaps in the losses.\nTo the best of our knowledge, this is the first result that shows the superiority of the decreasing learning rate over the doubling trick.\nIn addition, it suggests that, even for a fixed time horizon $T$, the decreasing learning rate tuning should be favored over the constant one.\n\nFinally, we showed that the regret of Decreasing Hedge on any stochastic instance is essentially characterized by the sub-optimality gap $\\Delta$.\nThis shows that adaptive algorithms, including algorithms achieving second-order regret bounds, can actually outperform Decreasing Hedge on some stochastic instances that exhibit a more refined form of ``easiness''.\n\n\n\\paragraph{A link with stochastic optimization.}\n\\label{par:connection_with_stochastic_optimization}\n\nOur results have a similar flavor to a well-known result \\citep{moulines2011nonasymptotic} about stochastic optimization: stochastic gradient descent (SGD) with learning rate $\\eta_t \\propto 1 \/ \\sqrt{t}$ (which is tuned for the convex case but not for the non-strongly convex case) and Polyak-Ruppert averaging achieves a fast $O(1 \/ (\\mu t))$ excess risk rate for $\\mu$-strongly convex problems, without the knowledge of $\\mu$.\nHowever, this link stops here since the two results are of a significantly different nature: the $O(1 \/ (\\mu t))$ rate is satisfied only by SGD with Polyak-Ruppert averaging, and it does not come from a regret bound;\neven in the $\\mu$-strongly convex case, it can be seen that SGD with step-size $\\eta_t \\propto 1 \/ \\sqrt{t}$ suffers a $\\Theta (\\sqrt{t})$ regret.\nIn fact, the opposite phenomenon occurs: in stochastic optimization, SGD uses a \\emph{larger} $\\Theta(1\/\\sqrt{t})$ step-size than the $\\Theta(1\/(\\mu t))$ step size which exploits the knowledge of strong convexity, but the effect of this larger step-size is balanced by the averaging.\nBy contrast, in the expert setting, Hedge uses a \\emph{smaller} $\\Theta(\\sqrt{(\\log M)\/t})$ learning rate than the constant, large enough learning rate which exploits the knowledge of the stochastic nature of the problem.\n\n\n\\paragraph{Acknowledgments.}\n\nThe authors wish to warmly thank all four Anonymous Reviewers for their helpful feedback and insights on this work.\nIn particular, the proof of Proposition~\\ref{prop:lowerbound-gap} was proposed by an Anonymous Referee, which allowed to shorten our initial proof.\n\n\\section{Proofs}\n\\label{sec:proofs}\n\nWe now provide the proofs of the results from the previous Sections, by order of appearance in the text.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:hedge-stochastic}}\n\\label{sec:proof-theorem-1}\n\nLet $t_0 = \\left\\lceil \\frac{8\\log M}{\\Delta^2} \\right\\rceil$, so that $\\sqrt{t_0} \\leq \\sqrt{1 + \\frac{8 \\log M}{\\Delta^2}} \\leq 1 + \\frac{\\sqrt{8 \\log M}}{\\Delta}$ (since $\\sqrt{a+b} \\leq \\sqrt{a} + \\sqrt{b}$ for $a,b \\geq 0$).\nThe worst-case regret bound of Hedge (Proposition~\\ref{prop:hedge-adversarial}) shows that for $1 \\leq T \\leq t_0$:\n\\begin{equation}\n R_{i^*,T}\n \\leq \\sqrt{T \\log M}\n \\leq \\sqrt{t_0 \\log M}\n \\leq \\sqrt{\\log M} + \\frac{2 \\sqrt{2} \\log M}{\\Delta}\n \\leq \\frac{4 \\log M}{\\Delta}\n \\label{eq:hedge-stochastic-proof-1}\n\\end{equation}\n(since $\\log M \\geq 1$ as $M \\geq 3$, $\\Delta \\leq 1$ and $2 \\sqrt{2} \\leq 3$),\nwhich establishes~\\eqref{eq:regret-stochastic-exp} for $T \\leq t_0$.\nIn order to prove~\\eqref{eq:regret-stochastic-exp} for $T \\geq t_0 +1$, we start by decomposing the regret with respect to $i^*$ as\n\\begin{equation}\n\\label{eq:hedge-stochastic-proof-2}\nR_{i^*,T}\n= \\widehat L_{T} - L_{i^*, T}\n= \\widehat L_{t_0} - L_{i^*, t_0} + \\sum_{t=t_0+1}^T (\\widehat \\ell_t - \\ell_{i^*, t})\n\\, .\n\\end{equation}\nSince $\\widehat L_{t_0} - L_{i^*, t_0} \\leq R_{t_0}$ is controlled by~\\eqref{eq:hedge-stochastic-proof-1}, it remains to upper bound the second term in~\\eqref{eq:hedge-stochastic-proof-2}.\nFirst, for every $t \\geq t_0 +1$,\n\\begin{equation}\n \\label{eq:instant-regret}\n \\widehat \\ell_t - \\ell_{i^*,t}\n = \\sum_{i \\neq i^*} v_{i,t} (\\ell_{i,t} - \\ell_{i^*,t})\n \\, .\n\\end{equation}\nSince $\\bm \\ell_t$ is independent of $\\bm v_t$ (which is $\\sigma (\\bm \\ell_1, \\dots, \\bm \\ell_{t-1})$-measurable),\ntaking the expectation in~\\eqref{eq:instant-regret} yields, denoting $\\Delta_i = \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}]$,\n\\begin{equation}\n \\label{eq:instant-expected-regret}\n \\mathbb E [\\widehat \\ell_t - \\ell_{i^*,t}]\n = \\sum_{i \\neq i^*} \\Delta_i \\mathbb E [v_{i,t}]\n \\, .\n\\end{equation}\nFirst, for every $i \\neq i^*$,\napplying Hoeffding's inequality to the i.i.d.\\@\\xspace centered variables $Z_{i,t} := - \\ell_{i,t} + \\ell_{i^*,t} + \\Delta_i$, which belong to $[-1 + \\Delta_i, 1 + \\Delta_i]$, yields\n\\begin{align}\n \\label{eq:stochastic-proof-hoeffding}\n \\P \\left( L_{i,t-1} - L_{i^*,t-1} < \\frac{\\Delta_i (t-1)}{2} \\right)\n &= \\P \\left( \\sum_{s=1}^{t-1} Z_{i,s} > \\frac{\\Delta_i (t-1)}{2} \\right) \\nonumber \\\\\n &\\leq e^{ - \\frac{t-1}{2} (\\Delta_i\/2)^2} \\nonumber \\\\\n &= e^{ - {(t-1) \\Delta_i^2}\/{8}}\n \\, .\n\\end{align}\nOn the other hand, if $L_{i,t-1} - L_{i^*,t-1} \\geq \\Delta_i (t-1) \/2$, then \n\\begin{align}\n \\label{eq:hedge-stochastic-proof-3}\n v_{i,t}\n &= \\frac{e^{- \\eta_t (L_{i,t-1} - L_{i^*,t-1})}}{1 + \\sum_{j \\neq i^*} e^{- \\eta_t (L_{j,t-1} - L_{i^*,t-1})}} \\nonumber \\\\\n &\\leq e^{- 2 \\sqrt{(\\log M) \/ t} \\times \\Delta_i (t-1) \/ 2} \\nonumber \\\\\n &\\leq e^{- \\Delta_i \\sqrt{(t-1) (\\log M)\/2}}\n\\end{align}\nsince $t \\leq 2(t-1)$.\nIt follows from~\\eqref{eq:hedge-stochastic-proof-3} and~\\eqref{eq:stochastic-proof-hoeffding} that, for $t \\geq t_0+1 \\geq 2$,\n\\begin{align}\n \\label{eq:hedge-stochastic-two-terms}\n \\mathbb E [v_{i,t}]\n &\\leq \\P \\left( L_{i,t-1} - L_{i^*,t-1} > \\frac{\\Delta_i (t-1)}{2} \\right) + e^{- \\Delta_i \\sqrt{(t-1)(\\log M)\/2}} \\nonumber \\\\\n &\\leq e^{- (t-1) \\Delta_i^2 \/ 8} + e^{- \\Delta_i \\sqrt{(t-1)(\\log M)\/2}}\n \\, .\n\\end{align}\nNow, a simple analysis of functions shows that the functions $f_1 (u) = u e^{-u}$ and $f_2 (u) = u e^{-u^2\/2}$ are decreasing on $[1, + \\infty)$.\nSince $\\Delta_i \\geq \\Delta$, this entails that\n\\begin{equation}\n \\label{eq:hedge-stochastic-proof-5}\n \\Delta_i e^{- (t-1) \\Delta_i^2 \/ 8}\n = \\frac{2}{\\sqrt{t-1}} f_2 \\left( \\frac{\\sqrt{t-1} \\Delta_i}{2} \\right)\n \\leq \\frac{2}{\\sqrt{t-1}} f_2 \\left( \\frac{\\sqrt{t-1} \\Delta}{2} \\right)\n = \\Delta e^{- (t-1) \\Delta^2 \/ 8}\n\\end{equation}\nprovided that $\\frac{\\sqrt{t-1} \\Delta}{2} \\geq 1$, \\ie $t \\geq 1 + \\frac{4}{\\Delta^2}$, which is the case since $t \\geq t_0 +1 \\geq 1 + \\frac{8 \\log M}{\\Delta^2}$.\nLikewise,\n\\begin{equation}\n \\label{eq:hedge-stochastic-proof-6}\n \\Delta_i e^{- \\Delta_i \\sqrt{(t-1) (\\log M) \/2}}\n \\leq \\Delta e^{- \\Delta \\sqrt{(t-1) (\\log M) \/2}}\n\\end{equation}\nif $\\Delta \\sqrt{(t-1) (\\log M) \/2} \\geq 1$, \\ie $t \\geq 1+ \\frac{2}{(\\log M) \\Delta^2}$, which is ensured by $t \\geq t_0 +1$.\nIt follows from~\\eqref{eq:instant-expected-regret}, \\eqref{eq:hedge-stochastic-two-terms}, \\eqref{eq:hedge-stochastic-proof-5} and~\\eqref{eq:hedge-stochastic-proof-6} that for every $t \\geq t_0 + 1$:\n\\begin{align}\n \\label{eq:hedge-stochastic-proof-simult}\n \\mathbb E [\\widehat \\ell_t - \\ell_{i^*,t}]\n &\\leq M \\Delta e^{- (t-1) \\Delta^2 \/ 8} + M \\Delta e^{- \\Delta \\sqrt{(t-1) (\\log M) \/2}} \\nonumber \\\\\n &= \\big( M e^{- t_0 \\Delta^2 \/ 8} \\big) \\big( \\Delta e^{- (t-t_0- 1) \\Delta^2 \/ 8} \\big) + \\big( M e^{- \\Delta \\sqrt{(t-1) (\\log M) \/8}} \\big) \\big( \\Delta e^{- \\Delta \\sqrt{(t-1) (\\log M) \/8}} \\big) \\nonumber \\\\\n &\\leq \\Delta e^{- (t-t_0- 1) \\Delta^2 \/ 8} + \\Delta e^{- \\Delta \\sqrt{(t-1)\/8}}\n\\end{align}\nwhere inequality~\\eqref{eq:hedge-stochastic-proof-simult} comes from the bound $M e^{-t_0 \\Delta^2 \/8} \\leq 1$ (since $t_0 \\geq \\frac{8 \\log M}{\\Delta^2}$) and from the fact that $M e^{- \\Delta \\sqrt{(t-1) (\\log M) \/8}} \\leq 1$ amounts to $t \\geq 1 + \\frac{8 \\log M}{\\Delta^2}$, that is, to $t \\geq t_0 +1$.\nSumming inequality~\\eqref{eq:hedge-stochastic-proof-simult} yields, for every $T \\geq t_0+1$,\n\\begin{align} \n \\mathbb E [ \\sum_{t = t_0 +1}^T (\\ell_{t} - \\ell_{i^*,t}) ]\n &\\leq \\sum_{t=t_0+1}^T \\left\\{ \\Delta e^{- (t-t_0- 1) \\Delta^2 \/ 8} + \\Delta e^{- \\Delta \\sqrt{(t-1)\/8}} \\right\\} \\nonumber \\\\\n &\\leq \\Delta \\sum_{t \\geq 0} e^{-t \\Delta^2\/8} + \\Delta \\sum_{t \\geq 1} e^{- (\\Delta \/ \\sqrt{8}) \\sqrt{t}} \\nonumber \\\\\n &\\leq \\Delta \\left( 1 + \\frac{8}{\\Delta^2} \\right) + \\Delta \\times \\frac{2}{(\\Delta \/ \\sqrt{8})^2} \\label{eq:hedge-stochastic-proof-7} \\\\\n \n \\leq \\frac{25}{\\Delta}\n\\end{align}\nwhere inequality~\\eqref{eq:hedge-stochastic-proof-7} comes from Lemma~\\ref{lem:tail} below.\nFinally, combining inequalities~\\eqref{eq:hedge-stochastic-proof-1} and~\\eqref{eq:hedge-stochastic-proof-7} yields the pseudo-regret bound $\\mathcal{R}_T \\leq \\frac{4 \\log M + 25}{\\Delta}$.\n\n\\begin{lemma}\n \\label{lem:tail}\n For every $\\alpha > 0$,\n \\begin{align}\n \\label{eq:tail-bound}\n \\sum_{t \\geq 1} e^{-\\alpha t} &\\leq \\frac{1}{\\alpha} \\\\\n \\label{eq:tail-bound-sqrt}\n \\sum_{t \\geq 1} e^{- \\alpha \\sqrt{t}} &\\leq \\frac{2}{\\alpha^2} \\, .\n \\end{align}\n\\end{lemma}\n\n\\begin{proof}\n Since the functions $t \\mapsto e^{-\\alpha {t}}$ and $t \\mapsto e^{-\\alpha \\sqrt{t}}$ are decreasing on $\\mathbf R^+$, we have\n \\begin{align*}\n &\\sum_{t \\geq 1} e^{-\\alpha t}\n \\leq \\int_0^{\\infty} e^{-\\alpha t} \\mathrm{d} t\n = \\frac{1}{\\alpha}\n \\, \n \\\\\n &\\sum_{t \\geq 1} e^{-\\alpha \\sqrt{t}}\n \\leq \\int_{0}^{+\\infty} e^{-\\alpha \\sqrt{t}} \\mathrm{d} t\n \\underset{u = \\alpha \\sqrt t}{=} \\frac{2}{\\alpha^2}\n \\int_{0}^{+\\infty} u e^{-u} \\mathrm{d} u\n = \\frac{2}{\\alpha^2}.\n \\end{align*}\n\\end{proof}\n\n\\begin{remark}\n \\label{rem:pseudoregret}\n While the upper bound of Theorem~\\ref{thm:hedge-stochastic} is stated for the pseudo-regret $\\mathcal{R}_T$, a similar upper bound holds for the expected regret $\\mathbb E [R_T]$.\n Indeed, under the assumptions of Theorem~\\ref{thm:hedge-stochastic}, for every $T \\geq \\frac{4 \\log M}{\\Delta^2}$, we have\n $\\mathbb E [R_T] \\leq \\mathcal{R}_T + \\frac{1.1}{\\Delta}$.\n\\end{remark}\n\n\n\\begin{proof}\n Note that $\\mathbb E [R_T] - \\mathcal{R}_T = \\mathbb E [L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T}]$.\n For every $a \\geq 0$, Hoeffding's inequality (applied to the i.i.d.\\@\\xspace centered variables $\\ell_{i^*,t} - \\ell_{i,t} + \\Delta_i \\in [-1 + \\Delta_i, 1 + \\Delta_i]$, $1\\leq t \\leq T$) entails \n \\begin{align}\n \\label{eq:proof-pseudo-regret-1}\n \\P \\left( L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T} \\geq a \\right)\n &\\leq \\sum_{i \\neq i^*} \\P \\left( L_{i^*,T} - L_{i,T} + \\Delta_i T \\geq \\Delta_i T + a \\right) \\nonumber \\\\\n &\\leq \\sum_{i \\neq i^*} e^{- (\\Delta_i T + a)^2 \/ (2T)} \\\\\n &\\leq M e^{- T \\Delta^2\/2} e^{-a^2\/(2T)} \\nonumber \\\\\n &\\leq e^{-T \\Delta^2\/4} e^{-a^2\/(2T)}\n \\label{eq:proof-pseudo-regret-2}\n \\, ,\n \\end{align}\n where inequality~\\eqref{eq:proof-pseudo-regret-2} comes from the fact that $M e^{-T \\Delta^2\/4} \\leq 1$ since $T \\geq \\frac{4 \\log M}{\\Delta^2}$.\n Since the random variable $L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T}$ is nonnegative, this implies that\n \\begin{align} \n \\mathbb E \\left[ L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T} \\right]\n &= \\int_0^{\\infty} \\P \\left( L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T} \\geq a \\right) \\mathrm{d} a \\nonumber \\\\\n &\\leq e^{-T \\Delta^2\/4} \\int_0^{\\infty} e^{-a^2\/(2T)} \\mathrm{d} a \\nonumber \\\\\n &= \\sqrt{\\frac{\\pi}{2}} \\cdot \\sqrt{T} e^{-T \\Delta^2\/4} \\nonumber \\\\\n &= \\frac{\\sqrt{\\pi}}{\\Delta} \\big[ \\Delta \\sqrt{T\/2} \\cdot e^{- (\\Delta \\sqrt{T\/2})^2\/2} \\big] \\nonumber \\\\\n &\\leq \\frac{\\sqrt{\\pi\/e}}{\\Delta}\n \\label{eq:proof-pseudo-regret-3}\n \\end{align}\n where inequality~\\eqref{eq:proof-pseudo-regret-3} comes from the fact that the function $u \\mapsto u e^{-u^2\/2}$ attains its maximum on $\\mathbf R^+$ at $u=1$.\n This concludes the proof, since $\\sqrt{\\pi\/e}\\leq 1.1$.\n\\end{proof}\n\n\n\\subsection{Proof of Proposition~\\ref{prop:lowerbound-gap}}\n\\label{sec:proof-lowerbound-gap}\n\nFix $M$, $\\Delta$ and $T$ as in Proposition~\\ref{prop:lowerbound-gap}.\nFor $i^* \\in \\{ 1, \\dots, M\\}$, denote $\\P_{i^*}$ the following distribution on $[0, 1]^{M \\times T}$: if $(\\ell_{i,t})_{1\\leq i \\leq M, 1\\leq t \\leq T} \\sim \\P_{i^*}$, then the variables $\\ell_{i,t}$ are independent Bernoulli variables, of parameter $\\frac{1}{2} - \\Delta$ if $i = i^*$ and $\\frac{1}{2}$ otherwise; also, denote by $\\mathbb E_{i^*}$ the expectation with respect to $\\P_{i^*}$.\nLet $\\mathcal{A} = (A_t)_{1 \\leq t \\leq T}$ be any Hedging algorithm, where $A_t: [0, 1]^{M \\times (t-1)} \\to \\mathcal{P}_M$ maps past losses $(\\bm \\ell_{1}, \\dots, \\bm \\ell_{t-1})$ to an element of the probability simplex $\\mathcal{P}_M \\subset \\mathbf R^M$ on $\\{ 1, \\dots, M \\}$.\nFor any $i^* \\in \\{ 1, \\dots, M \\}$, let $\\mathcal{R}_{T} (i^*, \\mathcal{A})$ denote the pseudo-regret of algorithm $\\mathcal{A}$ under the distribution $\\P_{i^*}$.\nSince $\\bm \\ell_t$ is independent of $\\bm v_t$ under $\\P_{i^*}$, we have\n\\begin{equation}\n \\label{eq:lowerbound-gap-pseudoregret}\n \\mathcal{R}_T (i^*, \\mathcal{A})\n = \\sum_{t=1}^T \\sum_{i \\neq i^*} \\mathbb E_{i^*} \\big[ v_{i,t} (\\ell_{i,t} - \\ell_{i^*,t}) \\big]\n = \\Delta \\sum_{t=1}^T \\sum_{i \\neq i^*} \\mathbb E_{i^*} [v_{i,t}]\n = \\Delta \\sum_{t=1}^T \\mathbb E_{i^*} [1 - v_{i^*,t}]\n\\end{equation}\nwith $\\bm v_t := A_t (\\bm\\ell_1, \\dots, \\bm\\ell_{t-1})$.\nIt follows from Equation~\\eqref{eq:lowerbound-gap-pseudoregret} that, for every $\\mathcal{A}$ and $i^*$, $\\mathcal{R}_T (i^*, \\mathcal{A})$ increases with $T$.\nHence, without loss of generality we may assume that {$T = \\lfloor (\\log M) \/ (16 \\Delta^2) \\rfloor$}.\nThe maximum pseudo-regret of $\\mathcal{A}$ on the instances $\\P_{i^*}$ is lower-bounded as follows:\n\\begin{equation}\n \\label{eq:lowerbound-gap-1}\n \\sup_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n \\geq \\frac{1}{M} \\sum_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n = \\frac{1}{M} \\sum_{1 \\leq i^* \\leq M} \\Delta \\sum_{t=1}^T \\mathbb E_{i^*} [ 1 - v_{i^*,t}]\n \\, .\n\\end{equation}\n\nWe now ``randomize'' the algorithm $\\mathcal{A}$, by replacing it with a randomized algorithm which picks expert $i$ at time $t$ with probability $v_{i,t}$.\nFormally, let $\\widetilde P = \\mathcal{U}}%{\\mathsf{Unif} ([0, 1])^{\\otimes T}$ be the distribution of $T$ independent uniform random variables on $[0, 1]$, and denote $\\widetilde \\P_{i^*} = \\P_{i^*} \\otimes \\widetilde P$ for $i^* \\in \\{ 1, \\dots, M \\}$.\nFurthermore, for every $\\bm v \\in \\mathcal{P}_M$, let $I_{\\bm v} : [0, 1] \\to \\{ 1, \\dots, M \\}$ be a measurable map such that $\\P (I_{\\bm v} (U) = i) = v_i$ for every $i \\in \\{ 1, \\dots, M \\}$, where $U \\sim \\mathcal{U}}%{\\mathsf{Unif} ([0 ,1])$.\nFor every sequence of losses $\\bm \\ell_1, \\dots, \\bm \\ell_T$ and random variables $U_1, \\dots, U_T$ and every $1 \\leq t \\leq T$, let $I_t = I_{\\bm v_t} (U_t)$, where $\\bm v_t = A_t (\\bm \\ell_1, \\dots, \\bm \\ell_t)$.\n\nDenote by $\\widetilde \\mathbb E_{i^*}$ the expectation with respect to $\\widetilde \\P_{i^*}$.\nBy definition of $I_{\\bm v}$, we have $\\mathbb E_{i^*} [ v_{i^*, t} ] = \\widetilde \\mathbb E_{i^*} [ \\indic{I_t = i^*} ]$ so that, denoting $N_i = \\sum_{i=1}^T \\indic{I_t = i}$ the number of times expert $i$ is picked,\n\\begin{equation*}\n \\sum_{t=1}^T \\mathbb E_{i^*} [1 - v_{i^*,t}] = \\widetilde \\mathbb E_{i^*} [ T - N_{i^*} ]\n \\geq \\P_{i^*} (N_{i^*} \\leq T\/2) \\cdot \\frac{T}{2}\n \\, .\n\\end{equation*}\nHence, letting $A_i \\subseteq [0, 1]^{M \\times T} \\times [0, 1]^T$ be the event $\\{ N_i > T\/2 \\}$,\nEquation~\\eqref{eq:lowerbound-gap-1} implies that\n\\begin{equation}\n \\label{eq:lowerbound-gap-2}\n \\sup_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n \\geq \\frac{\\Delta T}{2} \\times \\frac{1}{M} \\sum_{1\\leq i^* \\leq M} \\big( 1 - \\widetilde \\P_{i^*} (A_{i^*}) \\big)\n \\, .\n\\end{equation}\n\nIt now remains to upper bound $\\frac{1}{M} \\sum_{i^*} \\widetilde \\P_{i^*} (A_{i^*})$.\nTo do this, first note that the events $A_{i^*}$, $1 \\leq i^* \\leq M$, are pairwise disjoint.\nHence, Fano's inequality \\citep[see][p.2]{gerchinovitz2017fano} implies that, for every distribution $\\widetilde \\mathbb Q$ on $[0, 1]^{M \\times T} \\times [0, 1]^T$,\n\\begin{equation}\n \\label{eq:lowerbound-gap-fano}\n \\frac{1}{M} \\sum_{1\\leq i^* \\leq M} \\widetilde \\P_{i^*} ( A_{i^*})\n \\leq \\frac{1}{\\log M} \\bigg\\{ \\frac{1}{M} \\sum_{1 \\leq i^* \\leq M} \\kll{\\widetilde \\P_{i^*}}{\\widetilde \\mathbb Q} + \\log 2 \\bigg\\}\n\\end{equation}\nwhere $\\kll{\\P}{\\mathbb Q}$ denotes the Kullback-Leibler divergence between $\\P$ and $\\mathbb Q$.\nHere, we take $\\widetilde \\mathbb Q = \\mathbb Q \\otimes \\widetilde P$, where $\\mathbb Q$ is the product of Bernoulli distributions $\\mathcal{B} (1\/2)^{\\otimes T}$.\nThis choice leads to\n\\begin{equation*}\n \\kll{\\widetilde \\P_{i^*}}{\\widetilde \\mathbb Q}\n = \\kll{\\P_{i^*}}{\\mathbb Q}\n = T \\cdot \\kll{\\mathcal{B}(1\/2 - \\Delta)}{\\mathcal{B}(1\/2)}\n \\leq 4 T \\Delta^2\n \\leq \\frac{\\log M}{4}\n \\, ,\n\\end{equation*}\nwhere the first bound is obtained by comparing KL and $\\chi^2$ divergences \\citep[Lemma~2.7]{tsybakov2009nonparametric}.\nHence, inequality~\\eqref{eq:lowerbound-gap-fano} becomes (recalling that $M \\geq 4$)\n\\begin{equation*}\n \\frac{1}{M} \\sum_{1\\leq i^* \\leq M} \\widetilde \\P_{i^*} ( A_{i^*})\n \\leq \\frac{(\\log M)\/4}{\\log M} + \\frac{\\log 2}{\\log M}\n \\leq \\frac{3}{4}\n \\, ;\n\\end{equation*}\nplugging this into~\\eqref{eq:lowerbound-gap-2} yields, noting that $T = \\lfloor (\\log M) \/ (16 \\Delta^2) \\rfloor \\geq (\\log M) \/ (32 \\Delta^2)$ since $(\\log M)\/(16 \\Delta^2) \\geq 1$ (as $M \\geq 4$ and $\\Delta \\leq \\frac{1}{4}$),\n\\begin{equation*}\n \\sup_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n \\geq \\frac{\\Delta T}{2} \\times \\frac{1}{4}\n \\geq \\frac{\\log M}{256 \\Delta}\n \\, .\n\\end{equation*}\nThis concludes the proof.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:adv-gap} and Corollary~\\ref{cor:hedge-martingale}}\n\\label{sec:proof-thm-adv-gap}\n\n\nLet $t_0 $ be the smallest integer $t\\geq 1$ such that $M e^{- c_0 \\Delta \\sqrt{t \\log (M) \/ 8}} \\leq \\Delta$, namely $t_0 = \\left\\lceil \\frac{8}{c_0^2 \\Delta^2} \\frac{\\log^2 (M \/ \\Delta)}{\\log M} \\right\\rceil$.\n Note that $\\sqrt{t_0} \\leq \\sqrt{1 + \\frac{8}{c_0^2 \\Delta^2} \\frac{\\log^2 (M \/ \\Delta)}{\\log M}} \\leq 1 + \\frac{\\sqrt{8}}{c_0 \\Delta} \\frac{\\log (M \/ \\Delta)}{\\sqrt{\\log M}}$.\n Let $t_1 := t_0 \\vee \\tau_0$.\n For every $T \\leq t_1$, the regret bound in the assumption of Theorem~\\ref{thm:adv-gap} implies\n \\begin{align}\n \\label{eq:proof-adv-1}\n R_T\n &\\leq c_1 \\sqrt{T \\log M} \\nonumber \\\\ \n &\\leq c_1 \\sqrt{\\tau_0 \\log M} + c_1 \\sqrt{t_0 \\log M} \\nonumber \\\\ \n &\\leq c_1 \\sqrt{\\tau_0 \\log M} + c_1 \\sqrt{\\log M} + \\frac{\\sqrt{8} \\log(M\/\\Delta)}{c_0 \\Delta}\n \\end{align}\n which implies~\\eqref{eq:regret-gap-adv} with $c_2 = c_1 + \\frac{\\sqrt{8}}{c_0}$ and $c_3 = \\frac{\\sqrt{8}}{c_0}$ (since $1\\leq \\sqrt{\\log M} \\leq \\frac{\\log M}{\\Delta}$).\n From now on, assume that $T \\geq t_1 + 1$.\n Since $T \\geq \\tau_0$, we have $R_T = \\widehat L_T - L_{i^*,T}$, so that\n \\begin{equation}\n \\label{eq:proof-adv-2}\n R_T\n = \\widehat L_{t_1} - L_{i^*, t_1} + \\sum_{t=t_1 + 1}^T \\big( \\widehat \\ell_t - \\ell_{i^*, t} \\big)\n \\, .\n \\end{equation}\n In addition, we have for $t\\geq t_1 +1$\n \\begin{align} \n \\widehat \\ell_t - \\ell_{i^*, t}\n &= \\sum_{i \\neq i^*} v_{i,t} (\\ell_{i,t} - \\ell_{i^*, t}) \\nonumber \\\\\n &\\leq \\sum_{i \\neq i^*} v_{i,t} \\nonumber \\\\\n &= \\sum_{i\\neq i^*} \\frac{ e^{- \\eta_t (L_{i,t-1} - L_{i^*, t-1})}}{1 + \\sum_{j\\neq i^*} e^{- \\eta_t (L_{j,t-1} - L_{i^*, t-1})}} \\nonumber \\\\\n &\\leq \\sum_{i\\neq i^*} e^{- c_0 \\sqrt{(\\log M)\/t} \\times \\Delta (t-1)} \\label{eq:proof-adv-3} \\\\\n &\\leq M e^{- c_0 \\Delta \\sqrt{(t-1)(\\log M)\/2}} \\nonumber\n \\\\\n &\\leq \\big( M e^{- c_0 \\Delta \\sqrt{t_0 (\\log M)\/8}} \\big) e^{- c_0 \\Delta \\sqrt{(t-1)\/8}} \\label{eq:proof-adv-5} \\\\\n &\\leq \\Delta e^{- c_0 \\Delta \\sqrt{(t-1)\/8}} \\label{eq:proof-adv-6}\n \\end{align}\n where~\\eqref{eq:proof-adv-3} comes from the fact that $\\eta_t \\geq c_0 \\sqrt{(\\log M)\/t}$ and $L_{i,t-1} - L_{i^*,t-1} \\geq \\Delta (t-1)$ (since $t-1\\geq t_1 \\geq \\tau_0$),~\\eqref{eq:proof-adv-5} from the fact that $t-1 \\geq t_0$ and $\\log M \\geq 1$, and~\\eqref{eq:proof-adv-6} from the fact that $M e^{- c_0 \\Delta \\sqrt{t_0 (\\log M)\/8}} \\leq \\Delta$.\n Summing inequality~\\eqref{eq:proof-adv-6}, we obtain \n \\begin{align} \n \\sum_{t=t_1+1}^T (\\widehat \\ell_t - \\ell_{i^*, t}) \\nonumber\n &\\leq \\sum_{t=t_1+1}^T \\Delta e^{- c_0 \\Delta \\sqrt{(t-1)\/8}} \\\\\n &\\leq \\Delta \\sum_{t\\geq 1} e^{- c_0 \\Delta \\sqrt{t\/8}} \\nonumber \\\\\n &\\leq \\Delta \\times \\frac{2}{(c_0 \\Delta \/ \\sqrt{8})^2} \\label{eq:proof-adv-7} \\\\\n &= \\frac{16}{c_0^2 \\Delta}\n \\label{eq:proof-adv-tail}\n \\end{align}\n where~\\eqref{eq:proof-adv-7} follows from Lemma~\\ref{lem:tail}.\n Combining~\\eqref{eq:proof-adv-2},~\\eqref{eq:proof-adv-1} and~\\eqref{eq:proof-adv-tail} proves Theorem~\\ref{thm:adv-gap} with $c_2 = c_1 + \\frac{\\sqrt{8}}{c_0}$, $c_3 = \\frac{\\sqrt{8}}{c_0}$ and $c_4 = \\frac{16}{c_0^2}$.\n\n \\begin{proof}[Proof of Corollary~\\ref{cor:hedge-martingale}]\n Define $\\tau = \\sup \\{ t \\geq 0 , \\exists i \\neq i^* , L_{i,t} - L_{i^*, t} \\leq \\frac{\\Delta t}{2} \\}$.\n By Lemma~\\ref{lem:crossing-time} below, for every $\\varepsilon > 0$ we have, with probability at least $1-\\varepsilon$, $\\tau \\leq {8 ( \\log M + \\log \\varepsilon^{-1})}\/{\\Delta^2}$.\n By Theorem~\\ref{thm:adv-gap}, this implies that, with probability at least $1-\\varepsilon$,\n \\begin{align*}\n \\label{eq:proof-hedge-martingale-1}\n R_T\n &\\leq c_1 \\sqrt{\\tau \\log M} + \\frac{c_2 \\log M + c_3 \\log {\\Delta}^{-1} + c_4}{\\Delta \/ 2} \\\\\n &\\leq \\left( c_1 \\sqrt{8} + 2 c_2 \\right) \\frac{\\log M}{\\Delta} + c_1 \\frac{\\sqrt{8\\log M \\log \\varepsilon^{-1}}}{\\Delta} + 2 c_3 \\frac{\\log \\Delta^{-1}}{\\Delta} + \\frac{2 c_4}{\\Delta} \n \\end{align*}\n where $c_2, c_3, c_4$ are the constants of Theorem~\\ref{thm:adv-gap}.\n The bound~\\eqref{eq:regret-martingale-exp} on the pseudo-regret is obtained similarly from Theorem~\\ref{thm:adv-gap}, by using the fact that $\\mathcal{R}_T \\leq \\mathbb E [R_T]$ and\n \\begin{equation*}\n \\mathbb E [ \\sqrt{\\tau \\log M}] \\leq \\sqrt{\\mathbb E [\\tau] \\log M}\n \\leq \\sqrt{\\log M} \\sqrt{ 1 + \\frac{8 (\\log M + 1)}{\\Delta^2}}\n \\leq \\sqrt{\\log M} \\Big( 1 + \\frac{\\sqrt{8 \\log M} + 1}{\\Delta} \\Big)\n \\end{equation*}\n which is smaller than $(2 + \\sqrt{8}) ({\\log M})\/{\\Delta} \\leq 5 (\\log M)\/\\Delta$ since $M \\geq 3$ and $\\Delta \\leq 1$.\n \\end{proof}\n\n\\begin{lemma}\n \\label{lem:crossing-time}\n Let $(\\ell_{i,t})_{1\\leq i \\leq M, t\\geq 1}$ be as in Theorem~\\ref{thm:hedge-stochastic}.\n Denote $\\tau = \\sup \\{ t \\geq 0 , \\exists i \\neq i^* , L_{i,t} - L_{i^*, t} \\leq \\frac{\\Delta t}{2} \\}$.\n We have\n \\begin{equation}\n \\label{eq:crossing-time-exp}\n \\mathbb E [ \\tau ] \n \\leq 1 + \\frac{8 (\\log M + 1)}{\\Delta^2} \n \\, , \n \\end{equation}\n and for every $\\varepsilon \\in (0, 1)$,\n\\begin{equation}\n \\label{eq:crossing-time-prob}\n \\P \\Big( \\tau \\geq \\frac{8 (\\log M + \\log \\varepsilon^{-1})}{\\Delta^2} \\Big) \\leq \\varepsilon \\, .\n\\end{equation} \n\\end{lemma}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:crossing-time}]\n For every $i \\neq i^*$ and $t \\geq 1$, let $\\Delta_{i,t} := \\mathbb E [ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1}]$.\n Using the Hoeffding-Azuma's maximal inequality to the $(\\mathcal{F}_t)_{t\\geq 1}$-martingale difference sequence $Z_{i,t} = - (L_{i,t} - L_{i^*,t}) + \\Delta_{i,t}$ (such that $\\Delta_{i,t} - 1 \\leq Z_{i,t} \\leq \\Delta_{i,t} +1$), together with the fact that $\\Delta_{i,t} \\geq \\Delta$, implies that\n \\begin{equation}\n \\label{eq:proof-crossing-1}\n \\P \\left( \\exists t \\geq t_0, L_{i,t} - L_{i^*,t} \\leq \\frac{\\Delta t}{2} \\right)\n \\leq \\P \\left( \\sup_{t \\geq t_0} \\frac{1}{t} \\left( \\sum_{s=1}^t Z_{i,s} \\right) \\geq \\frac{\\Delta}{2} \\right)\n \\leq e^{- t_0 \\Delta^2\/8}\n \\, .\n \\end{equation}\n By a union bound, equation~\\eqref{eq:proof-crossing-1} implies that\n \\begin{equation}\n \\label{eq:proof-crossing-2}\n \\P \\left( \\tau \\geq t_0 \\right)\n \\leq M e^{-t_0 \\Delta^2\/8}\n \\, .\n \\end{equation}\n Solving for the probability level in~\\eqref{eq:proof-crossing-2} yields the high probability bound~\\eqref{eq:crossing-time-prob} on $\\tau$.\n The bound on $\\tau$ in expectation~\\eqref{eq:crossing-time-exp} ensues by integrating the high-probability bound over $\\varepsilon$.\n\\end{proof}\n\nWe recall Hoeffding-Azuma's maximal inequality for bounded martingale difference sequences \\citep{hoeffding1963probability,azuma1967weighted}.\nWhile it follows from a standard argument, we provide a short proof for completeness, since the inequality given in Proposition~\\ref{lem:hoeffding-maximal} below differs slightly from the one given in~\\citet{hoeffding1963probability}.\n\n\\begin{proposition}[Hoeffding-Azuma's maximal inequality]\n \\label{lem:hoeffding-maximal}\n Let $(Z_t)_{t \\geq 1}$ be a sequence of random variables adapted to a filtration $(\\mathcal{F}_t)_{t\\geq 1}$.\n Assume that $Z_t$ is a martingale difference sequence: $\\mathbb E [Z_t \\,|\\, \\mathcal{F}_{t-1}] = 0$ for any $t \\geq 1$, and that $A_t - 1 \\leq Z_t \\leq A_t+1$ almost surely, where $A_t$ is $\\mathcal{F}_{t-1}$-measurable.\n Then, denoting $S_n := \\sum_{t=1}^n Z_t$, we have for every $n \\geq 1$ and $a \\geq 0$:\n \\begin{equation}\n \\label{eq:hoeffding-maximal}\n \\P \\left( \\sup_{m \\geq n} \\frac{S_m}{m} \\geq a \\right)\n \\leq e^{- n a^2\/2}\n \\, .\n \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n Fix $\\lambda > 0$.\n By Hoeffding's inequality, $\\mathbb E [e^{\\lambda Z_t} \\,|\\, \\mathcal{F}_{t-1}] \\leq e^{\\lambda^2\/2}$, so that the sequence $M_t^{\\lambda} := \\exp \\big( \\lambda S_t - \\lambda^2 t \/ 2 \\big)$ is a positive supermartingale.\n Hence, Doob's supermartingale inequality implies that for $\\varepsilon \\in (0, 1]$:\n \\begin{equation}\n \\label{eq:proof-hoeffding-maximal-1}\n \\P \\Big( \\sup_{t \\geq 1} M_t^{\\lambda} \\geq \\frac{1}{\\varepsilon} \\Big)\n \\leq \\frac{\\mathbb E [M_0^\\lambda]}{1\/\\varepsilon} = \\varepsilon\n \\, .\n \\end{equation}\n Rearranging~\\eqref{eq:proof-hoeffding-maximal-1} and letting $\\lambda = \\sqrt{2 \\log (1\/\\varepsilon) \/ n}$ yields: with probability $1 - \\varepsilon$, for every $t \\geq n$,\n \\begin{equation}\n \\label{eq:proof-hoeffding-maximal-2}\n \\frac{S_t}{t}\n \\leq \\frac{\\log \\left( {1}\/{\\varepsilon} \\right)}{\\lambda t} + \\frac{\\lambda}{2}\n = \\sqrt{\\frac{\\log (1\/\\varepsilon)}{2}} \\left( \\frac{\\sqrt{n}}{t} + \\frac{1}{\\sqrt{t}} \\right)\n \\leq \\sqrt{\\frac{2 \\log (1\/\\varepsilon)}{n}}\n \\, .\n \\end{equation}\n Setting $\\varepsilon = e^{- n a^2\/2}$ in~\\eqref{eq:proof-hoeffding-maximal-2} gives the desired bound.\n\\end{proof}\n\n\n\\subsection{Proof of Proposition~\\ref{prop:lower-bound-hedge-cst}}\n\\label{sec:proof-lower-bound-hedge-cst}\n\nNote that, since the loss vectors $\\bm \\ell_t$ are in fact deterministic, $\\mathcal{R}_T = R_T$.\nDenoting $(v_{i,t})_{1 \\leq i \\leq M}$ the weights selected by the Constant Hedge algorithm at time $t$, and letting $c = c_0 \\sqrt{\\log M}$, we have\n\\begin{align}\n R_T\n &= \\sum_{t=1}^T \\sum_{i=2}^M v_{i, t} (\\ell_{i, t} - \\ell_{1, t}) \\nonumber \\\\\n &= \\sum_{t=1}^T \\sum_{i=2}^M \\frac{\\exp \\big( - \\frac{c}{\\sqrt{T}} (L_{i,t-1}-L_{1,t-1}) \\big)}{1 + \\sum_{2 \\leq i' \\leq M} \\exp \\big( - \\frac{c}{\\sqrt{T}} (L_{i',t-1}-L_{1,t-1}) \\big)} \\nonumber \\\\\n &= \\sum_{t=1}^T \\frac{ (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)}{1 + (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)} \\label{eq:proof-lower-cst1}\n \\, .\n\\end{align}\nNow, let $t_0 \\geq 0$ be the largest integer such that $(M-1) \\exp ( - \\frac{c}{\\sqrt{T}} t ) \\geq 1\/2$, namely\n\\begin{equation*}\n t_0 = \\Big\\lfloor \\frac{\\sqrt{T}}{c} \\log (2(M-1)) \\Big\\rfloor.\n\\end{equation*}\nIt follows from Equation~\\eqref{eq:proof-lower-cst1} that\n\\begin{equation}\n \\label{eq:proof-lower-cst2}\n R_T\n \\geq \\sum_{t=1}^{T \\wedge (t_0 + 1)} \\frac{ (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)}{1 + (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)} \n \\geq \\frac{1}{3} \\min (T, t_0 + 1)\n\\end{equation}\nwhere the second inequality comes from the fact that $\\frac{x}{1 + x} \\geq \\frac{1}{3}$ for $x \\geq \\frac{1}{2}$, \nwhich we apply to $x = (M - 1) \\exp ( - \\frac{c}{\\sqrt{T}} (t-1)) \\geq \\frac{1}{2}$ for $t \\leq T \\wedge (t_0+1) \\leq t_0 +1$.\nIn order to establish inequality~\\eqref{eq:lower-bound-hedge-cst}, it remains to note that\n\\begin{equation*}\n t_0 + 1\n \\geq \\frac{\\sqrt{T}}{c}\n \\log \\big( 2 (M-1) \\big)\n \\geq \\frac{\\sqrt{T \\log M}}{c_0}\n \\, ,\n\\end{equation*}\nsince $2(M-1) \\geq M$ and $c = \\sqrt{c_0 \\log M}$.\n\nNow, consider the Hedge algorithm with doubling trick. Assume that $T \\geq 2$, and let $k \\geq 1$ such that $T_k \\leq T < T_{k+1}$.\nSince $R_T = \\sum_{t=1}^T \\sum_{2\\leq i \\leq M} v_{i, t} (\\ell_{i, t} - \\ell_{1, t})$ and each of the terms in the sum is nonnegative, $R_T$ is lower bounded by the cumulative regret on the period $\\iint{T_{k-1}}{T_k - 1}$.\nDuring this period of length $T_{k-1}$, the algorithm reduces to the Hedge algorithm with constant learning rate $c_0 \\sqrt{\\log (M) \/ T_{k-1}}$, so that the above bound~\\eqref{eq:lower-bound-hedge-cst} applies; further bounding $T_{k-1} \\geq \\frac{T}{4}$ establishes~\\eqref{eq:lower-bound-hedge-doubling}.\n\n\\subsection{Proof of Proposition~\\ref{prop:second-order-bernstein}}\n\\label{sec:proof-second-order-bernstein}\n\nBy convexity of $x \\mapsto x^2$ and concavity of $x \\mapsto x^{\\beta}$,\nwe have:\n\\begin{align}\n \\label{eq:proof-second-bernstein-1}\n \\mathbb E [ (\\widehat \\ell_{t} - \\ell_{i^*,t})^2 ]\n &\\leq \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} (\\ell_{i,t} - \\ell_{i^*,t})^2 \\bigg] \\\\\n &= \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} \\mathbb E \\left[ (\\ell_{i,t} - \\ell_{i^*,t})^2 \\,|\\, \\mathcal{F}_{t-1} \\right] \\bigg] \\nonumber \\\\\n &\\leq B \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} \\right]^\\beta \\bigg] \\label{eq:proof-second-bernstein-2} \\\\\n &\\leq B \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} \\right] \\bigg]^\\beta \\label{eq:proof-second-bernstein-3} \\\\\n &= B \\mathbb E [\\widehat \\ell_t - \\ell_{i^*,t}]^\\beta\n\\end{align}\nwhere inequalities~\\eqref{eq:proof-second-bernstein-1} and~\\eqref{eq:proof-second-bernstein-3} come from Jensen's inequality, and~\\eqref{eq:proof-second-bernstein-2} from the Bernstein condition~\\eqref{eq:bernstein-condition}.\nTaking the expectation of the regret bound~\\eqref{eq:second-order-regret}, we obtain\n\\begin{align}\n \\label{eq:proof-second-1}\n \\mathbb E [R_{i^*,T}]\n &\\leq \\mathbb E \\Bigg[ C_1 \\sqrt{(\\log M) \\sum_{t=1}^T (\\widehat \\ell_t - \\ell_{i^*,t})^2} + C_2 \\log M \\Bigg] \\nonumber \\\\\n &\\leq C_1 \\sqrt{(\\log M) \\sum_{t=1}^T \\mathbb E \\big[ (\\widehat \\ell_t - \\ell_{i^*,t})^2 \\big]} + C_2 \\log M \\\\\n &\\leq C_1 \\sqrt{(\\log M) B \\sum_{t=1}^T \\mathbb E \\big[ \\widehat \\ell_t - \\ell_{i^*,t} \\big]^\\beta} + C_2 \\log M \\nonumber \\\\\n &= C_1 \\sqrt{B T \\log M} \\bigg( \\frac{1}{T} \\sum_{t=1}^T \\mathbb E \\big[ \\widehat \\ell_t - \\ell_{i^*,t} \\big]^\\beta \\bigg)^{1\/2} + C_2 \\log M \\nonumber \\\\\n &\\leq C_1 \\sqrt{B T \\log M} \\left( \\frac{\\mathbb E [R_{i^*,T}]}{T} \\right)^{\\beta\/2} + C_2 \\log M\n \\label{eq:proof-second-ineq}\n\\end{align}\nwhere inequalities~\\eqref{eq:proof-second-1} and~\\eqref{eq:proof-second-ineq} come from Jensen's inequality.\nLetting $r = {\\mathbb E [R_{i^*,T}]}\/{T}$ and $u = ({\\log M})\/{T}$,\ninequality~\\eqref{eq:proof-second-ineq} writes $r \\leq C_1 \\sqrt{B u} r^{\\beta\/2} + C_2 u$.\nThis implies that (depending on which of these two terms is larger) either $r \\leq 2 C_2 u$, or $r \\leq 2 C_1 \\sqrt{B u} r^{\\beta\/2}$, and the latter condition amounts to $r \\leq (2 C_1)^{2\/(2-\\beta)} (B u)^{1\/(2-\\beta)}$.\nThis entails that\n\\begin{equation*}\n r \\leq (2 C_1)^{\\frac{2}{2-\\beta}} (B u)^{\\frac{1}{2-\\beta}} + 2 C_2 u\n \\, ,\n\\end{equation*}\nwhich amounts to\n\\begin{equation}\n \\label{eq:proof-second-final}\n \\mathbb E [R_{i^*,T}]\n \\leq C_3 (B \\log M)^{\\frac{1}{2-\\beta}} T^{\\frac{1-\\beta}{2-\\beta}} + C_4 \\log M\n\\end{equation}\nwhere $C_3 = (2C_1)^{2\/(2-\\beta)} \\leq \\max (1, 4C_1^2)$ and $C_4 = 2C_2$.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:hedge-no-bernstein}}\n\\label{sec:proof-lowerbound-no-bernstein}\n\nConsider the constant losses $\\ell_{1, t} = 0$, $\\ell_{i,t} = \\Delta$ where $\\Delta = {1} \\wedge c_0^{-1} \\sqrt{(\\log M)\/T}$.\nThese losses satisfy the $(1, 1)$-Bernstein condition since, for every $i>1$,\n$\\mathbb E [ (\\ell_{i,t} - \\ell_{1,t})^2 ] = \\Delta^2 \\leq \\Delta = \\mathbb E [\\ell_{i,t} - \\ell_{1,t}]$.\nOn the other hand, the regret of the Hedge algorithm with learning rate $\\eta_t = c_0 \\sqrt{(\\log M)\/t}$ writes\n\\begin{align}\n \\mathcal{R}_T\n &= \\sum_{t=1}^T \\sum_{i\\neq 1} \\mathbb E [v_{i,t} (\\ell_{i,t} - \\ell_{1,t})] \\nonumber \\\\\n &= \\Delta \\sum_{t=1}^T \\frac{(M-1) e^{-\\eta_t \\Delta (t-1)}}{1 + (M-1) e^{-\\eta_t \\Delta (t-1)}} \\nonumber \\\\\n &\\geq \\frac{\\Delta}{3} \\sum_{t=1}^T \\bm 1 \\Big( (M-1) e^{-\\eta_t \\Delta (t-1)} \\geq \\frac{1}{2} \\Big) \\nonumber \\\\\n &\\geq \\frac{\\Delta}{3} \\sum_{t=1}^T \\bm 1 \\left( M e^{- c_0 \\Delta \\sqrt{(t-1) \\log M}} \\geq 1 \\right) \\label{eq:proof-hedge-no-bernstein-0} \\\\\n &\\geq \\frac{\\Delta}{3} \\times \\min \\left( \\frac{\\log M}{c_0^2 \\Delta^2}, T \\right) \\nonumber \\\\\n &= \\frac{1}{3} \\min \\Big( \\frac{1}{c_0} \\sqrt{T \\log M}, {T} \\Big)\n \\, , \\label{eq:proof-hedge-no-bernstein-0b}\n\\end{align}\nwhere~\\eqref{eq:proof-hedge-no-bernstein-0} relies on the inequalities $2(M-1) \\geq M$ and $({t-1})\/{\\sqrt{t}} \\leq \\sqrt{t-1}$ for $M \\geq 2, t\\geq 1$, while~\\eqref{eq:proof-hedge-no-bernstein-0b}\nis obtained by noting that ${ (\\log M)}\/({c_0^2 \\Delta^2}) \\geq T$ since $\\Delta \\leq c_0^{-1} \\sqrt{(\\log M)\/T}$ and substituting for $\\Delta$.\n\n\\subsection{Proof of Theorem~\\ref{thm:hedge-characterize-gap}}\n\\label{sec:proof-hedge}\n\nAssume that the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace, and denote $i^* = \\mathop{\\mathrm{argmin}}_{1\\leq i \\leq M} \\mathbb E [\\ell_{i,t}]$ (which is assumed to be unique), $\\Delta = \\min_{i \\neq i^*} \\Delta_i > 0$ where $\\Delta_i = \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}]$ and $j \\in \\{1, \\dots, M\\}$ such that $\\Delta_j = \\Delta$.\nThe Decreasing Hedge algorithm with learning rate $\\eta_t = c_0 \\sqrt{(\\log M)\/t}$ satisfies\n\\begin{align} \n \\mathcal{R}_T\n &= \\sum_{t=1}^T \\sum_{i \\neq i^*} \\mathbb E [v_{i,t}] \\Delta_i \\nonumber \\\\\n &\\geq \\Delta \\sum_{t=1}^T \\mathbb E \\left[ \\frac{\\sum_{i \\neq i^*} e^{-\\eta_t (L_{i,t-1} - L_{i^*,t-1})}}{1 + \\sum_{i \\neq i^*} e^{-\\eta_t (L_{i,t-1} - L_{i^*,t-1})}} \\right] \\nonumber \\\\\n &\\geq \\Delta \\sum_{t=1}^T \\mathbb E \\left[ \\frac{e^{-\\eta_t (L_{j,t-1} - L_{i^*,t-1})}}{1 + e^{-\\eta_t (L_{j,t-1} - L_{i^*,t-1})}} \\right] \\label{eq:proof-no-bernstein-b1} \\\\\n &\\geq \\frac{\\Delta}{3} \\sum_{t=1}^T \\mathbb E \\left[ \\bm 1 \\left( e^{-\\eta_t (L_{j,t-1} - L_{i^*,t-1})} \\geq \\frac{1}{2} \\right) \\right] \\nonumber \\\\\n &= \\frac{\\Delta}{3} \\sum_{t=1}^T \\P \\left( \\eta_t (L_{j,t-1} - L_{i^*,t-1}) \\leq \\log 2 \\right) \\label{eq:proof-no-bernstein-b2}\n\\end{align}\nwhere~\\eqref{eq:proof-no-bernstein-b1} relies on the fact that the function $x \\mapsto \\frac{x}{1+x}$ is increasing on $\\mathbf R^+$.\nDenoting $a = (\\log 2)\/(c_0 \\sqrt{\\log M})$, we have for every $1 \\leq t \\leq 1 + \\frac{a^2}{4 \\Delta^2}$:\n\\begin{align} \n \\P \\left( \\eta_t (L_{j,t-1} - L_{i^*,t-1}) > \\log 2 \\right)\n &= \\P \\left( L_{j,t-1} - L_{i^*,t-1} - \\Delta (t-1) > a \\sqrt{t} - \\Delta (t-1) \\right) \\nonumber \\\\\n &\\leq \\P \\left( L_{j,t-1} - L_{i^*,t-1} - \\Delta (t-1) > \\frac{a \\sqrt{t-1}}{2} \\right) \\label{eq:proof-no-bernstein-aDelta} \\\\\n &\\leq e^{- a^2\/8} \\label{eq:proof-no-bernstein-hoeffding}\n\\end{align}\nwhere inequality~\\eqref{eq:proof-no-bernstein-aDelta} stems from the fact that $\\Delta (t-1) \\leq \\frac{a \\sqrt{t-1}}{2}$ (since $t \\leq 1 + \\frac{a^2}{4 \\Delta^2}$), while~\\eqref{eq:proof-no-bernstein-hoeffding} is a consequence of Hoeffding's bound applied to the i.i.d.\\@\\xspace $[-1-\\Delta, 1-\\Delta]$-valued random variables $\\ell_{j,s} - \\ell_{i^*,s} - \\Delta$, $1\\leq s \\leq t-1$.\nAssuming that $c_0 \\geq 1$, we have $a \\leq \\sqrt{\\log 2} \\leq 1$, so that by concavity of the function $x \\mapsto 1- e^{-x\/8}$, $1-e^{-a^2\/8} \\geq (1-e^{-1\/8})a^2$.\nCombining this with inequalities~\\eqref{eq:proof-no-bernstein-b2} and~\\eqref{eq:proof-no-bernstein-hoeffding} and using the fact that $\\left\\lfloor 1+\\frac{a^2}{4 \\Delta^2} \\right\\rfloor \\geq \\frac{a^2}{4 \\Delta^2}$, we obtain for $T \\geq \\frac{1}{4 \\Delta^2} \\geq \\frac{a^2}{4 \\Delta^2}$:\n\\begin{align}\n \\label{eq:proof-no-bernstein-b3}\n \\mathbb E \\left[ R_T \\right]\n \\geq \\frac{\\Delta}{3} \\min \\left( \\frac{a^2}{4 \\Delta^2}, T \\right) (1-e^{-1\/8}) a^2\n = \\frac{(1-e^{-1\/8}) a^4}{12 \\Delta}\n \\geq \\frac{1}{450 c_0^4 (\\log M)^2 \\Delta}\n \\, ,\n\\end{align}\nwhere the last inequality comes from the fact that $(\\log 2)^4 (1-e^{-1\/8})\/12 \\geq \\frac{1}{450}$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe ground-based gravitational wave (GW) detectors LIGO and Virgo have achieved great success in observing the GWs from the merger of binary neuron stars (BNS), binary black holes (BBH), and neutron star-black hole binaries (NSBH)~\\cite{LIGOScientific:2016aoc,LIGOScientific:2018mvr,LIGOScientific:2020ibl,LIGOScientific:2021usb,LIGOScientific:2021djp}. The third-generation ground-based detectors such as ET~\\footnote{\\url{http:\/\/www.et-gw.eu\/}} and CE~\\footnote{\\url{https:\/\/cosmicexplorer.org\/}} are under design and construction. Together with the space-borne LISA~\\footnote{\\url{https:\/\/www.lisamission.org\/}} and Chinese proposed projects like Taiji~\\cite{Hu:2017mde,Ruan:2018tsw} and Tianqin~\\cite{TianQin:2015yph} , they will be prepared for the detections of GWs around 2035. All of these are laser interferometers (LIs) and they will form a network of GW detectors from ground to space in the future. \n\nA novel type of GW detector called Atom interferometers (AIs) was proposed a decade ago~\\cite{Dimopoulos:2007cj,Dimopoulos:2008sv,Graham:2012sy,Hogan:2015xla}. In the concept AIs, gravitational radiation is sensed through precise measurement of the light flight time between two distantly separated atomic inertial references, each in a satellite in Medium Earth orbit (MEO). Ensembles of ultra-cold atomic Sr atoms at each location serve as precise atomic clocks. Light flight time is measured by comparing the phase of laser beams propagating between the two satellites with the phase of lasers referenced to the Sr optical transitions~\\cite{Graham:2017pmn}. Compared to the LIs, AIs consist of only a single baseline thus the design and building should be easier and cheaper than the traditional LIs. The AI projects such as ground based ZAIGA~\\cite{Zhan:2019quq} in China, AION~\\cite{Badurina:2019hst} in the UK, MIGA~\\cite{Geiger:2015tma} in France, ELGAR~\\cite{Canuel:2019abg} in Europe, and the space-borne MAGIS~\\cite{Graham:2017pmn} and AEDGE~\\cite{AEDGE:2019nxb} have been proposed and in preparation. \n\nAIs are proposed to probe not only the gravitational waves but also the dark matter. For GWs, AIs focus on the Deci-Hz gap between LIGO\/Virgo and LISA.\nIn the mid-frequency range, one can observe the long inspiral period of BNS, BBH, and NSBH. During the long observation, the motion of the space-borne detector around the Sun as well as in Earth orbit would induce large Doppler and reorientation effects, providing a precise angular resolution. Based on the space-borne AEDGE, we compose a series of papers focusing on the GW detections by AIs. \nIn the first paper~\\cite{Cai:2021ooo} (hereafter Paper I) we forecast the bright sirens detected by AEDGE and their applications on cosmology. The specific analysis on the source localization for the dark sirens was conducted in the second paper~\\cite{Yang:2021xox} (hereafter Paper II). The single baseline of AEDGE reorients on a rapid time scale compared to the observation duration. As a detector reorients and\/or moves, the observed waveform and phase are modulated and Doppler-shifted. This allows efficient determination of sky position and polarization information~\\cite{Graham:2017lmg,Graham:2017pmn}. In Paper II, we show AEDGE can even localize the dark sirens in such a small comoving volume that the unique host galaxy can be identified. These dark sirens are called ``golden dark sirens''. The measurements of Hubble constant from the simulated golden dark BNS and BBH are also performed. \n\nMany investigations suggest compact binaries that emit GWs can have non-negligible eccentricities and may contribute observational features in the sensitivity band of ground and space-based detectors~\\cite{Antonini:2012ad,Samsing:2013kua,Thompson:2010dp,East:2012xq}. \nDifferent mechanisms of the dynamic formation of the compact binaries of black holes and neutron stars have been proposed to study their eccentricities~\\cite{Rodriguez:2017pec,Samsing:2017xmd,Samsing:2017oij,Samsing:2018ykz,Wen:2002km,Pratten:2020fqn,OLeary:2008myb,Lee:2009ca}. \nOrbital eccentricity is arguably the most robust discriminator for distinguishing between isolated and dynamical BBH formation scenarios~\\cite{Zevin:2021rtf}. Some studies indicate that a fraction of the binaries possess eccentricities larger than 0.1 at 10 Hz~\\cite{Wen:2002km,Silsbee:2016djf,Antonini:2017ash,Liu:2019gdc}. The source localization improvement by the eccentricity for the ground-based detector networks has been investigated in some detail in~\\cite{Sun:2015bva,Ma:2017bux,Pan:2019anf}. They found that the eccentricity has more distinct effects on localization for higher-mass binaries than for the lower ones. For the case of $100~M_\\odot$ BBH, the improvement factor is about 2 in general when the eccentricity changes from 0.0 to 0.4~\\cite{Pan:2019anf}. Such an improvement is not adequate to considerably shrink the uncertainty of host galaxies (redshift) of dark sirens in the LIGO\/Virgo band, and to provide a conclusive measurement of the expansion of the Universe (e.g. the Hubble constant). While as shown in~\\cite{Yang:2022tig}, the eccentricity can significantly improve the distance estimation and source localization in the mid-band. The multiple harmonics induced by eccentricity can break the degeneracy between parameters in the waveform. In addition, the higher modes can enter the detector band much earlier than the dominant mode, which can provide more angular information. At some specific orientations (inclination angles), the typical compact binaries can achieve $\\mathcal{O}(10^2-10^4)$ improvement for the distance inference and $1.5\\sim{3.5}$ orders of magnitude improvement for the sky localization. Such a huge improvement on the 3-D localization could dramatically shrink the uncertainty of the host galaxies of the dark sirens. Up to now, only GW190521 has been reported to be eccentric in the latest GW catalog GWTC-3~\\cite{Romero-Shaw:2020thy,Gayathri:2020coq}. Considering the fact that the nonvanishing eccentricity is more likely to exist at lower frequency, we expect the dark sirens observed by the mid-band detector have greater potential on probing the cosmic expansion history, dynamics of dark energy, and gravity theory. \n\nIn this paper, we extend our research in Paper II and take the eccentricity effects into account for the dark sirens with AEDGE. In Sec.~\\ref{sec:typical}, we follow the methodology of~\\cite{Yang:2022tig} to check the improvement of distance estimation and source localization by eccentricity for the typical BNS, NSBH, and BBH with AEDGE. In Sec.~\\ref{sec:mock}, we adopt the similar method in Paper II to construct the catalogs of GWs for AEDGE. We first update the construction of catalogs of dark sirens in Paper II in the case of vanishing eccentricity. Comparing to Paper II, we update the waveform and the merger rates of BNS and BBH. We also include the NSBH into the simulation. We refine the fisher matrix calculation to ensure the convergence of the numerical derivatives. These updates and improvements make the simulation more realistic and reliable than that in Paper II. We then take account of eccentricity effects to construct the catalogs of dark sirens. We show how many potential host galaxies would be within the 3-D localization GW sources, with or without eccentricity. We estimate the the population of eccentric dark sirens and pick out the ones whose host galaxies can be best identified. We randomly select the golden dark sirens which AEDGE can track considering the limit of its operational time. The corresponding measurements of the Hubble constant are obtained in Sec.~\\ref{sec:Hubble}. We give the conclusions and discussions in Sec.~\\ref{sec:conclusion}.\n\n\n\n\n\\section{The improvement of distance estimation and localization from eccentricity \\label{sec:typical}}\nBy adopting a similar strategy in~\\cite{Yang:2022tig}, we mock up five types of typical compact binaries in GWTC-3~\\cite{LIGOScientific:2021djp} with component mass ranging from $\\mathcal{O}(1\\sim100)~M_{\\odot}$, i.e., a GW170817-like BNS with $(m_1,m_2)=(1.46,1.27)~M_{\\odot}$, a GW200105-like NSBH with $(9.0,1.91)~M_{\\odot}$, a GW191129-like light-mass BBH with $(10.7,6.7)~M_{\\odot}$, a GW150914-like medium-mass BBH with $(35.6,30.6)~M_{\\odot}$, and a GW190426-like heavy-mass BBH with $(106.9,76.6)~M_{\\odot}$. Note the light, medium, and heavy mass are in the context of the stellar-mass binaries in GWTC-3. The redshifts (distances) are also consistent with the real events in the catalog. We sample 1000 random sets of the angular parameters from the uniform and isotropic distribution for each typical binary and assign six discrete initial eccentricities $e_0=0$, 0.01, 0.05, 0.1, 0.2, and 0.4 at $f_0=0.1$ Hz. Then we have $5\\times 6\\times 1000=3\\times10^4$ cases. For each case, we perform the fisher matrix calculation to infer the errors of distance and sky location. \n\nWe use {\\sc PyCBC}~\\cite{alex_nitz_2021_5347736} to generate the waveform with the non-spinning, inspiral-only EccentricFD waveform approximant available in {\\sc LALSuite}~\\cite{lalsuite}. EccentricFD corresponds to the enhanced post-circular (EPC) model in~\\cite{Huerta:2014eca}. \nTo the zeroth order in the eccentricity, the model recovers the TaylorF2 PN waveform at 3.5 PN order~\\cite{Buonanno:2009zt}. To the zeroth PN order, the model recovers the PC expansion of~\\cite{Yunes:2009yz}, including eccentricity corrections up to order $\\mathcal{O}(e^8)$.\nThe strain can be written as~\\cite{Huerta:2014eca}\n\\begin{equation}\n\\tilde{h}(f)=-\\sqrt{\\frac{5}{384}}\\frac{\\mathcal{M}_c^{5\/6}}{\\pi^{2\/3}d_L}f^{-7\/6}\\sum_{\\ell=1}^{10}\\xi_{\\ell}\\left(\\frac{\\ell}{2}\\right)^{2\/3}e^{-i\\Psi_{\\ell}} \\,.\n\\label{eq:epc}\n\\end{equation} \nThe waveform keeps up to 10 harmonics, which corresponds to a consistent expansion in the eccentricity to $\\mathcal{O}(e^8)$ both in the amplitude and in the phase~\\cite{Yunes:2009yz}. In the vanishing eccentricity case, only the dominant (quadrupole) mode $\\ell=2$ remains, which is identical to the circular TaylorF2 model. With nonvanishing eccentricities, the induced multiple harmonics make the distance and angular parameters nontrivially coupled, enabling us to break the degeneracy among these parameters. In addition, the frequency of each harmonics is $\\ell F$ with $F$ the orbital frequency. Thus the higher harmonics ($\\ell>2$) should enter the detector band much earlier than the dominant mode ($\\ell=2$), which can provide more angular information. The $\\xi_{\\ell}$'s depend on the antenna pattern functions (also called detector response functions) $F_{+,\\times}$. For the space-borne AEDGE, we should consider the motion of the detector thus $F_{+,\\times}$ are functions of time. We give the detailed calculation of the antenna pattern functions in appendix~\\ref{app:F}. \n\nWe have 11 parameters in the waveform, namely the chirp mass $\\mathcal{M}_c$, the symmetric mass ratio $\\eta$, the luminosity distance $d_L$, the inclination angle $\\iota$, the sky location ($\\theta$, $\\phi$), the polarization $\\psi$, the time and phase at coalescence ($t_c$, $\\phi_c$), the initial eccentricity $e_0$ at frequency $f_0$, the azimuthal component of inclination angles (longitude of ascending nodes axis) $\\beta$. To estimate the uncertainty and covariance of the waveform parameters, we adopt the Fisher matrix technique \n\\begin{equation}\n\\Gamma_{ij}=\\left(\\frac{\\partial h}{\\partial P_i},\\frac{\\partial h}{\\partial P_j}\\right)\\,,\n\\end{equation}\nwith $P_i$ one of the 11 waveform parameters.\nThe inner product is defined as\n\\begin{equation}\n(a,b)=4\\int_{f_{\\rm min}}^{f_{\\rm max}}\\frac{\\tilde{a}^*(f)\\tilde{b}(f)+\\tilde{b}^*(f)\\tilde{a}(f)}{2 S_n(f)}df\\,.\n\\label{eq:innerp}\n\\end{equation}\nFor the noise power spectral density (PSD) $S_n(f)$, we adopt the sensitivity curve of AEDGE in the resonant modes (see the envelope in figure 1 of~\\cite{Ellis:2020lxl}).\nThen the covariance matrix of the parameters is $C_{ij}=(\\Gamma^{-1})_{ij}$, from which the uncertainty of each parameter $\\Delta P_i=\\sqrt{C_{ii}}$. The error of the sky localization is~\\cite{Cutler:1997ta}\n\\begin{equation}\n\\Delta \\Omega=2\\pi |\\sin(\\theta)|\\sqrt{C_{\\theta\\theta}C_{\\phi\\phi}-C_{\\theta\\phi}^2}\\,.\n\\end{equation}\nWe calculate the partial derivatives $\\partial \\tilde{h}\/\\partial P_i$ numerically by $[\\tilde{h}(f,P_i+dP_i)-\\tilde{h}(f,P_i)]\/dP_i$, with $dP_i=10^{-n}$. For each parameter, we need to optimize $n$ to make the derivative converged so that the Fisher matrix calculation is reliable. \n\nFor each typical event, the chirp mass $\\mathcal{M}_c$, symmetric mass ratio $\\eta$, and distance $d_L$ are calculated from the component mass and redshift. The angular parameters $P_{\\rm ang}=\\{\\iota,~\\theta,~\\phi,~\\psi,~\\beta\\}$ are sampled from the uniform and isotropic distribution with 1000 sets for each typical event. We use the inclination angle $\\iota$ to represent to angular parameter since we find it is more relevant in terms of the results. Without loss of generality, we fix the coalescence time and phase to be $t_c=\\phi_c=0$. We choose the frequency band of AEDGE to be [0.1, 3] Hz, where the detector is the most sensitive. This range corresponds to lower and upper bounds of frequency in the integral of Eq.~(\\ref{eq:innerp})~\\footnote{This is also different with Paper II in which we naively set the lower bound of frequency to be 0.2 for BNS and 0.05 for BBH.}. However, we should consider the limited operation time of AEDGE for tracking the GWs. We set quadrupole ($\\ell=2$) as the reference mode and its frequency is double of the orbital's, $f_{\\ell=2}=2F$. Then the evolution of the binary orbit can be calculated in terms of the quadrupole frequency. To ensure the observational time of AEDGE for all harmonics is around 400 days ($\\sim1$ year), we set the starting frequency of quadrupole $f_{\\rm start}(\\ell=2)$ to be 0.2, 0.1, 0.059, 0.026, and 0.0105 Hz for the typical BNS, NSBH, light BBH, medium BBH, and heavy BBH, respectively. That is, for the strain Eq.~(\\ref{eq:epc}) we should neglect the contribution of all the harmonics when the quadrupole's frequency is smaller than $f_{\\rm start}(\\ell=2)$~\\footnote{Note in the mid band we can only observe the inspiral phase of these binaries, thus we do not need to care about the upper frequency limit at the innermost-stable circular orbit.}. Thus we multiply the strain by the step function\n\\begin{equation}\n\\tilde{h}_{\\rm AEDGE}(f)=\\tilde{h}(f)\\mathcal{H}(2f-\\ell f_{\\rm start}) \\,,\n\\label{eq:hAEDGE}\n\\end{equation}\nwith the unit step function\n\\begin{equation}\n\\mathcal{H}(x)=\n\\begin{cases}\n1 & {\\rm if}~x\\geq0 \\,, \\\\\n0 & {\\rm otherwise} \\,.\n\\end{cases}\n\\end{equation}\nFor the orbital phase evolution, we numerically solve Eqs. (3.11) and (4.24) in~\\cite{Yunes:2009yz} to obtain the time to coalescence $t(f)$ for a nonvanishing $e_0$. The time to coalescence at a specific frequency is smaller for a larger eccentricity. So for the fixed $f_{\\rm start}(\\ell=2)$, the observational time is shorter with a larger eccentricity.\n\nWe collect all the results of the fisher matrix for the $3\\times10^4$ cases. Same as~\\cite{Yang:2022tig}, for each typical event with a specific orientation, we define the ratios\n\\begin{equation}\nR_{\\Delta d_L}=\\frac{\\Delta d_L|_{e_0={\\rm nonzero}}}{\\Delta d_L|_{e_0=0}}~{\\rm and}~R_{\\Delta \\Omega}=\\frac{\\Delta \\Omega |_{e_0={\\rm nonzero}}}{\\Delta\\Omega|_{e_0=0}} \\,,\n\\end{equation} \nto show the improvement induced by eccentricity in that orientation. If $R<1$, there is an improvement in the relevant parameter. A smaller $R$ indicates a larger improvement. We show the scatter plots of $\\Delta d_L\/d_L$, $R_{\\Delta d_L}$, $\\Delta \\Omega$, and $R_{\\Delta \\Omega}$ against $\\iota$. To give the statistical results, we define the minimum, mean, and maximum value of $x$ in the 1000 orientations as $\\min(x)$, $\\mathbb{E}(x)$, and $\\max(x)$, respectively.\n\nIn figure~\\ref{fig:rep}, we only show the distance inference of GW170817-like BNS and source localization of GW190426-like heavy BBH to represent\nour main results. We just compare the cases with $e_0=$0, 0.1, and 0.4 to give a concise look. The complete results can be found in appendix~\\ref{app:sup}. As shown in left panel of figure~\\ref{fig:rep}, a nonvanishing eccentricity can significantly improve the distance inference in the near face-on orientations (small inclination angle). Among all 1000 orientations, the $\\max(\\Delta d_L\/d_L)$ of GW170817-like BNS is reduced from 27.74 ($e_0=0$) to $0.82$ ($e_0=0.1$) and $0.35$ ($e_0=0.4$). Comparing to $e_0=0$ case, the largest improvement ($\\min(R_{\\Delta d_L})$) corresponds to 47 and 115 times stricter with $e_0=0.1$ and $e_0=0.4$, respectively. The huge improvement of distance inference in the near face-on orientations is true for all the typical events. The binaries with larger component mass and eccentricity can achieve more improvement. As shown in appendix~\\ref{app:sup}, for the heavy BBH with $e_0=0.4$, $\\min(R_{\\Delta d_L})=0.0012$, corresponding to 833 times improvement. We also find that for the heavy BBH case, there is an overall improvement of distance inference in all orientations. Our results indicate that the eccentricity effects are more distinct for the larger mass compact binaries. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{ddL_BNS170817}\n\\includegraphics[width=0.49\\textwidth]{dOmega_BBHheavy} \n\\caption{The distance inference of GW170817-like BNS (left) and source localization of GW190426-like heavy BBH.}\n\\label{fig:rep}\n\\end{figure}\n\nFor the source localization, we find eccentricity can lead to significant improvement for the BBH cases which have larger component mass than BNS and NSBH cases. As shown in the right panel of figure~\\ref{fig:rep}, the localization of heavy BBH is significantly improved by the eccentricities in almost all orientations. The largest improvement $\\min(R_{\\Delta \\Omega})= 8.16\\times 10^{-4}$, corresponding to $1.23\\times10^3$ times tighter. Like the distance inference, the heavier binaries benefits more from the eccentricity for the source localization. \nThe details of the improvement by eccentricity can also be found in the figures summarized in appendix~\\ref{app:sup}.\n\nTo illustrate the improvement of distance inference and localization for these typical binaries with variable eccentricities, we show the largest improvement ($\\min(R)$ in 1000 orientations) of each case in Fig.~\\ref{fig:Rwe}. We can see generally a heavier binary with higher eccentricity can achieve more improvement of distance inference and source localization. With eccentricity $e_0=0.4$, these typical binaries can most achieve \n1.5--3 orders of magnitude\nimprovement for the distance inference (from BNS to heavy BBH). As for the source localization, BNS and NSBH can not benefit much from the eccentricity. While BBHs can most achieve 1.5--3 orders of magnitude improvement (from light BBH to heavy BBH). We should note some anomalies in figure~\\ref{fig:Rwe}. 1) For the distance inference, the typical BNS benefits more from eccentricities than the typical NSBH and light BBH do. 2) For the source localization, BNS behaves similarly with NSBH and both have almost no improvement from eccentricity. 3) The light BBH's tendency is very close to that of medium BBH when $e_0<0.2$ and then they diverge for larger eccentricity. 4) In the BNS, NSBH, and especially for the light BBH cases, the localization achieves largest improvement when $e_0=0.2$, a higher eccentricity ($e_0=0.4$) can even worsen the performance. These anomalies are caused by many factors. On the one hand, eccentricity adds more harmonics in GWs. These harmonics can enlarge the SNR and improve the parameter estimation. The higher modes which enter the detector band much earlier can provide more angular information. On the other hand, eccentricity shrinks the inspiral time within the frequency band, which could lower the SNR and hence worsen the parameter estimation and localization. In addition, due to the different starting frequencies, for each binaries the detector band cover different length of harmonics. For instance, in BNS case, at the starting frequency $f_{\\rm start}(\\ell=2)=0.2$, all harmonics including $\\ell=1$ falls inside the detector band (0.1--3Hz). But in NSBH case, at $f_{\\rm start}(\\ell=2)=0.1$, the $\\ell=1$ mode's frequency is 0.05, falling outside the detector band hence should be truncated. Moreover, we have two more parameters $e_0$ and $\\beta$ in the eccentric waveform, which could degrade the overall performance of the parameter estimation. All above factors compete with each other and make the parameter estimation (distance inference and localization) differ from case to case. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Rwe}\n\\caption{The largest improvement of distance inference (upper panel) and source localization (lower panel) for the typical compact binaries in 1000 orientations with variable eccentricities.}\n\\label{fig:Rwe}\n\\end{figure}\n\nHere we would like to provide some more explanations for the inverse tendency of the error of the distance and localization versus the inclination angle. We can see in figure~\\ref{fig:rep} that generally the the error of distance is larger in smaller orbital inclination. On the contrary, the source localization is better when inclination is smaller. For distance it is due to the degeneracy between distance and inclination angle. In the amplitude of GW waveform~$h\\sim \\mathcal{A}_++\\mathcal{A}_\\times$, the distance $d_L$ and inclination angle $\\iota$ are tangled in the plus and cross polarization with different form, $\\mathcal{A}_+\\sim\\frac{1}{d_L}\\frac{1+\\cos(\\iota)}{2}$ and $\\mathcal{A}_+\\sim\\frac{1}{d_L}\\cos(\\iota)$. In order to identify the inclination of the binary system using the polarizations of the gravitational wave, we must distinguish the contributions of the plus and cross polarizations. At small $\\iota$, the two amplitudes from plus and cross polarizations have nearly identical contributions to the overall gravitational-wave amplitude. This is the main factor that leads to the strong degeneracy in the measurement of the distance and inclination~\\cite{Usman:2018imj}. So we expect a larger degeneracy between $d_L$ and $\\iota$ in the near face-on orientations and hence larger errors for both distance and inclination angle. As for the source localization, there is no obvious degeneracy between the sky location parameters $(\\theta,\\phi)$ and inclination angle $\\iota$. However, at smaller $\\iota$ the SNR is larger. So the parameter estimation should be better than that at larger $\\iota$. \n\nWe have showed that eccentricity, which is more likely to exist in the mid-band that in LIGO\/Virgo band, can improve the distance inference and source localization of dark sirens with AEDGE significantly. Note GWs are best localized at smallest orbital inclination where the distance are worst determined. But eccentricity happens to improve the distance inference most significantly there. In addition, one of the main targets for the mid-band detector like AEDGE is the intermediate mass black holes (IMBH). While in this paper we showed that the heaviest BBH can benefit most from the eccentricity for the distance inference and source localization. These facts suggests that eccentricity is the perfect ingredient for AEDGE dark sirens as precise probes of the Universe \n\n\\section{The construction of dark sirens catalogs and the host galaxy identification \\label{sec:mock}}\nConsidering the fact that eccentricity plays an important role in the distance inference and source localization of dark sirens with the mid-band detector AEDGE, we should take eccentricity effects into the construction of the dark sirens catalogs. In this section, we first update the construction of the catalogs of dark sirens in Paper II which does not consider the eccentricity effects, i.e., $e_0=0$. We adopt the EccentricFD waveform in which the $e_0=0$ case is equivalent to TaylorF2 at 3.5 PN order. While, in paper II we only expand the waveform to 2 PN order in the phase. We also update the BNS and BBH merger rates from the latest GWTC-3, as well as the BBH population. In addition to BNS and BBH, we add NSBH catalog. More importantly, we refine the numerical derivatives for the fisher matrix calculation, which would make the results more stable and thus robust and reliable. Then we include the eccentricity effects into the construction of the catalogs, to assess the its influence on the population and localization of the binaries.\n\nWe follow Paper II and assume the formation of compact binaries tracks the star formation rate.\nThe merge rate per comoving volume at a specific redshift $R_m(z_m)$ is related to the formation rate of massive binaries and the time delay distribution $P(t_d,\\tau)=\\frac{1}{\\tau}\\exp(-t_d\/\\tau)$ with an e-fold time of $\\tau=100$ Myr~\\cite{Vitale:2018yhm},\n\\begin{equation}\nR_m(z_m)=\\int_{z_m}^{\\infty}dz_f\\frac{dt_f}{dz_f}R_f(z_f)P(t_d) \\,.\n\\label{eq:Rm}\n\\end{equation}\nHere $t_m$ (or the corresponding redshift $z_m$) and $t_f$ are the look-back time when the systems merged and formed. $t_d=t_f-t_m$ is the time delay. $R_f$ is the formation rate of massive binaries and we assume it is proportional to the Madau-Dickinson (MD) star formation rate~\\cite{Madau:2014bja},\n\\begin{equation}\n\\psi_{\\rm MD}=\\psi_0\\frac{(1+z)^{\\alpha}}{1+[(1+z)\/C]^{\\beta}} \\,,\n\\label{eq:psiMD}\n\\end{equation}\nwith parameters $\\alpha=2.7$, $\\beta=5.6$ and $C=2.9$. The normalization factor $\\psi_0$ is determined by the local merger rates. We adopt the local merger rates of BNS, NSBH, and BBH inferred from GWTC-3, with $\\mathcal{R}_{\\rm BNS}=105.5^{+190.2}_{-83.9}~\\rm Gpc^{-3}~\\rm yr^{-1}$, $\\mathcal{R}_{\\rm NSBH}=45^{+75}_{-33}~\\rm Gpc^{-3}~\\rm yr^{-1}$, and $\\mathcal{R}_{\\rm BBH}=23.9^{+14.3}_{-8.6}~\\rm Gpc^{-3}~\\rm yr^{-1}$~\\cite{LIGOScientific:2021psn}. Note we assume the observed NSBH GW200105 and GW200115 are representatives of the population of NSBH. Then we convert the merger rate per comoving volume in the source frame to merger rate density per unit redshift in the observer frame\n\\begin{equation}\nR_z(z)=\\frac{R_m(z)}{1+z}\\frac{dV(z)}{dz} \\,,\n\\label{eq:Rz}\n\\end{equation}\nwhere $dV\/dz$ is the comoving volume element. \n\nHaving the merger rates as redshift, we can sample the redshift distribution of BNS, NSBH, and BBH. Like Paper II, we use the median merger rates to construct the catalogs. We have 11 parameters in the waveform (for vanishing eccentricity there are 9 except $e_0$ and $\\beta$). The luminosity distance $d_L$ is calculated from the sampled redshift by assuming a fiducial cosmological model $\\Lambda$CDM with $H_0=67.72~\\rm km~s^{-1}~Mpc^{-1}$ and $\\Omega_m=0.3104$, corresponding to the mean values obtained from the latest \\textit{Planck} experiment~\\cite{Planck:2018vyg}. The sky localization ($\\theta$, $\\phi$), inclination angle $\\iota$, and polarization $\\psi$ are drawn from isotropic distribution. Without loss of generality we set the time and phase at coalescence to be $t_c=\\phi_c=0$. As for the chirp mass and symmetric mass ration, we consider different strategy for these three binary types. In the BNS case, we assume a uniform distribution of mass in [1, 2.5] $M_{\\odot}$, which is consistent with the assumption for the prediction of the BNS merger rate in GWTC-3~\\cite{LIGOScientific:2021psn}. In the NSBH case, since the merger rate is inferred by assuming the observed NSBH GW200105 and GW200115 are representatives of the population of NSBH, we just randomly choose the component mass of these two events. As for the BBH case, we adopt the same strategy in Paper II with BBH population in GWTC-3. We draw the distribution of component mass of BBH from the histogram of mass distribution of BBH in GWTC-3~\\footnote{We first infer the histograms of primary mass $m_1$ and mass ratio $q$ from GWTC-3. The distribution of $m_1$ and $q$ are sampled accordingly. Then the second mass is just $m_2=m_1q$. We should make sure that $m_2\\ge3~M_{\\odot}$.}. The primary mass and mass ratio peak around 30--40 $M_{\\odot}$ and 0.7. \n\nWe sample the mergers of BNS, NSBH, and BBH in 5 years since the operation time of AEDGE is supposed to be 5--10 years~\\cite{AEDGE:2019nxb}. We set the frequency band and starting frequency to be same as in section~\\ref{sec:typical}. This means the observational time for each event is around 1 year. For each sampled merger, we assume four discrete eccentricities, i.e., $e_0=0$, 01, 0.2, and 0.4 at $f_0=0.1$ Hz. We select the mergers with SNR>8 as the candidate events that could be detected (within the detection range) by AEDGE in 5 years. For each events, we adopt the fisher matrix to derive their distance errors and source localizations. By assigning a uniform eccentricity for each event, we would like to assess the influence of eccentricity on the population and localization of the GWs that could be detected by AEDGE. We will give a discussion about the distribution of eccentricity and the realistic population of eccentric binaries later.\n\nFigure~\\ref{fig:hist} shows the cumulative histogram of events within the detection range of AEDGE in 5 years. The highest redshift AEDGE can reach for BNS and NSBH are around 0.13 and 0.45, respectively. For BBH, the horizon is much larger but we set a cut-off at $z=2$ since the for higher redshift we usually can not obtain the spectroscopic measurement of the redshift. In addition, the large uncertainty of localization makes the BBH at high redshift useless for our purpose in this paper. In the circular case, the total numbers are 106, 1105, and 95369 for BNS, NSBH, and BBH ($z\\leq2$), respectively. The numbers of BNS and BBH are smaller than that in Paper II, which is due to the different choice of the merger rates and the lower limit of frequency band of AEDGE (we adopt $f_{\\rm min}=0.05$ Hz for BBH in Paper II, while in this paper $f_{\\rm min}=0.1$ Hz). We note that a larger eccentricity leads a smaller population of the events. This is due to the fact that eccentricity reduces the inspiral (orbital evolution) time of binaries in the frequency band (0.1--3 Hz). The smaller observational time leads smaller accumulation of SNR, especially for the dominant quadrupole mode. So the GWs whose SNR are just a little above the detection threshold when $e_0=0$ may not be detected if they have nonvanshing eccentricities. In the NSBH case, the largest redshift AEDGE can reach is smaller for eccentric events. Comparing to BNS and BBH, the population of NSBH decrease the most with eccentricity. The reason is that we choose GW200105 and GW200115 as the representatives of NSBH population. The component masses in the NSBH catalog are fixed to be the same as either of these two typical events. So, the high-redshift eccentric events are definitely to be below the SNR threshold. While for BNS and BBH, there may be a larger sampled component mass to compensate for the low SNR. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{hist}\n\\caption{The cumulative histogram of events which are within the detection range of AEDGE in 5 years. Note we set a cut-off at $z=2$ for BBH.}\n\\label{fig:hist}\n\\end{figure}\n\nFigures~\\ref{fig:err_dL} and~\\ref{fig:err_Omega} show the error of distance and localization of the binaries that are within the detection range of AEDGE in 5 years. Eccentricity can significantly improve the overall distance inference of the binaries in the catalogs. For the source localization, BNS and NSBH can not benefit obviously from eccentricity. The localizations of eccentric events are even worsen in some cases. However, the source localization of BNS and NSBH are $\\mathcal{O}(10^{-4})~\\rm deg^{2}$ level even without eccentricity. While BBH's localization is considerably improved by eccentricity. The optimal localization at low redshift is improved to be better than $\\mathcal{O}(10^{-3})~\\rm deg^{2}$. We find that, in some cases, with $e_0=0.2$ the binaries can achieve the most improvements. All of these features can be expected based on the results in section~\\ref{sec:typical}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{err_dL}\n\\caption{The distance error of the events which are within the detection range of AEDGE in 5 years.}\n\\label{fig:err_dL}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{err_Omega}\n\\caption{The localization of the events which are within the detection range of AEDGE in 5 years.}\n\\label{fig:err_Omega}\n\\end{figure}\n\nTo assess the galaxy identification of the binaries in the catalogs, we should calculate their 3-D localization volumes which can be obtained from the errors of distance and localization in figures~\\ref{fig:err_dL} and~\\ref{fig:err_Omega}. We follow the method in~\\cite{Yu:2020vyy} to convert $\\Delta d_L$ and $\\Delta\\Omega$ to the 99\\% confidence ellipsoid of the localization. We use $V_{\\rm loc}$ to denote the 3-D volume of the localization. To estimate the numbers of potential host galaxies in the localization volume, we assume the galaxy is uniformly distributed in the comoving volume and the number density $n_g= 0.01~\\rm Mpc^{-3}$. This number is derived by taking the Schechter function parameters in B-band $\\phi_*=1.6\\times 10^{-2} h^3 {\\rm Mpc^{-3}}, \\alpha=-1.07, L_*=1.2\\times 10^{10} h^{-2} L_{B,\\odot}$ and $h=0.7$, integrating down to 0.12 $L_*$ and comprising 86\\% of the total luminosity~\\cite{Chen:2016tys}. Then the threshold localization volume is $V_{\\rm th}=100~\\rm Mpc^3$. If $V_{\\rm loc}\\leq V_{\\rm th}$, the host galaxy of the dark sirens can be identified uniquely and we call these golden dark sirens.\n\nFigure~\\ref{fig:V_loc} shows the the 99\\% confidence level (C.L.) of the 3-D localization of the events that are within the detection range of AEDGE in 5 years. We can see in the circular case, several BNS and NSBH events at low redshift can be localized within $V_{\\rm th}$. As for BBH, a few events can be localized with only one potential host galaxy. However, through the improvement from eccentricity, the eccentric BBH at low redshift can be well localized to become the golden dark sirens. The number of the golden dark sirens in the catalogs are summarized in table~\\ref{tab:np}. We also show the number of dark sirens whose potential host galaxies' count $n_p$ are less than 10. The result shows BBH can benefit the most from the eccentricity. In the circular case, it is almost impossible to detect the golden dark BBH. While nonvanishing eccentricities significantly increase the possibility to detect the golden BBH at low redshift. Note in the NSBH case, eccentricity would worsen the results compared to the circular case. The $e_0=0.2$ case gives an overall better result than other cases. All of these results are consistent with the expectation in section~\\ref{sec:typical}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{V_loc}\n\\caption{The 3-D localization volumes of the events which are within the detection range of AEDGE in 5 years. The horizontal dashed line corresponds to the threshold volume that the unique host galaxy can be identified.}\n\\label{fig:V_loc}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \n\\hline\n & \\multicolumn{2}{c|}{$e_0=0$} & \\multicolumn{2}{c|}{$e_0=0.1$} & \\multicolumn{2}{c|}{$e_0=0.2$} & \\multicolumn{2}{c|}{$e_0=0.4$} \\\\ \n\\hline\nBinary type & Golden~ & $1