本文研究了A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex，Math.Comp.75（2006），1449- 1466].对于给定的半有理多胞形P和有理子空间L，我们对给定的多项式函数h在平行于子空间L的多胞形P的所有格片上积分，并求和。我们首先发展了一个算法理论的参数中间生成函数。然后，我们研究了这些中间和的Ehrhart理论，即，结果的依赖性作为一个函数的膨胀的多面体。我们提供了一个算法来计算所得到的Ehrhart拟多项式的形式显式的步骤多项式。这些公式对于实膨胀（而不仅仅是整数膨胀）自然有效，因此提供了一个直接的方法来研究实埃赫哈特理论。
英文摘要：
We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.
Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra
本文研究凸多胞体内的格点计数的计算问题，当每一点都有一个权值时，我们给出了一个计算变维数有理简单多面体的加权Ehrhart拟多项式的最高次系数的有效算法，当格点的权值由一个多项式函数h给出时.我们的技术是基于A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex，Math.Comp.75（2006），pp. 1449- 1466]在未加权情况下（即，h = 1）。与Barvinok的方法相比，我们的方法是局部的，在生成函数的水平上获得近似，处理一般加权的情况下，并提供封闭形式的系数作为阶跃多项式的扩张。为了证明我们的方法的实用性，我们报告的计算实验表明，即使我们简单的实现可以与最先进的软件竞争。
In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n…
Intermediate Sums on Polyhedra II: Bidegree and Poisson Formula
本文继续研究了A。Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex，Math.Comp.75（2006），1449-1466]。通过众所周知的分解，考虑仿射锥s+c的情况就足够了，其中s是任意实顶点，c是有理多面体锥。对于给定的有理子空间L，我们对给定的多项式函数h在仿射锥s + c的所有平行于子空间L的格片上积分，并求和积分。我们研究了这些中间和的中间生成函数$S^L(s+c)(ξ)$，并揭示了二度结构的参数s和$ξ$，这是隐含地使用在算法中，我们的文件[计算的最高系数加权Ehrhart拟多项式的有理多面体，发现。12（2012），435-469]和[Intermediate sums on polyhedra：Computation and real Ehrhart theory，Mathematika 59（2013），1-22]。二度结构是关键的一个新的证明Baldoni-Berline-Vergne逼近定理离散生成函数[局部欧拉-麦克劳林展开的Barvinok估值和Ehrhart系数的合理多面体，Contemp。452（2008），15-33]，使用关于参数s和连续性论证的傅里叶分析。我们的研究还支持即将发表的论文，其中我们研究了多参数多面体族的中间和。
Let $V$ be a real vector space of dimension $n$ and let $M\subset V$ be a lattice. Let $P\subset V$ be an $n$-dimensional polytope with vertices in $M$, and let $φ\colon V\rightarrow \CC $ be a homogeneous polynomial function of degree $d$ (i.e., an element of $\Sym^{d} (V^{*})$). For $q\in \ZZ_{>0}$ and any face $F$ of $P$, let $D_{φ,F} (q)$ be the sum of $φ$ over the lattice points in the dilate $qF$. We define a generating function $G_φ(q,y) \in \QQ [q] [y]$ packaging together the various $D_{φ,F} (q)$, and show that it satisfies a functional equation that simultaneously generalizes Ehrhart…
A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes that the $h^*$-vector of a spanning lattice polytope has no gaps, i. e., $h^*_i =0$ implies $h^*_{i+1}=0$. This generalizes a recent result by Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud-Goto conjecture. We also discuss how this relates to unimodality questions of lattice polytopes and previously achieved decomposition results on lattice polytopes of given degree.
We prove that the problem of minimizing the number of integer points inparallel translations of a rational convex polytope in $\mathbb{R}^6$ is NP-hard. We apply this result to show that given a rational convex polytope $P \subset \mathbb{R}^6$, finding the largest integer $t$ s.t. the expansion $tP$ contains fewer than $k$ integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations.
We characterize all signed Minkowski sums that define generalized permutahedra, extending results of Ardila-Benedetti-Doker (2010). We use this characterization to give a complete classification of all positive, translation-invariant, symmetric Minkowski linear functionals on generalized permutahedra. We show that they form a simplicial cone and explicitly describe their generators. We apply our results to prove that the linear coefficients of Ehrhart polynomials of generalized permutahedra, which include matroid polytopes, are non-negative, verifying conjectures of De Loera-Haws-Koeppe (2009)…
We show how to compute the Ehrhart polynomial of the free sum of two lattice polytopes containing the origin $P$ and $Q$ in terms of the enumerative combinatorics of $P$ and $Q$. This generalizes work of Beck, Jayawant, McAllister, and Braun, and follows from the observation that the weighted $h^*$-polynomial is multiplicative with respect to the free sum. We deduce that given a lattice polytope $P$ containing the origin, the problem of computing the number of lattice points in all rational dilates of $P$ is equivalent to the problem of computing the number of lattice points in all integer dil…
Let $P(b)\subset R^d$ be a semi-rational parametric polytope, where $b=(b_j)\in R^N$ is a real multi-parameter. We study intermediate sums of polynomial functions $h(x)$ on $P(b)$, $$ S^L (P(b),h)=\sum_{y}\int_{P(b)\cap (y+L)} h(x) \mathrm dx, $$ where we integrate over the intersections of $P(b)$ with the subspaces parallel to a fixed rational subspace $L$ through all lattice points, and sum the integrals. The purely discrete sum is of course a particular case ($L=0$), so $S^0(P(b), 1)$ counts the integer points in the parametric polytopes. The chambers are the open conical subsets of $R^N$ s…
Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation
School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）；Center for Combinatorics, LPMC, Nankai University（南开大学组合数学中心）
本文提出一种利用常数项和有限域精确计算 Birkhoff 多胞形 Ehrhart 级数的方法，通过根过滤器、对称函数和广义 Todd 系数实现高效计算，并推广到 World Cup 问题，给出直至阶数 12 的完整级数。
For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family are the Gaussian polytopes defined as convex hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $\mathbb R^d$. The second family are convex hulls of random walks with exchangeable increments satisfying certain mild general position assumptio…
Let a polyhedron $P$ be defined by one of the following ways: (i) $P = \{x \in R^n \colon A x \leq b\}$, where $A \in Z^{(n+k) \times n}$, $b \in Z^{(n+k)}$ and $rank\, A = n$; (ii) $P = \{x \in R_+^n \colon A x = b\}$, where $A \in Z^{k \times n}$, $b \in Z^{k}$ and $rank\, A = k$. And let all rank order minors of $A$ be bounded by $Δ$ in absolute values. We show that the short rational generating function for the power series $$ \sum\limits_{m \in P \cap Z^n} x^m $$ can be computed with the arithmetic complexity $ O\left(T_{SNF}(d) \cdot d^{k} \cdot d^{\log_2 Δ}\right), $ where $k$ and $Δ$ a…
Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the $q\rightarrow 1$ limit of the $q$-hypergeometric Nahm sums. They make an appearance in various fields: hyperbolic geometry, knot theory, quiver representation theory, topological strings and conformal field theory. In this paper we focus primarily on Nahm sums and Nahm equations that arise in relation with symmetric quivers. For a large class of them, we prove that quiver A-polynomials -- specialized resultants of the Nahm equations, are tempered (the so-called K-theoretic condition). This implies that t…
Fano and Reflexive Polytopes from Feynman Integrals
Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS（理论物理研究所，巴黎萨克雷大学，法国原子能和替代能源委员会，法国国家科学研究中心）；Department of Physics and Astronomy, Ghent University（根特大学物理与天文学系）
