The Weighted Euler-Maclaurin Formula for a simple integral polytope

Sum-integral interpolators and the Euler-Maclaurin formula for polytopes

A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space $V$, namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in $V$. The paper introduces the notion of an interpolator between these two families of meromorphic functions. We prove that every rigid complement map in $V$ gives rise to an effectively computable \SI-interpolator (and a local Euler-MacLaurin formula), an \IS-inte…

Local Euler-Maclaurin formula for polytopes

We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C) on W with constant rational coefficients, which is unchanged when C is translated by an integral vector. Then for every convex rational polytope P in a rational euclidean space V and every polynomial function f (x) on V, the sum of the values of f(x) at the integral points of P is equal to the sum, for all faces F of P, of the integral over F of the function …

On a Tauberian Theorem of Ingham and Euler-Maclaurin Summation

We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler-Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the comp…

Riemann sums over polytopes

We show that the Euler-MacLaurin formula for Riemann sums has an n-dimensional analogue in which intervals on the line get replaced by convex polytopes.

Resurgence of the Euler-MacLaurin summation formula

The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula. Using a mild resurgence hypothesis for the function to be summed, we give a Borel summable t…

块计数约束调和和：谱展开与块定向欧拉-麦克劳林公式

Block-count constrained harmonic sums: spectral expansion and block-directed Euler-Maclaurin

本文精确刻画了固定数字块出现次数下调和和的性质，通过语言分解与随机基数展开建立谱理论，并给出介于欧拉-麦克劳林与泰勒展开之间的块定向插值公式。

计算光滑多面体的参数化加权厄尔哈特多项式

Computing parametric weighted Ehrhart polynomials of smooth polytopes

该研究证明整多面体形变时加权厄尔哈特与$h^*$-多项式系数是分段多项式函数，提出并在SageMath中实现了基于加权欧拉-麦克劳林公式的光滑多面体计算算法，并讨论了加权$h^*$-多项式系数符号的相关问题。

Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a …

The iterated logarithmic algebra II: Sheffer sequences

An extension of the theory of the Iterated Logarithmic Algebra gives the logarithmic analog of a Sheffer or Appell sequence of polynomials. This leads to several examples including Stirling's formula and a logarithmic version of the Euler-MacLaurin summation formula. Grâce à une généralisation de la théorie de l'algèbre des logarithmes itérés, on definit un analogue logarithmique des suites de polynômes de Sheffer et d'Appell. Quelques exemples d'applications permettent de déduire la formule de Stirling ainsi qu'un version logarithmique de la formule de sommation de Euler--MacLaurin.

Algebraic Birkhoff Factorization and the Euler-Maclaurin Formula on Cones

We equip the space of lattice cones with a coproduct which makes it a connected cograded colagebra. The exponential sum and exponential integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties-- reminiscent of the inclusion-exclusion principle for the cardinal on finite sets-- of such linear maps and establish a compatibility of these properties with respect to the convolution quotient of the coalgebra. Implementing the Algebraic Birkhoff Factorization procedure on the…

Generalized asymptotic Euler's relation for certain families of polytopes

According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the number of all faces of P for some positive integer m and for some 0 < i < m+1. We show some classes of polytopes for which the above proportion is asymptotically equal to 1/m.

An algebraic construction of sum-integral interpolators

This paper presents an algebraic construction of Euler-Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin formulas of Berline-Vergne While the approach of this paper originates in the theory of toric varieties, and recovers previous results about characteristic classes of toric varieties, the present paper is self-contained and does not rely on results from toric geometry. We aim in particular to exhibit in a combinatorial way ingredients such as such Todd classes and cycle-…

加权Ehrhart级数与MacMahon公式的B型类似物

Weighted Ehrhart series and a type-$\mathsf{B}$ analogue of a formula of MacMahon

本文通过多面体的$h^*$-多项式给出了带符号多重集排列上主指标与下降统计量联合分布的生成多项式公式，并建立了A、B型广义欧拉数与多面体之间的联系，进而将结果推广到着色排列。

The Ehrhart Function for Symbols

We derive an Ehrhart function for symbols from the Euler-MacLaurin formula with remainder.

A Short Proof of Euler--Poincaré Formula

"V - E + F = 2", the famous Euler's polyhedral formula, has a natural generalization to convex polytopes in every finite dimension, also known as the Euler-Poincaré Formula. We provide another short inductive proof of the general formula. Our proof is self-contained and it does not use shellability of polytopes.

Graphs, Skeleta and Reconstruction of Polytopes

A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Grünbaum's 1967 book. This survey paper includes varied results and open questions by many researchers on simplicial polytopes, nearly simple polytopes, cubical polytopes, zonotopes, crosspolytopes, and Eulerian posets.

Orientations, lattice polytopes, and group arrangements II: Modular and integral flow polynomials of graphs

We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law for the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative inte…

Weighted lattice point sums in lattice polytopes, unifying Dehn--Sommerville and Ehrhart--Macdonald

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…

Angle sums on polytopes and polytopal complexes

We will study the angle sums of polytopes, listed in the $α$-vector, working to exploit the analogy between the f-vector of faces in each dimension and the alpha-vector of angle sums. The Gram and Perles relations on the $α$-vector are analogous to the Euler and Dehn-Sommerville relations on the f-vector. First we describe the spaces spanned by the the alpha-vector and the $α$-f-vectors of certain classes of polytopes. Families of polytopes are constructed whose angle sums span the spaces of polytopes defined by the Gram and Perles equations. This shows that the dimension of the affine span of…

Approximating sums by integrals only: multiple sums and sums over lattice polytopes

The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of $f$. It was shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms…

Euler Characteristic of real nondegenerate tropical complete intersections

We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining a nondegenerate tropical complete intersection are equal to 1. The intersection multiplicity numbers we use are sums of mixed volumes of polytopes which are dual to cells of the tropical hypersurfaces. We show that the Euler characteristic of a real nondegenerate tropical complete intersection depends only on the Newton polytopes of the tropical polynomials…

The stringy Euler number of Calabi-Yau hypersurfaces in toric varieties and the Mavlyutov duality

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $Δ$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $Δ$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $Δ$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(Δ, Δ^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of M…

Möbius Polynomials of Face Posets of Convex Polytopes

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their $f$-vectors and some additional constraints.

Calabi-Yau超曲面在Toric变种中的Hilbert多项式与多面体边界的格点计数

Hilbert Polynomials of Calabi Yau Hypersurfaces in Toric Varieties and Lattice Points in Polytope Boundaries

本文研究了Calabi-Yau超曲面在Toric变种中的Hilbert多项式，通过多面体边界的格点计数和K-理论中的欧拉类计算，揭示了其与组合学中包含-排除原理的平行性。

A glimpse into continuous combinatorics of posets, polytopes, and matroids

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…

Euler flag enumeration of Whitney stratified spaces

The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi…

关于右角双曲3-多面体有限体积上欧拉子图关联的双曲链环

Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

研究了右角双曲3-多面体有限体积上欧拉子图关联的双曲链环，证明了其与多面体顶点和哈密顿回路的关系，并给出了链环的构造与性质。

Moment-angle complexes and polyhedral products for convex polytopes

Let P be a convex polytope not simple in general. In the focus of this paper lies a simplicial complex K_P which carries complete information about the combinatorial type of P. In the case when P is simple, K_P is the same as dP*, where P* is a polar dual polytope. Using the canonical embedding of a polytope P into nonnegative orthant, we introduce a moment-angle space Z_P for a polytope P. It is known, that in the case when P is simple the space Z_P is homeomorphic to the polyhedral product (D^2,S^1)^{K_P}. When P is not simple, we prove that the space Z_P is homotopically equivalent to the s…

广义角向量、几何格与旗角

Generalized angle vectors, geometric lattices, and flag-angles

本文通过几何格的代数组合理论证明了广义角向量 Euler--Poincaré 型关系的存在性与唯一性，并引入多面体旗角，利用关联代数确定了内/外旗角向量的线性关系。