Ehrhart Polynomials Research Articles

Examples and counterexamples in Ehrhart theory

This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and h^{*} -polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as well as unimodality, log-concavity, and real-rootedness for h^{*} -polynomials.We survey inequalities that arise when the polytope has different normality properties. We include statements previously unknown in the Ehrhart theory setting as well as some original contributions to this topic. We address numerous variations of the conjecture asserting that IDP polytopes have a unimodal h^{*} -polynomial and construct concrete examples showing that these variations of the conjecture are false. Emphasis is put on polytopes arising within algebraic combinatorics.Furthermore, we describe and construct polytopes having pathological properties on their Ehrhart coefficients and roots, and we indicate for the first time a connection between the notions of Ehrhart positivity and h^{*} -real-rootedness. We investigate the log-concavity of the sequence of evaluations of an Ehrhart polynomial at the non-negative integers. We conjecture that IDP polytopes have a log-concave Ehrhart series. Many additional problems and challenges are proposed.

Valuative invariants for large classes of matroids

AbstractWe study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a stressed subset. This framework provides a new combinatorial characterization of the class of (elementary) split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which, in turn, can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on these matroids depend solely on the behavior of the invariant on a tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan–Lusztig polynomials, the Whitney numbers of the first and second kinds, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's ‐polynomials, as well as Chow rings of matroids and their Hilbert–Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

Chern-Simons theory, Ehrhart polynomials, and representation theory

The Hilbert space of level q Chern-Simons theory of gauge group G of the ADE type quantized on T2 can be represented by points that lie on the weight lattice of the Lie algebra g up to some discrete identifications. Of special significance are the points that also lie on the root lattice. The generating functions that count the number of such points are quasi-periodic Ehrhart polynomials which coincide with the generating functions of SU(q) representation of the ADE subgroups of SU(2) given by the McKay correspondence. This coincidence has roots in a string/M theory construction where D3(M5)-branes are put along an ADE singularity. Finally, a new perspective on the McKay correspondence that involves the inverse of the Cartan matrices is proposed.

Some characterizations of the complex projective space via Ehrhart polynomials

Let [Formula: see text] be the Ehrhart polynomial associated to an integral multiple [Formula: see text] of the standard simplex [Formula: see text]. In this paper, we prove that if [Formula: see text] is an [Formula: see text]-dimensional polarized toric manifold with associated Delzant polytope [Formula: see text] and Ehrhart polynomial [Formula: see text] such that [Formula: see text], for some [Formula: see text], then [Formula: see text] (where [Formula: see text] is the hyperplane bundle on [Formula: see text]) in the following three cases: (1) arbitrary [Formula: see text] and [Formula: see text], (2) [Formula: see text] and [Formula: see text] and (3) [Formula: see text] under the assumption that the polarization [Formula: see text] is asymptotically Chow semistable.

Ehrhart theory of paving and panhandle matroids

Abstract We show that the base polytope P M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of P M , starting with Katzman’s formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni’s work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.

Connection between Graphs' Chromatic and Ehrhart Polynomials

Connection between Graphs' Chromatic and Ehrhart Polynomials

The Distribution of Roots of Ehrhart Polynomials for the Dual of Root Polytopes of Type C

The Distribution of Roots of Ehrhart Polynomials for the Dual of Root Polytopes of Type C

Contact invariants of [formula omitted]-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to•the Ehrhart (quasi-)polynomial of its toric diagram;•the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity;•the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.

Some Computational Study of the Root Distribution of Ehrhart Polynomials

In this paper, we investigate the root distribution of the Ehrhart polynomials of lattice polytopes. When the lattice polytope is reflexive, the roots of the Ehrhart polynomial distribute symmetrically with respect to the line [Formula: see text]. A special case of this distribution is when all the roots lie on this line. Our main concern is to find out which reflexive polytopes satisfy this special condition. Such lattice polytopes are called CL-polytopes. Another special case opposite to this is when all the roots are real. Such lattice polytopes are called real polytopes. The first topic of this paper is the Ehrhart polynomials of equatorial spheres, which are related to graded posets. We discuss the CL-ness of the equatorial Ehrhart polynomials for complete graded posets and zig-zag posets. The second topic is to investigate the relation of the root distribution of the Ehrhart polynomials of a reflexive polytope [Formula: see text] and its dual [Formula: see text]. We discuss the CL-ness and realness of [Formula: see text] and [Formula: see text] in pair, of dimensions up to 4. Throughout this paper, we investigate these problems by computer calculation.

On the moments of moments of random matrices and Ehrhart polynomials

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.

When ties are possible: Weak Condorcet winners and Arrovian rationality

We use Ehrhart polynomials to estimate the likelihood of each three-candidate social ranking produced by pairwise majority rule assuming an even number of voters and the Impartial Anonymous Culture condition. We then calculate the probability the ranking is transitive and the probability of a weak Condorcet winner. Finally, we determine the weak Condorcet efficiency of various voting rules. We prove that Baldwin, Nanson, Copeland, and ranked pairs are weak Condorcet efficient, as is Borda unless there are no ties. Simulations show that among the rest, Dowdall is typically the most efficient rule for small voter groups and anti-plurality the least.

The Residue Theorem

Cauchy's Residue Theorem, commonly known as the residue Theorem, is a key theorem in complex variables because it enables people to determine the enclosed curve line of integrals for functionalities. Real integrals and infinite series may also be calculated using it. Our ability to handle complicated analysis is improved by the residue theorem. By connecting the internal Ehrhart polynomials to the closures, we illustrate these tetrahedron Ehrhart-Macdonald reciprocity laws. We calculate the Earhart coefficient of the codimensions to demonstrate our methodology. We conclude by demonstrating how to use our ideas to locate any paste applied with a convex lattice.

Ehrhart Positivity of Tesler Polytopes and Berline–Vergne’s Valuation

For {varvec{a}} in {mathbb {R}}_{ge 0}^{n}, the Tesler polytope {text {Tes}}_{n}({varvec{a}}) is the set of upper triangular matrices with non-negative entries whose hook sum vector is {varvec{a}}. Motivated by a conjecture of Morales, we study the questions of whether the coefficients of the Ehrhart polynomial of {text {Tes}}_n(1,1,dots ,1) are positive. We attack this problem by studying a certain function constructed by Berline–Vergne and its values on faces of a unimodularly equivalent copy of {text {Tes}}_n(1,1,dots ,1). We develop a method of obtaining the dot products appeared in formulas for computing Berline–Vergne’s function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline–Vergne’s function has positive values on codimension 2 and 3 faces of the polytopes we consider. As a consequence, we prove that the third and fourth coefficients of the Ehrhart polynomial of {text {Tes}}_{n}(1,dots ,1) are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of {text {Tes}}_{n}(1,dots ,1) including all the integral Tesler polytopes.

Pruned Inside-Out Polytopes, Combinatorial Reciprocity Theorems and Generalized Permutahedra

Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We show (quasi-)polynomiality and reciprocity results for the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017), Billera-Jia-Reiner (2009), and Karaboghossian (2022). Applying this reciprocity theorem to hypergraphic polytopes allows to give a geometric proof of a combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). This proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.

Lattice zonotopes of degree 2

The Ehrhart polynomial{text {ehr}}_P (n) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers nge 0. The degree of P is defined as the degree of its h^*-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.

The bipermutahedron

The harmonic polytope and the bipermutahedron are two related polytopes that arose in the Lagrangian geometry of matroids. We study the bipermutahedron. We show that it is a simple polytope whose faces are in bijection with the vertex-labeled and edge-labeled multigraphs with no isolated vertices; the generating function for its \(f\)-vector is a simple evaluation of the three variable Rogers-Ramanujan function. We introduce the biEulerian polynomial, which counts bipermutations according to their number of descents, and equals the \(h\)-polynomial of the bipermutahedral fan. We construct a unimodular triangulation of the product \(\Delta \times \cdots \times \Delta\) of triangles that is combinatorially equivalent to (the triple cone over) the bipermutahedral fan. Ehrhart theory then gives us a formula for the biEulerian polynomial, which we use to show that this polynomial is real-rooted and that the \(h\)-vector of the bipermutahedral fan is log-concave and unimodal. We describe all the deformations of the bipermutahedron; that is, the ample cone of the bipermutahedral toric variety. We prove that among all polytopes in this family, the bipermutahedron has the largest possible symmetry group. Finally, we show that the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.Mathematics Subject Classifications: 52B20, 52B05, 05A15Keywords: Polytope, bipermutahedron, bipermutations, descents, \(f\)-vector, \(h\)-vector, unimodular triangulation, Ehrhart polynomial, real-rooted polynomial, deformation cone

Period collapse in Ehrhart quasi-polynomials of \(\{1,3\}\)-graphs

A graph whose nodes have degree \(1\) or \(3\) is called a \(\{1,3\}\)-graph. Liu and Osserman associated a polytope to each \(\{1,3\}\)-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set \(\{0,\frac14,\frac12,1\}\), which implies that the period of their Ehrhart quasi-polynomials is either \(1, 2\), or \(4\). We show that the period of the Ehrhart quasi-polynomial of these polytopes is 2 if the graph is a tree, the period is at most 2 if the graph is cubic, and the period is \(4\) otherwise. In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.Mathematics Subject Classifications: 05C30, 05C76, 52B20Keywords: Ehrhart polynomials, period collapse

Low dimensional flow polytopes and their toric ideals

The toric ideal of a d-dimensional flow polytope has an initial ideal generated by square-free monomials of degree at most d. The toric ideal of a flow polytope of dimension at most four has an initial ideal generated by square-free monomials of degree at most two, with the only exception of the four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal generated by a square-free cubic monomial. The proof is based on a method to classify certain compressed flow polytopes, and a construction of a quadratic pulling triangulation of them. Along the way compressed flow polytopes are classified up to dimension four, and their Ehrhart polynomials are computed.

Ehrhart polynomials of rank two matroids

Over a decade ago De Loera, Haws and Köppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding h⁎-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater than or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that h⁎-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have log-concave and unimodal coefficients. In particular, this shows that the h⁎-polynomial of the second hypersimplex is real-rooted, thereby strengthening a result of De Loera, Haws and Köppe.

The Root Distributions of Ehrhart Polynomials of Free Sums of Reflexive Polytopes

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 $A_3^\vee \oplus (A_1^\vee)^{\oplus n}$ for any $n$ satisfy this property. We also perform computational experiments for other types of free sums of $A_n^\vee$'s or $A_n$'s.