Equivariant Ehrhart theory

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system.     

Rational Ehrhart quasi-polynomials

Ehrhart?s famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors. It turns out that the number of integral points can still be written as a rational quasi-polynomial. Furthermore, the coefficients of this rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation.     

Characteristic and Ehrhart Polynomials

Let A be a subspace arrangement and let (A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from , where is the type B Weyl arrangement, then (A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of (A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in while the second deals with all finite Weyl groups but only their hyperplane arrangements.     

Ehrhart Series and Lattice Triangulations

We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and “local contributions” of the links of its nonunimodular faces. We also compute new examples of nonunimodal h ^*-vectors of reflexive polytopes. Supported by the Clay Mathematics Institute.     

Universal inequalities in Ehrhart theory

In this paper, we show the existence of universal inequalities for the h*-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the h*-polynomial that are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients h* ₁ and h* ₂ of the h*-vector (h* ₀, h* ₁,..., h* _d) of a lattice polytope of any degree satisfy Scott’s inequality if h* ₃ = 0.     

Ehrhart polynomials of cyclic polytopes

The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In (Coefficients and roots of Ehrhart polynomials, preprint), the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial of it is equal to its volume plus the Ehrhart polynomial of its lower envelope and proved the case when the dimension d=2. In our article, we prove the conjecture for any dimension.     

Ehrhart Series for Connected Simple Graphs

The Ehrhart ring of the edge polytope ${\mathcal{P}_G}$ for a connected simple graph G is known to coincide with the edge ring of the same graph if G satisfies the odd cycle condition. This paper gives for a graph which does not satisfy the condition, a generating set of the defining ideal of the Ehrhart ring of the edge polytope, described by combinatorial information of the graph. From this result, two factoring properties of the Ehrhart series are obtained; the first one factors out bipartite biconnected components, and the second one factors out a even cycle which shares only one edge with other part of the graph. As an application of the factoring properties, the root distribution of Ehrhart polynomials for bipartite polygon trees is determined.     

Norm Bounds for Ehrhart Polynomial Roots

M. Beck et al. found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1+(d+1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.     

Ehrhart polynomials and stringy Betti numbers

We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of δ-vectors of reflexive polytopes. The first author was partially supported by NSF grant DMS 0500127 and the second author was supported by a Graduate Research Fellowship from the NSF     

Mochizuki's indigenous bundles and Ehrhart polynomials

Mochizuki's work on torally indigenous bundles [1] yields combinatorial identities by degenerating to different curves of the same genus. We rephrase these identities in combinatorial language and strengthen them, giving relations between Ehrhart quasi-polynomials of different polytopes. We then apply the theory of Ehrhart quasi-polynomials to conclude that the number of dormant torally indigenous bundles on a general curve of a given type is expressed as a polynomial in the characteristic of the base field. In particular, we conclude the same for the number vector bundles of rank two and trivial determinant whose Frobenius-pullbacks are maximally unstable, as well as self-maps of the projective line with prescribed ramification. The second author was supported by a fellowship from the Japan Society for the Promotion of Science during the preparation of this paper.     

Weighted Ehrhart theory and orbifold cohomology

We introduce the notion of a weighted δ-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted δ-vectors from a combinatorial perspective. We present a version of Ehrhart Reciprocity and prove a change of variables formula. We deduce a new geometric interpretation of the coefficients of the Ehrhart δ-vector. More specifically, they are sums of dimensions of orbifold cohomology groups of a toric stack.     

Maximal periods of (Ehrhart) quasi-polynomials

A quasi-polynomial is a function defined of the form q(k)=cd(k)kd+cd−1(k)kd−1+?+c₀(k), where c₀,c₁,…,cd are periodic functions in k∈Z. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the cj(k) for Ehrhart quasi-polynomials. For generic polytopes, McMullen's bounds seem to be sharp, but sometimes smaller periods exist. We prove that the second leading coefficient of an Ehrhart quasi-polynomial always has maximal expected period and present a general theorem that yields maximal periods for the coefficients of certain quasi-polynomials. We present a construction for (Ehrhart) quasi-polynomials that exhibit maximal period behavior and use it to answer a question of Zaslavsky on convolutions of quasi-polynomials.     

Notes on the Roots of Ehrhart Polynomials

We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n², where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n². For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes. We further give a characterisation of the roots of Ehrhart polyomials in the three-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.     

Roots of Ehrhart polynomials arising from graphs

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle |z+\fracd4| ￡ \fracd4|z+\frac{d}{4}| \le \frac{d}{4} or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip -\fracD2 ￡ Re(a) ￡ \fracD2-1-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.     

The Ehrhart Polynomial of the Birkhoff Polytope

The nth Birkhoff polytope is the set of all doubly stochastic n × n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these polytopes, which have been known for n $\leq$ 8. We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function counting the integer points in the dilated polytope. One reason to be interested in this counting function is that the leading term of the Ehrhart polynomial is—up to a trivial factor—the volume of the polytope. We implemented our methods in the form of a computer program, which yielded the Ehrhart polynomial (and hence the volume) of the ninth Birkhoff polytope, as well as the volume of the tenth Birkhoff polytope.     

Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.     

Counterexamples of the Conjecture on Roots of Ehrhart Polynomials

On roots of Ehrhart polynomials, Beck et al. conjecture that all roots α of the Ehrhart polynomial of an integral convex polytope of dimension d satisfy −d≤ℜ(α)≤d−1. In this paper, we provide counterexamples for this conjecture.     

Computing the Ehrhart polynomial of a convex lattice polytope

We prove that computation of any fixed number of highest coefficients of the Ehrhart polynomial of an integral polytope can be reduced in polynomial time to computation of the volumes of faces. This research was supported by the United States Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute of Cornell University, Contract DAAL03-91-C-0027.     

Characteristic polynomials, Ehrhart quasi-polynomials, and torus groups

For k a positive integer, we consider the problem of counting solutions x to the equation kx=0, where x is to lie in a given subset K of a torus group. This problem is considered for three classes of subsets K.     

Lattice polytopes,Hecke operators,and the Ehrhart polynomial

Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.