Related papers: Boundary $h^\ast$-vectors and unimodular triangula…

For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known…

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured…

If $P$ is a lattice polytope (i.e., $P$ is the convex hull of finitely many integer points in $\mathbb{R}^d$) of dimension $d$, Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|nP \cap \mathbb{Z}^d|$ is a…

The cosmological polytope of a graph $G$ was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram $G$. Basic results in the theory of positive…

The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix…

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…

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gr\"obner basis techniques, half-open decompositions and methods for…

We prove that the Ehrhart $h^*$-vector is unimodal for unimodular triangulations whose boundary is an induced subcomplex.

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…

In this note we investigate the convex hull of those $n \times n$-permutation matrices that correspond to symmetries of a regular $n$-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein…

The Ehrhart quasipolynomial of a rational polytope $P$ encodes the number of integer lattice points in dilates of $P$, and the $h^*$-polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition…

We study semigroup algebras arising from lattice polytopes, compute their volume polynomials (particularizing work of Hochster), and establish strong Lefschetz properties (generalizing work of the first three authors). This resolves several…

There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to…

For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are to (1) determine if its (Ehrhart) $h^\ast$-polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients.…

In Ehrhart theory, the $h^*$-vector of a rational polytope often provide insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic…

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov;…

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) $h^\ast$-polynomial. This conjecture can be viewed as a strengthening of a…

Using a ribbon structure of the graph, we construct a dissection of the symmetric edge polytope of a graph into unimodular simplices. Our dissection is shellable, and one can interpret the elements of the resulting $h$-vector via graph…

The Ehrhart polynomial of a lattice polytope $P$ encodes information about the number of integer lattice points in positive integral dilates of $P$. The $h^\ast$-polynomial of $P$ is the numerator polynomial of the generating function of…

Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including…