Related papers: Lower bounds on the coefficients of Ehrhart polyno…
A complete classification of the $\delta$-vectors of lattice polytopes whose normalized volumes are at most $4$ is known. In the present paper, we will classify all the $\delta$-vectors of lattice polytopes with normalized volumes $5$.
The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart…
We give an algorithm to compute weighted Ehrhart functions of lattice polytopes for polynomial weights using Lagrange interpolation. We show how to compute generating functions of polynomials using those of unit cubes and Eulerian numbers,…
Ehrhart theory is the study of the enumeration of lattice points in lattice polytopes. Equivariant Ehrhart theory is a generalization of Ehrhart theory that takes into account the action of a finite group acting via affine transformations…
We investigate the sign patterns of coefficients in the Ehrhart polynomial of the Cartesian product between the $r$-th pyramid over the Reeve tetrahedron and the hypercube $[0, n]^n$. This investigation yields partial results on the sign…
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 which are independent of both the…
The Ehrhart polynomial $\text{ehr}_P(n)$ of a lattice polytope $P$ counts the number of integer points in the $n$-th integral dilate of $P$. The $f^*$-vector of $P$, introduced by Felix Breuer in 2012, is the vector of coefficients of…
There is currently a growing interest in understanding which lattice simplices have unimodal local $h^\ast$-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for…
Ehrhart's conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including…
The Ehrhart polynomial $ehr_P (n)$ of a lattice polytope $P$ gives the number of integer lattice points in the $n$-th dilate of $P$ for all integers $n\geq 0$. The degree of $P$ is defined as the degree of its $h^\ast$-polynomial, a…
We study the Ehrhart theory of quadratic irrational polytopes that undergo vector dilations. That is, for a given polytope with vertices in $\mathbb{Q}(\sqrt{D})$, and a different dilation factor for each facet, we show that the leading…
In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in $\mathbb Z^3$. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of…
We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the $\beta$-invariant of the matroid. This yields a lattice-point counting formula for the…
Ehrhart polynomials are extensively-studied structures that interpolate the discrete volume of the dilations of integral $n$-polytopes. The coefficients of Ehrhart polynomials, however, are still not fully understood, and it is not known…
A lattice polytope translated by a rational vector is called an almost integral polytope. In this paper we investigate Ehrhart quasi-polynomials of almost integral polytopes. We study the relationship between the shape of the polytopes and…
In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led us to define certain polynomials analogous to Ehrhart…
We show that, if the interior of a lattice d-polytope P contains at least one lattice point, then it contains a lattice point whose coefficient of asymmetry with respect to P is at most b for some number b depending on d only. As an…
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…
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…
In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its…