Related papers: Roots of Ehrhart Polynomials of Smooth Fano Polyto…

The conjecture on roots of Ehrhart polynomials, stated by Matsui et al. \cite[Conjecture 4.10]{MHNOH}, says that all roots $\alpha$ of the Ehrhart polynomial of a Gorenstein Fano polytope of dimension $d$ satisfy $-\frac{d}{2} \leq…

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 {\it et al.}\ that all roots $\alpha$ of Ehrhart…

Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes…

The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this paper, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the…

Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope $\Pc \subset \RR^d$ of dimension $d$ such that (i) its Ehrhart polynomial $i(\Pc, n)$ possesses $d$ distinct roots; (ii) $i(\Pc, n)$…

The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We…

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 n^2, where n is the…

In this paper, we study the Ehrhart polynomial of the dual of the root polytope of type C of dimension $d$, denoted by $C_d^*$. We prove that the roots of the Ehrhart polynomial of $C_d^*$ have the same real part $-1/2$, and we also prove…

The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In math.CO/0402148, the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial…

The Ehrhart polynomial of the $d$-th hypersimplex $\Delta(d,n)$ of order $n$ is studied. By computational experiments and a known result for $d=2$, we conjecture that the real part of every roots of the Ehrhart polynomial of $\Delta(d,n)$…

We study a class of polynomials that has all of its roots on the critical line and shares many properties with Ehrhart polynomials. Braun showed that the roots of Ehrhart polynomials are bounded quadratically and Higashitani provided…

We investigate the Ehrhart polynomial for the class of 0-symmetric convex lattice polytopes in Euclidean $n$-space $\mathbb{R}^n$. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima are closely related…

It is shown that, for each $d \geq 4$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that each of the coefficients of $n, n^{2}, \ldots, n^{d-2}$ of its Ehrhart polynomial $i(\mathcal{P},n)$ is negative.…

Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…

It is shown that, for each $d \geq 4$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that each of the coefficients of $n, n^{2}, \ldots, n^{d-2}$ of its Ehrhart polynomial $i(\mathcal{P},n)$ is negative.

Over a decade ago De Loera, Haws and K\"oppe 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…

M. Beck, J. De Loera, M. Develin, J. Pfeifle and R. Stanley 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…

A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ --- that is, a "polynomial" in which the coefficients are…

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 and it turns out that the number of integral…

The Ehrhart quasipolynomial of a rational polytope $\mathsf{P}$ encodes fundamental arithmetic data of $\mathsf{P}$, namely, the number of integer lattice points in positive integral dilates of $\mathsf{P}$. Ehrhart quasipolynomials were…