Roots of Ehrhart polynomials and symmetric $\delta$-vectors

TL;DR

This paper investigates the roots of Ehrhart polynomials of Gorenstein Fano polytopes, confirming a conjecture about their roots' real parts for certain dimensions and real roots, using analysis of SSNN polynomials.

Contribution

It extends the analysis to SSNN polynomials, a broader class, and verifies the conjecture for real roots and dimensions up to five.

Findings

Conjecture holds for real roots.

Conjecture verified for dimensions up to 5.

Analysis of SSNN polynomials supports the conjecture.

Abstract

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 $-\frac{d}{2} \leq \Re(\alpha) \leq \frac{d}{2} -1$. In this paper, we observe the behaviors of roots of SSNN polynomials which are a wider class of the polynomials containing all the Ehrhart polynomials of Gorenstein Fano polytopes. As a result, we verify that this conjecture is true when the roots are real numbers or when $d \leq 5$.