Roots of Ehrhart polynomials of Gorenstein Fano polytopes

TL;DR

This paper constructs Gorenstein Fano polytopes with Ehrhart polynomials having roots with specific properties, revealing detailed root structures related to the polytopes' geometry.

Contribution

It provides explicit constructions of Gorenstein Fano polytopes with Ehrhart polynomials exhibiting prescribed distributions of real and imaginary roots.

Findings

Ehrhart polynomials can have roots with real parts exactly -1/2.

Constructed polytopes have Ehrhart polynomials with all real roots in (-1, 0).

The number of imaginary roots in the Ehrhart polynomial can be controlled by the construction.

Abstract

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) $i(\Pc, n)$ possesses exactly $2k$ imaginary roots; (iii) $i(\Pc, n)$ possesses exactly $d - 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to $- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$.