The root distributions of Ehrhart polynomials of free sums of reflexive polytopes

TL;DR

This paper investigates the distribution of roots of Ehrhart polynomials for free sums of reflexive polytopes, showing many roots lie on a specific line in the complex plane, with computational experiments supporting these findings.

Contribution

It establishes conditions under which roots of Ehrhart polynomials of free sums of certain reflexive polytopes lie on the line Re(z) = -1/2, and provides computational evidence for other cases.

Findings

Roots of Ehrhart polynomials for specific free sums lie on Re(z) = -1/2.

Proved cases where roots lie on the canonical line for certain reflexive polytopes.

Performed computational experiments supporting the root distribution patterns.

Abstract

In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n}$ for any $n$ satisfy this property. We also perform computational experiments for other types of free sums of $A_n^\vee$'s or $A_n$'s.