Roots of the Ehrhart polynomial of hypersimplices

TL;DR

This paper investigates the roots of Ehrhart polynomials of hypersimplices, conjecturing their real parts are negative and bounded, and proves the conjecture for specific cases, providing insights into their root distribution.

Contribution

The paper proves the conjecture for the case d=3 and establishes bounds on the roots' real parts for large n and fixed d, advancing understanding of hypersimplex Ehrhart roots.

Findings

Confirmed the conjecture for d=3.

Bounded the real parts of roots between -n/d and 1.

Provided evidence for the conjecture in the case 4 ≤ d ≪ n.

Abstract

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 $\Delta(d,n)$ is negative and larger than $- \frac{n}{d}$ if $n \geq 2d$. In this paper, we show that the conjecture is true when $d=3$ and that every root $a$ of the Ehrhart polynomial of $\Delta(d,n)$ satisfies $-\frac{n}{d} < {\rm Re} (a) < 1$ if $4 \leq d \ll n$.