Roots of the Ehrhart polynomial of hypersimplices

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 $n\ge 2d$. 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 $-\frac{n}{d}<\mathrm{Re}(a)<1$ if $4\le d\ll n$.

18 pages, 8 figures

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