Roots of Ehrhart Polynomials of Smooth Fano Polytopes

TL;DR

This paper proves Golyshev's conjecture for smooth polytopes up to dimension five, showing their Ehrhart polynomial roots have real part -1/2, and provides explicit root descriptions, with counterexamples in dimension six.

Contribution

It offers an elementary proof of Golyshev's conjecture for dimensions up to five and explicitly describes the roots of Ehrhart polynomials for these polytopes.

Findings

Roots of Ehrhart polynomials for smooth polytopes in dimensions ≤5 have real part -1/2

Explicit descriptions of roots in each dimension are provided

Counterexamples show the result does not extend to dimension six

Abstract

V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots $z\in\C$ of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.