Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

TL;DR

This paper constructs a specific smooth Fano polytope, derived from symmetric edge polytopes of odd cycles, whose Ehrhart polynomial has a root with a real part exceeding the dimension, challenging existing conjectures.

Contribution

It provides a counterexample to two conjectures on Ehrhart polynomial roots by analyzing smooth Fano polytopes from odd cycle edge polytopes.

Findings

Ehrhart polynomial of the 127-cycle polytope has a root with real part greater than the dimension.

Counterexample disproves two conjectures on Ehrhart polynomial roots.

Demonstrates the existence of smooth Fano polytopes with roots outside expected bounds.

Abstract

The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this paper, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.