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

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.

4 pages, We changed the order of the auhors and omitted a lot of parts of the paper. (If you are interested in omitted parts, then please read v1)

References in corpus (4)

• Roots of Ehrhart polynomials arising from graphs
• Centrally symmetric configurations of integer matrices
• Smooth Fano polytopes arising from finite directed graphs
• Counterexamples of the conjecture on roots of Ehrhart polynomials

Cited by in corpus (8)

• Many faces of symmetric edge polytopes
• Interlacing Ehrhart Polynomials of Reflexive Polytopes
• Symmetric edge polytopes and matching generating polynomials
• The $h^{\ast }$-polynomials of locally anti-blocking lattice polytopes and their $\gamma$-positivity
• A toric deformation method for solving Kuramoto equations
• Ehrhart polynomial roots of reflexive polytopes
• Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering
• Graph edge contraction and subdivisions for adjacency polytopes