Hard Lefschetz properties and distribution of spectra in singularity theory and Ehrhart theory

TL;DR

This paper explores the distribution of spectra in singularity and Ehrhart theories, establishing a hard Lefschetz property for Laurent polynomials and polytopes, and providing criteria that relate to a conjecture by Katzarkov-Kontsevitch-Pantev.

Contribution

It introduces a new combinatorial criterion for the hard Lefschetz property in both singularity and Ehrhart theories, linking spectral distributions to geometric and algebraic structures.

Findings

Established a hard Lefschetz property for Laurent polynomials and polytopes.

Provided combinatorial criteria for the Lefschetz property.

Gave insights related to the Katzarkov-Kontsevitch-Pantev conjecture.

Abstract

We discuss the distribution of the spectrum at infinity of a convenient and nondegenerate Laurent polynomial (singularity side) and the distribution of the Newton spectrum of a polytope (Ehrhart theory side). To this end, we study a hard Lefschetz property for Laurent polynomials and for polytopes and we give combinatorial criteria for this property to be true. This provides informations about a conjecture by Katzarkov-Kontsevitch-Pantev.