Gorenstein polytopes and their stringy E-functions

TL;DR

This paper proves that the stringy E-function of a Gorenstein polytope is a polynomial, confirming a conjecture, and explores structural properties like joins and irreducibility related to nef-partitions.

Contribution

It establishes the polynomial nature of the stringy E-function for Gorenstein polytopes and introduces concepts of joins and irreducibility, linking them to nef-partition decompositions.

Findings

Stringy E-function is a polynomial, not just rational.

Comparison results for lattice points in faces of Gorenstein polytopes.

Introduction of irreducible Gorenstein polytopes and their relation to nef-partitions.

Abstract

Inspired by ideas from algebraic geometry, Batyrev and the first named author have introduced the stringy E-function of a Gorenstein polytope. We prove that this a priori rational function is actually a polynomial, which is part of a conjecture of Batyrev and the first named author. The proof relies on a comparison result for the lattice point structure of a Gorenstein polytope P, a face F of P and the face of the dual Gorenstein polytope corresponding to F. In addition, we study joins of Gorenstein polytopes and introduce the notion of an irreducible Gorenstein polytope. We show how these concepts relate to the decomposition of nef-partitions.