Binomial regular sequences and free sums

TL;DR

This paper explores the algebraic structures underlying Ehrhart series of free sums of rational polytopes, focusing on binomial regular sequences, prime ideals, and normality in affine monoid algebras.

Contribution

It provides an algebraic framework for understanding Ehrhart series of free sums, characterizing when binomial regular sequences generate prime ideals and preserve normality.

Findings

Characterization of when binomial regular sequences generate prime ideals

Conditions under which normality is preserved in residue class rings

Algebraic derivation of combinatorial results on Ehrhart series

Abstract

Recently several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from structural results on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhart series. We characterize when a binomial regular sequence generates a prime ideal or even normality is preserved for the residue class ring.