On factorization invariants and Hilbert functions

TL;DR

This paper links factorization invariants in semigroups to Hilbert functions of graded modules, extending known periodicity and quasilinearity results from numerical to finitely generated semigroups.

Contribution

It introduces a new framework connecting factorization invariants with Hilbert functions, broadening the understanding of their behavior in finitely generated semigroups.

Findings

Catenary degree and delta set are eventually periodic in finitely generated semigroups.

Omega-primality is shown to be eventually quasilinear in this broader setting.

The paper establishes a method to determine invariants via Hilbert functions.

Abstract

Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical semigroups (additive subsemigroups of the natural numbers), several factorization invariants are known to admit predictable behavior for sufficiently large semigroup elements. In particular, the catenary degree and delta set invariants are both eventually periodic, and the omega-primality invariant is eventually quasilinear. In this paper, we demonstrate how each of these invariants is determined by Hilbert functions of graded modules. In doing so, we extend each of the aforementioned eventual behavior results to finitely generated semigroups, and provide a new framework through which to study factorization structures in this setting.