Bounds for syzygies of monomial curves

TL;DR

This paper establishes an upper bound on the Betti numbers of semigroup rings based on the width of the numerical semigroup, advancing the Herzog-Stamate conjecture and covering most cases for 4-generated semigroups.

Contribution

It provides a width-dependent upper bound for Betti numbers and proves the Herzog-Stamate bound for nearly all 4-generated semigroups.

Findings

Upper bound for Betti numbers depending on semigroup width

Progress towards Herzog-Stamate conjecture

Verification of the bound for almost all 4-generated semigroups

Abstract

Let G be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of G which depends only on the width of G, that is, the difference between the largest and the smallest generator of G. In this way, we make progress towards a conjecture of Herzog and Stamate. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.