Let Γ ⊆ N Γ ⊆ N be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of Γ Γ which depends only on the width of Γ Γ , that is, the difference between the largest and the smallest generator of Γ Γ . In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. 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.
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Caviglia et al. (2024) studied this question.