Caviglia–et al. generalized Betti number conjecture based on width

Establish that for every numerical semigroup S and every homological degree t ≥ 1, the t-th Betti number β_t of the binomial defining ideal I_S of the semigroup ring k[S] equals β_t = t · (wd(S)+1 choose t+1), where wd(S) is the width of S.

Background

In 2024, Caviglia et al. proposed a strengthening of the width-based bound by predicting a uniform formula for all Betti numbers of the defining ideal, solely in terms of the width.

The present paper computes β_1 for Sally-type semigroups via Hochster’s formula and shows these semigroups satisfy the width bound, contributing evidence for such width-based predictions in a significant subclass.