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.

References

... and in 2024 Caviglia et al. generalized this conjecture to $\displaystyle \beta_t=t{wd(S)+1 \choose t+1}$, where $\beta_t$ is the $t{\text{th}$ Betti number of $I_S$.

— Numerical Semigroups of Sally Type  (2507.11738 - Dubey et al., 15 Jul 2025) in Section 1, Introduction

We make a stronger conjecture, based on the techniques utilised in the proof of Lemma \ref{lem:alphad-2n}. Macaulay2 computations verify the conjecture for small $m$ and $n$.

\begin{conjecture} Let $4\leq n<m-1$, $S=\Gamma_m(n,n+1)$, and $R=K[S]$. Let $Q$ be a polynomial ring in $m-2$ variables and $I_S\subset Q$ be the defining ideal of $S$. Then the Betti numbers of $Q/I_S$ are $\beta_0=1$ and

\beta_d=d\binom{m-2}{d+1}-2\binom{m-3}{d-1}+(n-1)\binom{m-n}{d-n+2}-(n-3)\binom{m-n-1}{d-n+1}\;\text{for } d\geq 1. \qedhere

\end{conjecture}

— The Betti numbers and Golodness of numerical semigroup rings of Sally type  (2609.26752 - Batavia et al., 22 Sep 2026) in Section 6, “Further Questions and Conjectures”