Numerical Semigroups of Sally Type
Saipriya Dubey, Kriti Goel, Nil Sahin, Srishti Singh, Hema Srinivasan
TL;DR
This work analyzes Sally-type numerical semigroups $S^e(m,n)$ to bound and explicitly determine the minimal number of generators of their defining ideals. It derives a Frobenius-number formula, establishes a precise Gorenstein criterion (only for $(m,n)=(2,3)$), and computes $\mu(I^e(m,n))$ via Hochster's formula, supported by a constructive minimal generating set and a GAP-based algorithm. The authors provide a detailed combinatorial framework using Stanley-Reisner complexes, plus an extensive appendix with exact beta-number computations for key parameter ranges. The results confirm a width- bound conjecture in this class and furnish practical tools (algorithms and GAP code) for computing first Betti numbers of semigroup rings.
Abstract
Judith Sally proved in 1980 that the associated graded ring of one-dimensional Gorenstein local rings of multiplicity $e$ and embedding dimension $e-2$ are Cohen-Macaulay. She showed that the defining ideal of the associated graded ring of such rings is generated by ${e-2 \choose 2}$ elements. Numerical semigroup rings are a big class of one-dimensional Cohen-Macaulay rings. In 2014, Herzog and Stamate proved that the numerical semigroup $<e,e+1,e+4,\ldots,2e-1 >$ defines a Gorenstein semigroup ring satisfying Sally's conditions above and such semigroups are called Gorenstein Sally Semigroups. We call a numerical semigroup $S$ as Sally type if $<S >= < e,e+1,\ldots,e+m-1, e+m+1,\ldots, e+n-1,e+n+1, \ldots 2e-1>$ for some $2 \leq m <n \leq e-2$. In this paper, we give a formula for its Frobenius number along with a necessary and sufficient condition for it to be Gorenstein. We compute the minimal number of generators for the defining ideal of the semigroup ring $k[S]$. Additionally, we present an algorithm and a GAP code used in applying Hochster's combinatorial formula to compute the first Betti number of $k[S]$.