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The Betti numbers and Golodness of numerical semigroup rings of Sally type
Abstract
We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel--Şahin--Singh--Srinivasan. Our approach uses Apéry resolutions and their specialisations. For the family Γm(n)=⟨ m,m+1,…,m+n̂,…,2m−1⟩, 2≤ n<m, where the hat denotes omission, we also compute the Poincaré series of the residue field and characterise Golodness. For fixed multiplicity m and 3≤ n<m, we show that these rings share the same rational residue field Poincaré series, but exhibit arbitrarily late departures from Serre's upper bound.
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BibTeX
@misc{batavia2026betti,
title = {{The Betti numbers and Golodness of numerical semigroup rings of Sally type}},
author = {Manav Batavia and Omkar Javadekar and Manohar Kumar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26752},
eprint = {2609.26752},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26752},
}
RIS
TY - UNPB
TI - The Betti numbers and Golodness of numerical semigroup rings of Sally type
AU - Manav Batavia
AU - Omkar Javadekar
AU - Manohar Kumar
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26752
UR - https://arxiv.org/abs/2609.26752
AB - We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel--Şahin--Singh--Srinivasan. Our approach uses Apéry resolutions and their specialisations. For the family Γ_m(n)=⟨ m,m+1,…,m+n̂,…,2m−1⟩, 2≤ n<m, where the hat denotes omission, we also compute the Poincaré series of the residue field and characterise Golodness. For fixed multiplicity m and 3≤ n<m, we show that these rings share the same rational residue field Poincaré series, but exhibit arbitrarily late departures from Serre's upper bound.
ER -