Families of numerical semigroups and a special case of the Huneke-Wiegand conjecture

TL;DR

This paper proves a special case of the Huneke-Wiegand conjecture for numerical semigroup rings generated by generalized arithmetic sequences, using arithmetic sequence detection and visualization techniques.

Contribution

It extends the verification of the conjecture to a new class of numerical semigroups generated by generalized arithmetic sequences.

Findings

Confirmed the conjecture for semigroups generated by generalized arithmetic sequences.

Developed visualization methods to identify necessary arithmetic sequences.

Demonstrated the approach's effectiveness in this specific case.

Abstract

The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. Garc\'ia-S\'anchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings $\Bbbk[\Gamma]$ can be answered in the affirmative by locating certain arithmetic sequences within the numerical semigroup $\Gamma$. In this paper, we use their approach to prove the Huneke-Wiegand conjecture in the case where $\Gamma$ is generated by a generalized arithmetic sequence and showcase how visualizations can be leveraged to find the requisite arithmetic sequences.