Defining ideals of some numerical semigroup rings with arithmetic pseudo-Frobenius numbers

TL;DR

This paper investigates the defining ideals of numerical semigroup rings, proving they are determinantal under certain conditions related to pseudo-Frobenius numbers forming an arithmetic sequence, thus addressing a conjecture in the field.

Contribution

It establishes that the defining ideal is determinantal when pseudo-Frobenius numbers form an arithmetic sequence, under specific numerical semigroup conditions.

Findings

Defining ideal is determinantal under given conditions.

Partially resolves a conjecture by Cuong, Kien, Truong, and Matsuoka.

Provides new insights into the structure of numerical semigroup rings.

Abstract

In this paper, we study defining ideals of numerical semigroup rings. Let $H$ be a numerical semigroup with multiplicity $a_0$ and embedding dimension $n$. Assuming $a_0/2+1\leq n$, we prove that the defining ideal of $H$ is determinantal when the set of pseudo-Frobenius numbers forms an arithmetic sequence of length $n-1$. This partly resolves a conjecture of Cuong, Kien, Truong and, Matsuoka.