Pseudo-Frobenius numbers and defining ideals in stretched numerical semigroup rings

TL;DR

This paper explores the relationship between pseudo-Frobenius numbers and defining ideals in stretched numerical semigroup rings, resolving a conjecture and providing conditions for Cohen-Macaulay tangent cones.

Contribution

It proves a conjecture linking pseudo-Frobenius numbers to defining ideals in stretched semigroup rings and offers numerical criteria for Cohen-Macaulay tangent cones.

Findings

Resolved a conjecture under the stretched condition.

Provided numerical conditions for Cohen-Macaulay tangent cones.

Connected pseudo-Frobenius numbers with the structure of defining ideals.

Abstract

The pseudo-Frobenius numbers of a numerical semigroup $H$ are deeply connected to the structure of the defining ideal of its semigroup ring $k[H]$. In this paper, we resolve a certain conjecture related to this connection under the assumption that $k[H]/(t^a)$ is stretched, where $a$ is the multiplicity of $H$. Furthermore, we provide numerical conditions for the tangent cone of $k[H]$ to be Cohen-Macaulay.