Kou Takahashi | Searcharxiv

arXiv2026

A criterion for determinantal presentations of numerical semigroup rings

We study numerical semigroups whose defining ideals admit determinantal presentations by the $2\times2$ minors of a $2\times n$ matrix. A conjecture of Cuong, Kien, Matsuoka, and Truong relates such presentations to arithmetic progressions of pseudo-Frobenius numbers. We prove a numerical criterion for this determinantal presentation, reducing the conjecture to a problem on factorizations of maximal elements of an Apéry set in the numerical semigroup. As an application, we verify the conjecture in embedding dimension four under a unique factorization assumption on an Apéry set.

math.AC↗

arXiv2025

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

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.

math.AC↗