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arXiv · 2609.18136

A criterion for determinantal presentations of numerical semigroup rings

Satoshi Murai·Kou Takahashi
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Abstract

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.

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Satoshi Murai, Kou Takahashi. 2026-09-16. A criterion for determinantal presentations of numerical semigroup rings. https://arxiv.org/abs/2609.18136

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