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arXiv · 2309.07793

On faces of the Kunz cone and the numerical semigroups within them

Levi Borevitz·Tara Gomes·Jiajie Ma·Harper Niergarth·Christopher O'Neill·Daniel Pocklington·Rosa Stolk·Jessica Wang·Shuhang Xue
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Abstract

A numerical semigroup is a cofinite subset of the non-negative integers that is closed under addition and contains 0. Each numerical semigroup $S$ with fixed smallest positive element $m$ corresponds to an integer point in a rational polyhedral cone $\mathcal C_m$, called the Kunz cone. Moreover, numerical semigroups corresponding to points in the same face $F \subseteq \mathcal C_m$ are known to share many properties, such as the number of minimal generators. In this work, we classify which faces of $\mathcal C_m$ contain points corresponding to numerical semigroups. Additionally, we obtain sharp bounds on the number of minimal generators of $S$ in terms of the dimension of the face of $\mathcal C_m$ containing the point corresponding to $S$.

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Levi Borevitz, Tara Gomes, Jiajie Ma, Harper Niergarth, Christopher O'Neill, Daniel Pocklington, Rosa Stolk, Jessica Wang, Shuhang Xue. 2023-09-14. On faces of the Kunz cone and the numerical semigroups within them. https://arxiv.org/abs/2309.07793

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