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arXiv · 2401.06025

Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone

Cole Brower·Joseph McDonough·Christopher O'Neill
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Abstract

A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal C_m \subseteq \mathbb R^{m-1}$, and the face of $\mathcal C_m$ containing that integer point determines certain algebraic properties of $S$. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of $\mathcal C_m$. We characterize several aspects of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are known to index the faces of $\mathcal C_m$, and our results culminate in a method of "walking" the face lattice of the Kunz cone in a manner analogous to that of a Gr\"obner walk. We apply our results in several contexts, including a wealth of computational data obtained from the aforementioned "walks" and a proof of a recent conjecture concerning which numerical semigroups achieve the highest minimal presentation cardinality when one fixes the smallest positive element and the number of generators.

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Cole Brower, Joseph McDonough, Christopher O'Neill. 2024-01-11. Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone. https://arxiv.org/abs/2401.06025

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