Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone
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| Format: | Preprint |
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2024
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| _version_ | 1866912244067991552 |
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| author | Brower, Cole McDonough, Joseph O'Neill, Christopher |
| author_facet | Brower, Cole McDonough, Joseph O'Neill, Christopher |
| contents | 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 Gröbner 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone Brower, Cole McDonough, Joseph O'Neill, Christopher Combinatorics Commutative Algebra 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 Gröbner 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. |
| title | Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2401.06025 |