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.
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Keywords: numerical semigroup, poset, polyhedral cone
Brower, Cole 1; McDonough, Joseph 2; O’Neill, Christopher 1
CC-BY 4.0
@article{ALCO_2025__8_3_597_0,
author = {Brower, Cole and McDonough, Joseph and O{\textquoteright}Neill, Christopher},
title = {Numerical semigroups, polyhedra, and posets {IV:} walking the faces of the {Kunz} cone},
journal = {Algebraic Combinatorics},
pages = {597--617},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {3},
year = {2025},
doi = {10.5802/alco.425},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.425/}
}
TY - JOUR AU - Brower, Cole AU - McDonough, Joseph AU - O’Neill, Christopher TI - Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone JO - Algebraic Combinatorics PY - 2025 SP - 597 EP - 617 VL - 8 IS - 3 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.425/ DO - 10.5802/alco.425 LA - en ID - ALCO_2025__8_3_597_0 ER -
%0 Journal Article %A Brower, Cole %A McDonough, Joseph %A O’Neill, Christopher %T Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone %J Algebraic Combinatorics %D 2025 %P 597-617 %V 8 %N 3 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.425/ %R 10.5802/alco.425 %G en %F ALCO_2025__8_3_597_0
Brower, Cole; McDonough, Joseph; O’Neill, Christopher. Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone. Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 597-617. doi : 10.5802/alco.425. https://alco.centre-mersenne.org/articles/10.5802/alco.425/
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