{
  "id": "2401.06025",
  "version": "v1",
  "published": "2024-01-11T16:33:45.000Z",
  "updated": "2024-01-11T16:33:45.000Z",
  "title": "Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone",
  "authors": [
    "Cole Brower",
    "Joseph McDonough",
    "Christopher O'Neill"
  ],
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "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 points 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-11T16:33:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical semigroup",
      "kunz cone",
      "kunz fan",
      "highest minimal presentation cardinality",
      "polyhedral cone complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}