{
  "id": "2309.07793",
  "version": "v1",
  "published": "2023-09-14T15:30:43.000Z",
  "updated": "2023-09-14T15:30:43.000Z",
  "title": "On faces of the Kunz cone and the numerical semigroups within them",
  "authors": [
    "Levi Borevitz",
    "Tara Gomes",
    "Jiajie Ma",
    "Harper Niergarth",
    "Christopher O'Neill",
    "Daniel Pocklington",
    "Rosa Stolk",
    "Jessica Wang",
    "Shuhang Xue"
  ],
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "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 $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-14T15:30:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical semigroup",
      "kunz cone",
      "minimal generators",
      "rational polyhedral cone",
      "cofinite subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}