{
  "id": "2208.14090",
  "version": "v1",
  "published": "2022-08-30T09:08:47.000Z",
  "updated": "2022-08-30T09:08:47.000Z",
  "title": "Bounds for invariants of numerical semigroups and Wilf's Conjecture",
  "authors": [
    "Marco D'Anna",
    "Alessio Moscariello"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Given coprime positive integers $g_1 < \\ldots < g_e$, the Frobenius number $F=F(g_1,\\ldots,g_e)$ is the largest integer not representable as a linear combination of $g_1,\\ldots,g_e$ with non-negative integer coefficients. Let $n$ denote the number of all representable non-negative integers less than $F$; Wilf conjectured that $F+1 \\le e n$. We provide bounds for $g_1$ and for the type of the numerical semigroup $S=\\langle g_1,\\ldots,g_e \\rangle$ in function of $e$ and $n$, and use these bounds to prove that $F+1 \\le q e n$, where $q= \\left \\lceil \\frac{F+1}{g_1} \\right \\rceil$, and $F+1 \\le e n^2$. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup $S=\\langle g_1,\\ldots,g_e \\rangle$ is almost-symmetric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-30T09:08:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A99",
      "11B75",
      "20M14"
    ],
    "keywords": [
      "numerical semigroup",
      "wilfs conjecture",
      "invariants",
      "frobenius number",
      "non-negative integer coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}