{
  "id": "2011.09187",
  "version": "v1",
  "published": "2020-11-18T10:20:09.000Z",
  "updated": "2020-11-18T10:20:09.000Z",
  "title": "The Buchweitz set of a numerical semigroup",
  "authors": [
    "S. Eliahou",
    "J. I. García-García",
    "D. Marín-Aragón",
    "A. Vigneron-Tenorio"
  ],
  "categories": [
    "math.CO",
    "math.AC",
    "math.NT"
  ],
  "abstract": "Let $A \\subset {\\mathbb Z}$ be a finite subset. We denote by $\\mathcal{B}(A)$ the set of all integers $n \\ge 2$ such that $|nA| > (2n-1)(|A|-1)$, where $nA=A+\\cdots+A$ denotes the $n$-fold sumset of $A$. The motivation to consider $\\mathcal{B}(A)$ stems from Buchweitz's discovery in 1980 that if a numerical semigroup $S \\subseteq {\\mathbb N}$ is a Weierstrass semigroup, then $\\mathcal{B}({\\mathbb N} \\setminus S) = \\emptyset$. By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (1893). In this paper, we prove that for any numerical semigroup $S \\subset {\\mathbb N}$ of genus $g \\ge 2$, the set $\\mathcal{B}({\\mathbb N} \\setminus S) $ is finite, of unbounded cardinality as $S$ varies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-18T10:20:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H55",
      "11P70",
      "20M14"
    ],
    "keywords": [
      "numerical semigroup",
      "buchweitz set",
      "finite subset",
      "condition fails",
      "weierstrass semigroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}