{
  "id": "2310.12530",
  "version": "v1",
  "published": "2023-10-19T07:11:32.000Z",
  "updated": "2023-10-19T07:11:32.000Z",
  "title": "The Frobenius problem over real number fields",
  "authors": [
    "Alex Feiner",
    "Zion Hefty"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT",
    "math.RA"
  ],
  "abstract": "Given a number field $K$ that is a subfield of the real numbers, we generalize the notion of the classical Frobenius problem to the ring of integers $\\mathfrak{O}_K$ of $K$ by describing certain Frobenius semigroups, $\\mathrm{Frob}(\\alpha_1,\\dots,\\alpha_n)$, for appropriate elements $\\alpha_1,\\dots,\\alpha_n\\in\\mathfrak{O}_K$. We construct a partial ordering on $\\mathrm{Frob}(\\alpha_1,\\dots,\\alpha_n)$, and show that this set is completely described by the maximal elements with respect to this ordering. We also show that $\\mathrm{Frob}(\\alpha_1,\\dots,\\alpha_n)$ will always have finitely many such maximal elements, but in general, the number of maximal elements can grow without bound as $n$ is fixed and $\\alpha_1,\\dots,\\alpha_n\\in\\mathfrak{O}_K$ vary. Explicit examples of the Frobenius semigroups are also calculated for certain cases in real quadratic number fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-19T07:11:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D07",
      "11H06",
      "06F05"
    ],
    "keywords": [
      "real number fields",
      "maximal elements",
      "real quadratic number fields",
      "frobenius semigroups",
      "classical frobenius problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}