{
  "id": "2202.12054",
  "version": "v1",
  "published": "2022-02-24T12:20:34.000Z",
  "updated": "2022-02-24T12:20:34.000Z",
  "title": "On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms",
  "authors": [
    "Alfred Geroldinger",
    "Franz Halter-Koch",
    "Qinghai Zhong"
  ],
  "categories": [
    "math.NT",
    "math.AC",
    "math.CO"
  ],
  "abstract": "Let $G$ be an additive finite abelian group and $\\Gamma \\subset \\operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \\cdot \\ldots \\cdot g_{\\ell}$ over $G$ is a ($\\Gamma$-)weighted zero-sum sequence if there are $\\gamma_1, \\ldots, \\gamma_{\\ell} \\in \\Gamma$ such that $\\gamma_1 (g_1) + \\ldots + \\gamma_{\\ell} (g_{\\ell})=0$. We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group $\\Gamma$) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-24T12:20:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20M13",
      "11B50",
      "11B75",
      "11E12",
      "11R27"
    ],
    "keywords": [
      "weighted zero-sum sequence",
      "binary quadratic forms",
      "galois number fields",
      "norm monoids",
      "galois algebraic number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}