{
  "id": "1708.04800",
  "version": "v1",
  "published": "2017-08-16T08:21:33.000Z",
  "updated": "2017-08-16T08:21:33.000Z",
  "title": "Number systems over orders",
  "authors": [
    "Attila Pethő",
    "Jörg Thuswaldner"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{K}$ be a number field of degree $k$ and let $\\mathcal{O}$ be an order in $\\mathbb{K}$. A generalized number system over $\\mathcal{O}$ (GNS for short) is a pair $(p,\\mathcal{D})$ where $p \\in \\mathcal{O}[x]$ is monic and $\\mathcal{D}\\subset\\mathcal{O}$ is a complete residue system modulo $p(0)$. If each $a \\in \\mathcal{O}[x]$ admits a representation of the form $a \\equiv \\sum_{j =0}^{\\ell-1} d_j x^j \\pmod{p}$ with $\\ell\\in\\mathbb{N}$ and $d_0,\\ldots, d_{\\ell-1}\\in\\mathcal{D}$ then the GNS $(p,\\mathcal{D})$ is said to have the finiteness property. Using fundamental domains $\\mathcal{F}$ of the action of $\\mathbb{Z}^k$ on $\\mathbb{R}^k$ we define classes $\\mathcal{G}_\\mathcal{F} := \\{ (p, D_\\mathcal{F}) \\;:\\; p \\in \\mathcal{O}[x] \\}$ of GNS whose digit sets $D_\\mathcal{F}$ are defined in terms of $\\mathcal{F}$ in a natural way. We are able to prove general results on the finiteness property of GNS in $\\mathcal{G}_\\mathcal{F}$ by giving an abstract version of the well-known \"dominant condition\" on the absolute coefficient $p(0)$ of $p$. In particular, depending on mild conditions on the topology of $\\mathcal{F}$ we characterize the finiteness property of $(p(x\\pm m), D_\\mathcal{F})$ for fixed $p$ and large $m\\in\\mathbb{N}$. Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-16T08:21:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "52C22"
    ],
    "keywords": [
      "finiteness property",
      "general results",
      "number field",
      "complete residue system modulo",
      "power integral bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}