{
  "id": "1609.09787",
  "version": "v1",
  "published": "2016-09-30T16:05:12.000Z",
  "updated": "2016-09-30T16:05:12.000Z",
  "title": "Universally and existentially definable subsets of global fields",
  "authors": [
    "Kirsten Eisentraeger",
    "Travis Morrison"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We show that rings of $S$-integers of a global function field $K$ of odd characteristic are first-order universally definable in $K$. This extends work of Koenigsmann and Park who showed the same for $\\mathbb{Z}$ in $\\mathbb{Q}$ and the ring of integers in a number field, respectively. We also give another proof of a theorem of Poonen and show that the set of non-squares in a global field of characteristic $\\neq 2$ is diophantine. Finally, we show that the set of pairs $(x,y)$ in $(K^{\\times})^2$ such that $x$ is not a norm in $K(\\sqrt{y})$ is diophantine over $K$ for any global field $K$ of characteristic $\\neq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-30T16:05:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "11R37"
    ],
    "keywords": [
      "global field",
      "existentially definable subsets",
      "global function field",
      "extends work",
      "number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}