{
  "id": "math/0703907",
  "version": "v1",
  "published": "2007-03-30T06:24:05.000Z",
  "updated": "2007-03-30T06:24:05.000Z",
  "title": "Characterizing integers among rational numbers with a universal-existential formula",
  "authors": [
    "Bjorn Poonen"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-30T06:24:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "11R52"
    ],
    "keywords": [
      "rational numbers",
      "universal-existential formula",
      "characterizing integers",
      "universal quantifiers",
      "existential quantifiers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3907P"
    }
  }
}