{
  "id": "1306.6760",
  "version": "v1",
  "published": "2013-06-28T09:13:12.000Z",
  "updated": "2013-06-28T09:13:12.000Z",
  "title": "An existential 0-definition of F_q[[t]] in F_q((t))",
  "authors": [
    "Will Anscombe",
    "Jochen Koenigsmann"
  ],
  "comment": "9 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that the valuation ring F_q[[t]] in the local field F_q((t)) is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an existential-F_q-definable bounded neighbouhood of 0. Then we `tweak' this set by subtracting, taking roots, and applying Hensel's Lemma in order to find an existential-F_q-definable subset of F_q[[t]] which contains tF_q[[t]]. Finally, we use the fact that F_q is defined by the formula x^q-x=0 to extend the definition to the whole of F_q[[t]] and to rid the definition of parameters. Several extensions of the theorem are obtained, notably an existential 0-definition of the valuation ring of a non-trivial valuation with divisible value group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-28T09:13:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C60",
      "12L12"
    ],
    "keywords": [
      "existential",
      "definition",
      "parameters",
      "local field",
      "henselian topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6760A"
    }
  }
}