{
  "id": "1706.03302",
  "version": "v1",
  "published": "2017-06-11T03:08:03.000Z",
  "updated": "2017-06-11T03:08:03.000Z",
  "title": "On existential definitions of C.E. subsets of rings of functions of characteristic 0",
  "authors": [
    "Russell Miller",
    "Alexandra Shlapentokh"
  ],
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (2) Every c.e. set of integers has a finite-fold Diophantine definition over the ring of integral functions of any function field of characteristic $0$. (3) All c.e. subsets of polynomial rings over totally real number fields have finite-fold Diophantine definitions. (These are the first examples of infinite rings with this property.) (4) Let $K$ be a one-variable function field over a number field and let $p$ be any prime of $K$. Then the valuation ring of $p$ has a Diophantine definition. (5) Let $K$ be a one-variable function field over a number field and let $S$ be a finite set of its primes. Then all c.e. subsets of $O_{K,S}$ are existentially definable. (Here $O_{K,S}$ is the ring of $S$-integers or a ring of integral functions.)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-11T03:08:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "12L05",
      "03B25"
    ],
    "keywords": [
      "existential definitions",
      "finite-fold diophantine definition",
      "integral functions",
      "characteristic",
      "one-variable function field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}