{
  "id": "1601.07158",
  "version": "v1",
  "published": "2016-01-26T20:38:39.000Z",
  "updated": "2016-01-26T20:38:39.000Z",
  "title": "As Easy as $\\mathbb Q$: Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields",
  "authors": [
    "Kirsten Eisentraeger",
    "Russell Miller",
    "Jennifer Park",
    "Alexandra Shlapentokh"
  ],
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "Hilbert's Tenth Problem over the field $\\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\\mathbb Q$ having the property that Hilbert's Tenth Problem for $R$, denoted $HTP(R)$, is Turing equivalent to $HTP(\\mathbb Q)$. We are able to put several additional constraints on the rings $R$ that we construct. Given any computable nonnegative real number $r \\leq 1$ we construct such a ring $R = Z[\\frac1p : p \\in S]$ with $S$ a set of primes of lower density $r$. We also construct examples of rings $R$ for which deciding membership in $R$ is Turing equivalent to deciding $HTP(R)$ and also equivalent to deciding $HTP(\\mathbb Q)$. Alternatively, we can make $HTP(R)$ have arbitrary computably enumerable degree above $HTP(\\mathbb Q)$. Finally, we show that the same can be done for subrings of number fields and their prime ideals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-26T20:38:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "12L05",
      "03D45"
    ],
    "keywords": [
      "number fields",
      "biggest open problems",
      "turing equivalent",
      "arbitrary computably enumerable degree",
      "construct examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}