{
  "id": "2104.02520",
  "version": "v1",
  "published": "2021-04-06T14:53:23.000Z",
  "updated": "2021-04-06T14:53:23.000Z",
  "title": "$\\mathbb Q\\setminus\\mathbb Z$ is diophantine over $\\mathbb Q$ with 32 unknowns",
  "authors": [
    "Geng-Rui Zhang",
    "Zhi-Wei Sun"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by proving that $\\mathbb Q\\setminus\\mathbb Z$ is diophantine over $\\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\\ldots,x_{n})\\in\\mathbb Q[t,x_1,\\ldots,x_{n}]$ such that for any rational number $t$ we have $$t\\not\\in\\mathbb Z\\iff \\exists x_1\\cdots\\exists x_{n}[P(t,x_1,\\ldots,x_{n})=0]$$ where variables range over $\\mathbb Q$, equivalently $$t\\in\\mathbb Z\\iff \\forall x_1\\cdots\\forall x_{n}[P(t,x_1,\\ldots,x_{n})\\not=0].$$ In this paper we prove further that we may even take $n=32$ and require deg$\\,P<6\\times10^{11}$, which provides the best record in this direction. Combining this with a result of Sun, we get that there is no algorithm to decide for any $P(x_1,\\ldots,x_{41})\\in\\mathbb Z[x_1,\\ldots,x_{41}]$ whether $$\\forall x_1\\cdots\\forall x_9\\exists y_1\\cdots\\exists y_{32}[P(x_1,\\ldots,x_9,y_1,\\ldots,y_{32})=0].$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-06T14:53:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D35",
      "11U05",
      "03D25",
      "11D99",
      "11S99"
    ],
    "keywords": [
      "diophantine",
      "best record",
      "variables range",
      "rational number",
      "celebrated theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}