{
  "id": "1511.06689",
  "version": "v1",
  "published": "2015-11-19T19:44:09.000Z",
  "updated": "2015-11-19T19:44:09.000Z",
  "title": "A hypothetical algorithm which computes an upper bound for the heights of solutions of a Diophantine equation whose solution set is finite",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let f(n)=1 if n=1, 2^(2^(n-2)) if n \\in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \\in {6,7,8,...}. We conjecture that if a system T \\subseteq {x_i+1=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \\leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite. We show that the conjecture and Matiyasevich's conjecture on finite-fold Diophantine representations are jointly inconsistent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-19T19:44:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05"
    ],
    "keywords": [
      "diophantine equation",
      "solution set",
      "upper bound",
      "hypothetical algorithm",
      "finite-fold diophantine representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}