{
  "id": "1805.04085",
  "version": "v1",
  "published": "2018-05-10T17:41:57.000Z",
  "updated": "2018-05-10T17:41:57.000Z",
  "title": "Diophantine problems in solvable groups",
  "authors": [
    "Albert Garreta",
    "Alexei Miasnikov",
    "Denis Ovchinnikov"
  ],
  "comment": "21 pages",
  "categories": [
    "math.GR",
    "math.LO",
    "math.NT"
  ],
  "abstract": "We study systems of equations in different classes of solvable groups. For each group $G$ in one of these classes we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by systems of equations (e-interpretable). This leads to the conjecture that $\\mathbb{Z}$ is e-interpretable in $G$ and that the Diophantine problem in $G$ is undecidable. %This stems from a long standing conjecture which states the same for the ring $O$. We further prove that $\\mathbb{Z}$ is e-interpretable in any generalized Heisenberg group and in any finitely generated nonabelian free (solvable-by-nilpotent) group. The latter applies in particular to the case of free solvable groups and to the already known case of free nilpotent groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-10T17:41:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F70",
      "20F10",
      "03B25",
      "03D35",
      "20F18",
      "20F16"
    ],
    "keywords": [
      "diophantine problem",
      "free nilpotent groups",
      "algebraic integers",
      "free solvable groups",
      "study systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}