{
  "id": "1301.5029",
  "version": "v1",
  "published": "2013-01-21T22:20:13.000Z",
  "updated": "2013-01-21T22:20:13.000Z",
  "title": "Markoff-Rosenberger triples in arithmetic progression",
  "authors": [
    "Enrique González-Jiménez",
    "José M. Tornero"
  ],
  "comment": "To appear in Journal of Symbolic Computation",
  "journal": "J. Symbolic Comput. 53 (2013), 53-63",
  "doi": "10.1016/j.jsc.2012.11.003",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 = dxyz (a generalization of the well--known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x^2+y^2+z^2 = dxyz over quadratic fields and the classic Markoff equation x^2+y^2+z^2 = 3xyz over an arbitrary number field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-21T22:20:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progression",
      "markoff-rosenberger triples",
      "arbitrary number field",
      "classic markoff equation",
      "complete decision algorithm"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.5029G"
    }
  }
}