{
  "id": "math/9903106",
  "version": "v2",
  "published": "1999-03-17T22:51:13.000Z",
  "updated": "1999-03-18T17:05:26.000Z",
  "title": "Solving the sextic by iteration: A complex dynamical approach",
  "authors": [
    "Scott Crass",
    "Peter Doyle"
  ],
  "comment": "19 pages, 2 figures",
  "journal": "International Mathematics Research Notices, 1997, No.2, 83-99",
  "categories": [
    "math.DS"
  ],
  "abstract": "Recently, Peter Doyle and Curt McMullen devised an iterative solution to the fifth degree polynomial. At the method's core is a rational mapping of the Riemann sphere with the icosahedral symmetry of a general quintic. Moreover, this map posseses \"reliable\" dynamics: for almost any initial point, the its trajectory converges to one of the periodic cycles that comprise an icosahedral orbit. This symmetry-breaking provides for a reliable or \"generally-convergent\" quintic-solving algorithm: with almost any fifth-degree equation, associate a rational mapping that has reliable dynamics and whose attractor consists of points from which one computes a root. An algorithm that solves the sixth-degree equation requires a dynamical system with the symmetry of the alternating group on six things. This group does not act on the Riemmann sphere, but does act on the complex projective plane--this is the Valentiner group. The present work exploits the resulting 2-dimensional geometry in finding a Valentiner-symmetric rational mapping whose elegant dynamics experimentally appear to be reliable in the above sense---transferred to the 2-dimensional setting. This map provides the central feature of a conjecturally-reliable sextic-solving algorithm analogous to that employed in the quintic case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "1999-03-18T17:05:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex dynamical approach",
      "rational mapping",
      "fifth degree polynomial",
      "elegant dynamics experimentally appear",
      "icosahedral symmetry"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......3106C"
    }
  }
}