{
  "id": "math/0510444",
  "version": "v2",
  "published": "2005-10-20T17:45:57.000Z",
  "updated": "2005-12-13T17:23:39.000Z",
  "title": "Heights and preperiodic points of polynomials over function fields",
  "authors": [
    "Robert L. Benedetto"
  ],
  "comment": "9 pages; added references, corrected minor typos, updated definition of isotrivial for dynamical systems, added Proposition 5.1 to clarify the main proof",
  "journal": "IMRN 2005:62, 3855-3866",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let K be a function field in one variable over an arbitrary field F. Given a rational function f(z) in K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of f all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be non-preperiodic points of canonical height zero. In this paper, we show that for polynomial f, such points exist only if f is isotrivial. In fact, such K-rational points exist only if f is defined over the constant field of K after a K-rational change of coordinates.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-12-13T17:23:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "11D45",
      "37F10"
    ],
    "keywords": [
      "function field",
      "canonical height zero",
      "polynomial",
      "rational function",
      "arbitrary field"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10444B"
    }
  }
}