{
  "id": "math/0311030",
  "version": "v2",
  "published": "2003-11-04T10:50:44.000Z",
  "updated": "2004-04-22T09:21:03.000Z",
  "title": "A lower bound for the height of a rational function at $S$-unit points",
  "authors": [
    "Pietro Corvaja",
    "Umberto Zannier"
  ],
  "comment": "Plain TeX 18 pages. Version 2; minor changes. To appear on Monatshefte fuer Mathematik",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Gamma$ be a finitely generated subgroup of the multiplicative group $\\G_m^2(\\bar{Q})$. Let $p(X,Y),q(X,Y)\\in\\bat{Q}$ be two coprime polynomials not both vanishing at $(0,0)$; let $\\epsilon>0$. We prove that, for all $(u,v)\\in\\Gamma$ outside a proper Zariski closed subset of $G_m^2$, the height of $p(u,v)/q(u,v)$ verifies $h(p(u,v)/q(u,v))>h(1:p(u,v):q(u,v))-\\epsilon \\max(h(uu),h(v))$. As a consequence, we deduce upper bounds for (a generalized notion of) the g.c.d. of $u-1,v-1$ for $u,v$ running over $\\Gamma$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-04-22T09:21:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J25"
    ],
    "keywords": [
      "rational function",
      "unit points",
      "lower bound",
      "proper zariski closed subset",
      "deduce upper bounds"
    ],
    "note": {
      "typesetting": "Plain TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....11030C"
    }
  }
}