{
  "id": "math/0401231",
  "version": "v1",
  "published": "2004-01-19T15:02:29.000Z",
  "updated": "2004-01-19T15:02:29.000Z",
  "title": "Linear equations with unknowns from a multiplicative group in a function field",
  "authors": [
    "Jan-Hendrik Evertse",
    "Umberto Zannier"
  ],
  "comment": "15 pages, LaTeX file",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let k be an algebraically closed field of characteristic 0, let K/k be a transcendental extension of arbitrary transcendence degree and let G be a multiplicative subgroup of (K^*)^n such that (k^*)^n is contained in G, and G/(k^*)^n has finite rank r. We consider linear equations a1x1+...+anxn=1 (*) with fixed non-zero coefficients a1,...,an from K, and with unknowns (x1,...,xn) from the group G. Such a solution is called degenerate if there is a subset of a1x1,...,anxn whose sum equals 0. Two solutions (x1,...,xn), (y1,...,yn) are said to belong to the same (k^*)^n-coset if there are c1,...,cn in k^* such that y1=c1*x1,...,yn=cn*xn. We show that the non-degenerate solutions of (*) lie in at most 1+C(3,2)^r+C(4,2)^r+...+C(n+1,2)^r (k^*)^n-cosets, where C(a,b) denotes the binomial coefficient a choose b.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-19T15:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D72"
    ],
    "keywords": [
      "linear equations",
      "function field",
      "multiplicative group",
      "arbitrary transcendence degree",
      "fixed non-zero coefficients a1"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1231E"
    }
  }
}