{
  "id": "1701.09051",
  "version": "v1",
  "published": "2017-01-31T14:20:33.000Z",
  "updated": "2017-01-31T14:20:33.000Z",
  "title": "Linear independence of values of G-functions",
  "authors": [
    "Stéphane Fischler",
    "Tanguy Rivoal"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given any non-polynomial $G$-function $F(z)=\\sum\\_{k=0}^\\infty A\\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\\_n^{[s]}(z)=\\sum\\_{k=0}^\\infty \\frac{A\\_k}{(k+n)^s}z^k$ for any integers $s\\geq 0$ and $n\\geq 1$. For any fixed algebraic number $\\alpha$ such that $0 \\textless{} \\vert \\alpha \\vert \\textless{} R$ and any number field $\\mathbb{K}$ containing $\\alpha$ and the $A\\_k$'s, we define $\\Phi\\_{\\alpha, S}$ as the $\\mathbb{K}$-vector space generated by the values $F\\_n^{[s]}(\\alpha)$, $n\\ge 1$ and $0\\leq s\\leq S$. We prove that $u\\_{\\mathbb{K},F}\\log(S)\\leq \\dim\\_{\\mathbb{K}}(\\Phi\\_{\\alpha, S})\\leq v\\_F S$ for any $S$, with effective constants $u\\_{\\mathbb{K},F}\\textgreater{}0$ and $v\\_F\\textgreater{}0$, and that the family $(F\\_n^{[s]}(\\alpha))\\_{1\\le n \\le v\\_F, s \\ge 0}$ contains infinitely many irrational numbers. This theorem applies in particular when $F$ is an hypergeometric series with rational parameters or a multiple polylogarithm, and it encompasses a previous result by the second author and Marcovecchio in the case of polylogarithms. The proof relies on an explicit construction of Pad\\'e-type approximants. It makes use of results of Andr\\'e, Chudnovsky and Katz on $G$-operators, of a new linear independence criterion \\`a la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-31T14:20:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "g-functions",
      "number field",
      "linear independence criterion",
      "saddle point method",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}