{
  "id": "1608.02455",
  "version": "v1",
  "published": "2016-08-08T14:39:24.000Z",
  "updated": "2016-08-08T14:39:24.000Z",
  "title": "Zeroes and rational points of analytic functions",
  "authors": [
    "Georges Comte",
    "Yosef Yomdin"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "For an analytic function $f(z)=\\sum_{k=0}^\\infty a_kz^k$ on a neighbourhood of a closed disc $D\\subset {\\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the graph $\\Gamma_f$ of $f_{\\vert D}$ and algebraic curves of degree $d$ is polynomially bounded in $d$. In particular, we show these assumptions are satisfied for random power series, for some explicit classes of lacunary series, and for solutions of linear differential equations with coefficients in ${\\bf Q}[z]$. As a consequence, for any function $f$ in these families, $\\Gamma_f$ has less than $\\beta \\log^\\alpha T$ rational points of height at most $T$, for some $\\alpha, \\beta >0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-08T14:39:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14P05",
      "11G50"
    ],
    "keywords": [
      "analytic function",
      "rational points",
      "random power series",
      "linear differential equations",
      "intersection points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}