{
  "id": "math/0601043",
  "version": "v1",
  "published": "2006-01-03T13:32:36.000Z",
  "updated": "2006-01-03T13:32:36.000Z",
  "title": "Variation of argument and Bernstein index for holomorphic functions on Riemann surfaces",
  "authors": [
    "Yulij Ilyashenko"
  ],
  "comment": "11 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "An upper bound of the variation of argument of a holomorphic function along a curve on a Riemann surface is given. This bound is expressed through the Bernstein index of the function multiplied by a geometric constant. The Bernstein index characterizes growth of the function from a smaller domain to a larger one. The geometric constant in the estimate is explicitly given. This result is applied in \\cite {GI} to the solution of the restricted version of the infinitesimal Hilbert 16th problem, namely, to upper estimates of the number of zeros of abelian integrals in complex domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-03T13:32:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58F21"
    ],
    "keywords": [
      "holomorphic function",
      "riemann surface",
      "bernstein index characterizes growth",
      "geometric constant",
      "infinitesimal hilbert 16th problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1043I"
    }
  }
}