{
  "id": "math/0509727",
  "version": "v1",
  "published": "2005-09-30T15:59:21.000Z",
  "updated": "2005-09-30T15:59:21.000Z",
  "title": "Upper bounds of topology of complex polynomials in two variables",
  "authors": [
    "Alexey Glutsyuk"
  ],
  "comment": "51 pages",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "The paper deals with a complex polynomial $H$ in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial $H$ has at least two distinct critical values. We prove quantitative versions of this statement. Supposing $H$ appropriately normalized (by affine coordinate changes in the image and in the source) we prove upper bounds for the following quantities: - the sum of the coefficients of the lower terms; - the minimal size of a bidisc containing all the nontrivial topology of a given level curve $S_t=\\{ H=t\\}$; - the minimal lengths of representatives of cycles in $H_1(S_t,\\zz)$ vanishing along appropriate paths from $t$ to the critical values of $H$; - the intersection indices of the latter cycles. All these results (expect for the latter bound) are used in my joint work with Yu.S.Ilyashenko \"Restricted version of the Hilbert 16-th problem\" (available on the arxiv). In the latter paper we obtain an explicit upper bound of the number of zeros for a wide class of Abelian integrals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-09-30T15:59:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D05",
      "14D05",
      "30C15",
      "30C10"
    ],
    "keywords": [
      "complex polynomial",
      "explicit upper bound",
      "affine coordinate changes",
      "multiple zero lines",
      "generic highest homogeneous part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9727G"
    }
  }
}