{
  "id": "1012.5201",
  "version": "v1",
  "published": "2010-12-23T13:48:59.000Z",
  "updated": "2010-12-23T13:48:59.000Z",
  "title": "Upper bounds for the number of zeroes for some Abelian integrals",
  "authors": [
    "Armengol Gasull",
    "J. Tomás Lázaro",
    "Joan Torregrosa"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Consider the vector field $x'= -yG(x, y), y'=xG(x, y),$ where the set of critical points $\\{G(x, y) = 0\\}$ is formed by $K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree $n$ and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of $K$ and $n.$ Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for $K\\le4$ we recover or improve some results obtained in several previous works.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-23T13:48:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C08",
      "34C07",
      "34C23",
      "37C27"
    ],
    "keywords": [
      "abelian integral",
      "upper bounds",
      "general polynomial perturbation",
      "period annulus",
      "orthogonal directions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.5201G"
    }
  }
}