{
  "id": "2007.04757",
  "version": "v1",
  "published": "2020-07-09T13:06:10.000Z",
  "updated": "2020-07-09T13:06:10.000Z",
  "title": "The local Poincaré problem for irreducible branches",
  "authors": [
    "José Cano",
    "Pedro Fortuny Ayuso",
    "Javier Ribón"
  ],
  "comment": "18 pages, 5 figures, accepted for publication in Revista Matem\\'atica Iberoamericana",
  "categories": [
    "math.AG",
    "math.CA",
    "math.CV",
    "math.DS"
  ],
  "abstract": "Let ${\\mathcal F}$ be a germ of holomorphic foliation defined in a neighborhood of the origin of ${\\mathbb C}^{2}$ that has a germ of irreducible holomorphic invariant curve $\\gamma$. We provide a lower bound for the vanishing multiplicity of ${\\mathcal F}$ at the origin in terms of the equisingularity class of $\\gamma$. Moreover, we show that such a lower bound is sharp. Finally, we characterize the types of dicritical singularities for which the multiplicity of $\\mathcal{F}$ can be bounded in terms of that of $\\gamma$ and provide an explicit bound in this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-09T13:06:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S05",
      "32S65",
      "14H20"
    ],
    "keywords": [
      "irreducible branches",
      "lower bound",
      "irreducible holomorphic invariant curve",
      "equisingularity class",
      "multiplicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}