{
  "id": "1911.12803",
  "version": "v1",
  "published": "2019-11-28T17:35:04.000Z",
  "updated": "2019-11-28T17:35:04.000Z",
  "title": "Separatrices for real analytic vector fields in the plane",
  "authors": [
    "Eduardo Cabrera",
    "Rogério Mol"
  ],
  "comment": "16 pages",
  "categories": [
    "math.DS",
    "math.CA",
    "math.CV"
  ],
  "abstract": "Let $X$ be a germ of real analytic vector field at $({\\mathbb R}^{2},0)$ with an algebracally isolated singularity. We say that $X$ is a topological generalized curve if there are no topological saddle-nodes in its reduction of singularities. In this case, we prove that if either the order $\\nu_{0}(X)$ or the Milnor number $\\mu_{0}(X)$ is even, then $X$ has a formal separatrix, that is, a formal invariant curve at $0 \\in {\\mathbb R}^{2}$. This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-28T17:35:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S65",
      "37F75",
      "34Cxx",
      "14P15"
    ],
    "keywords": [
      "real analytic vector field",
      "formal invariant curve",
      "singularity",
      "formal separatrix",
      "milnor number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}