{
  "id": "1907.08585",
  "version": "v1",
  "published": "2019-07-19T17:25:36.000Z",
  "updated": "2019-07-19T17:25:36.000Z",
  "title": "Measuring the local non-convexity of real algebraic curves",
  "authors": [
    "Miruna-Stefana Sorea"
  ],
  "comment": "32 pages, 34 figures",
  "categories": [
    "math.AG",
    "math.CO",
    "math.GT",
    "math.MG",
    "math.OC"
  ],
  "abstract": "The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider sufficiently small levels of a real bivariate polynomial in a small enough neighbourhood of a strict local minimum at the origin of the real affine plane. We introduce and describe a new combinatorial object, called the Poincare-Reeb graph, whose role is to encode the shape of such curves and to allow us to quantify their non-convexity. Moreover, we prove that in this setting the Poincare-Reeb graph is a plane tree and can be used as a tool to study the asymptotic behaviour of level curves near a strict local minimum. Finally, using the real polar curve, we show that locally the shape of the levels stabilises and that no spiralling phenomena occur near the origin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-19T17:25:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14P25",
      "14P05",
      "14H20",
      "14B05",
      "05C05",
      "14Q05",
      "14Q05"
    ],
    "keywords": [
      "real algebraic curves",
      "local non-convexity",
      "strict local minimum",
      "real algebraic plane curves",
      "poincare-reeb graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}