{
  "id": "1806.00521",
  "version": "v1",
  "published": "2018-06-01T19:45:21.000Z",
  "updated": "2018-06-01T19:45:21.000Z",
  "title": "The lemniscate tree of a random polynomial",
  "authors": [
    "Michael Epstein",
    "Boris Hanin",
    "Erik Lundberg"
  ],
  "comment": "18 pages, 6 figures",
  "categories": [
    "math.PR",
    "math.CO",
    "math.CV"
  ],
  "abstract": "To each generic complex polynomial $p(z)$ there is associated a labeled binary tree (here referred to as a \"lemniscate tree\") that encodes the topological type of the graph of $|p(z)|$. The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the singular components of those level sets $|p(z)|=t$ passing through a critical point. In this paper, we address the question \"How many branches appear in a typical lemniscate tree?\" We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class and second for the lemniscate tree arising from a random polynomial generated by i.i.d. zeros. From a more general perspective, these results take a first step toward a probabilistic treatment (within a specialized setting) of Arnold's program of enumerating algebraic Morse functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-01T19:45:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "60G60",
      "31A15",
      "14P25",
      "05A15",
      "60C05",
      "60F05"
    ],
    "keywords": [
      "random polynomial",
      "generic complex polynomial",
      "enumerating algebraic morse functions",
      "labeled binary tree",
      "arnolds program"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}