{
  "id": "2304.07207",
  "version": "v1",
  "published": "2023-04-14T15:47:00.000Z",
  "updated": "2023-04-14T15:47:00.000Z",
  "title": "A Combinatorial Presentation for Branched Coverings of the 2-Sphere",
  "authors": [
    "Arcelino Bruno Lobato Do Nascimento"
  ],
  "comment": "29 pages, 41 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "William Thurston (1946-2012) gave a combinatorial characterization for generic branched self-coverings of the two-sphere by associating a planar graph to them 10.48550/arXiv.1502.04760. By generalizing the notion of local balancing, the author extends the Thurston result to encompass any branched covering of the two-sphere. As an application, we supply a lower bound for the number of equivalence classes of real rational functions for each given ramification profile. Furthermore, as a consequence, we obtain a new proof for a theorem ( 10.2307/3062151 , 10.4007/annals.2009.170.863 , 10.1090/S0894-0347-09-00640-7 ) that corresponds to a special case of a reality problem in enumerative geometry which was known as the B. \\& M. Shapiro Conjecture, now it is a theorem \\cite{MR2552110}. The theorem version that we prove concerns generic rational functions, assuring that if all critical points of that function are real, then we can transform it into a rational map with real coefficients by post-composition with an automorphism of $\\mathbb{C}\\mathbb{P}^{1}$. The proof we present is constructive and founded on elementary arguments.}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-14T15:47:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M12",
      "57M15",
      "05C15"
    ],
    "keywords": [
      "combinatorial presentation",
      "branched covering",
      "concerns generic rational functions",
      "real rational functions",
      "planar graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}