{
  "id": "1510.06291",
  "version": "v1",
  "published": "2015-10-20T01:50:16.000Z",
  "updated": "2015-10-20T01:50:16.000Z",
  "title": "Certain rational functions on the Riemann sphere with more than three branch points",
  "authors": [
    "Ji-jian Song",
    "Bin Xu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GT",
    "math.AG",
    "math.GR"
  ],
  "abstract": "Consider a collection $\\Lambda$ of partitions of a positive integer $d$ with form $$(a_1,\\cdots, a_p),\\,(b_1,\\cdots, b_q),\\,(m_1+1,1,\\cdots,1),\\cdots, (m_l+1,1,\\cdots,1),$$ where $(m_1,\\cdots, m_l)$ is a partition of $p+q-2>0$. We prove that there exists a rational function on the Riemann sphere with branch data $\\Lambda$ if and only if $\\max\\bigl(m_1,\\cdots,m_l\\bigr)$ is less than $d/{\\rm GCD}(a_1,\\cdots, a_p,b_1,\\cdots, b_q)$, which generalizes a theorem of G. Boccara in Discrete Math.(1982). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-20T01:50:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann sphere",
      "rational function",
      "branch points",
      "branch data",
      "belyi functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151006291S"
    }
  }
}