{
  "id": "math/0512370",
  "version": "v1",
  "published": "2005-12-15T16:08:51.000Z",
  "updated": "2005-12-15T16:08:51.000Z",
  "title": "Elementary proof of the B. and M. Shapiro conjecture for rational functions",
  "authors": [
    "Alexandre Eremenko",
    "Andrei Gabrielov"
  ],
  "comment": "21 pages",
  "journal": "Notions of positivity and the geometry of polynomials, trends in mathematics, Springer, Basel, 2011, p. 167-178",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation L such that L(g) is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-12-15T16:08:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "14N10",
      "14P99",
      "26C15"
    ],
    "keywords": [
      "elementary proof",
      "shapiro conjecture",
      "fractional linear transformation",
      "real rational function",
      "fuchsian differential equations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12370E"
    }
  }
}