{
  "id": "1807.02140",
  "version": "v1",
  "published": "2018-07-05T18:34:09.000Z",
  "updated": "2018-07-05T18:34:09.000Z",
  "title": "Distances between zeroes and critical points for random polynomials with i.i.d. zeroes",
  "authors": [
    "Zakhar Kabluchko",
    "Hauke Seidel"
  ],
  "comment": "28 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\\xi_0,\\xi_1,\\ldots,\\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\\to\\infty$, with high probability there is a critical point of $Q_n$ which is very close to $\\xi_0$. We localize the position of this critical point by proving that the difference between $\\xi_0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\\xi_0))$ and variance of order $\\log n \\cdot n^{-3}$. Here, $f(z)= \\mathbb E[1/(z-\\xi_k)]$ is the Cauchy-Stieltjes transform of the $\\xi_k$'s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-05T18:34:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "60G57",
      "60B10"
    ],
    "keywords": [
      "critical point",
      "random polynomial",
      "approximately complex gaussian distribution",
      "random variables",
      "asymptotic regime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}