{
  "id": "1610.06248",
  "version": "v1",
  "published": "2016-10-19T23:55:58.000Z",
  "updated": "2016-10-19T23:55:58.000Z",
  "title": "Pairing between zeros and critical points of random polynomials with independent roots",
  "authors": [
    "Sean O'Rourke",
    "Noah Williams"
  ],
  "comment": "38 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $p_n$ be a random, degree $n$ polynomial whose roots are chosen independently according to the probability measure $\\mu$ on the complex plane. For a deterministic point $\\xi$ lying outside the support of $\\mu$, we show that almost surely the polynomial $q_n(z):=p_n(z)(z - \\xi)$ has a critical point at distance $O(1/n)$ from $\\xi$. In other words, conditioning the random polynomials $p_n$ to have a root at $\\xi$, almost surely forces a critical point near $\\xi$. More generally, we prove an analogous result for the critical points of $q_n(z):=p_n(z)(z - \\xi_1)\\cdots (z - \\xi_k)$, where $\\xi_1, \\ldots, \\xi_k$ are deterministic. In addition, when $k=o(n)$, we show that the empirical distribution constructed from the critical points of $q_n$ converges to $\\mu$ in probability as the degree tends to infinity, extending a recent result of Kabluchko.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-19T23:55:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical point",
      "random polynomials",
      "independent roots",
      "probability measure",
      "deterministic point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}