{
  "id": "1104.5360",
  "version": "v1",
  "published": "2011-04-28T11:37:25.000Z",
  "updated": "2011-04-28T11:37:25.000Z",
  "title": "On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients",
  "authors": [
    "Friedrich Götze",
    "Dmitry Zaporozhets"
  ],
  "comment": "8 pages",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "Consider a random polynomial $G_n(z)=\\xi_nz^n+...+\\xi_1z+\\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\\log(1+\\log(1+|\\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\\to\\infty$, to two centered circles and is uniform in the argument as $n\\to\\infty$. The radii of the circles are $|\\xi_0/\\xi_\\tau|^{1/\\tau}$ and $|\\xi_\\tau/\\xi_n|^{1/(n-\\tau)}$, where $\\xi_\\tau$ denotes the coefficient with the maximum modulus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-28T11:37:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15"
    ],
    "keywords": [
      "random polynomial",
      "complex roots",
      "heavy-tailed coefficients",
      "distribution",
      "maximum modulus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.5360G"
    }
  }
}