{
  "id": "1211.3958",
  "version": "v1",
  "published": "2012-11-16T17:18:46.000Z",
  "updated": "2012-11-16T17:18:46.000Z",
  "title": "Asymptotic Results for Random Polynomials on the Unit Circle",
  "authors": [
    "Gabriel H. Tucci",
    "Philip A. Whiting"
  ],
  "comment": "14 pages. arXiv admin note: text overlap with arXiv:1202.3184",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study the asymptotic behavior of the maximum magnitude of a complex random polynomial with i.i.d. uniformly distributed random roots on the unit circle. More specifically, let $\\{n_k\\}_{k=1}^{\\infty}$ be an infinite sequence of positive integers and let $\\{z_{k}\\}_{k=1}^{\\infty}$ be a sequence of i.i.d. uniform distributed random variables on the unit circle. The above pair of sequences determine a sequence of random polynomials $P_{N}(z) = \\prod_{k=1}^{N}{(z-z_k)^{n_k}}$ with random roots on the unit circle and their corresponding multiplicities. In this work, we show that subject to a certain regularity condition on the sequence $\\{n_k\\}_{k=1}^{\\infty}$, the log maximum magnitude of these polynomials scales as $s_{N}I^{*}$ where $s_{N}^{2}=\\sum_{k=1}^{N}{n_{k}^{2}}$ and $I^{*}$ is a strictly positive random variable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-16T17:18:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit circle",
      "asymptotic results",
      "uniform distributed random variables",
      "log maximum magnitude",
      "complex random polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.3958T"
    }
  }
}