{
  "id": "1601.06417",
  "version": "v1",
  "published": "2016-01-24T18:59:37.000Z",
  "updated": "2016-01-24T18:59:37.000Z",
  "title": "Pairing of Zeros and Critical Points for Random Polynomials",
  "authors": [
    "Boris Hanin"
  ],
  "comment": "v1 comments welcome",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "Let p_N be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure mu on the Riemann sphere S^2. This article proves that if we condition p_N to have a zero at some fixed point xi in , then, with high probability, there will be a critical point w_xi a distance 1/N away from xi. This 1/N distance is much smaller than the one over root N typical spacing between nearest neighbors for N i.i.d. points on S^2. Moreover, with the same high probability, the argument of w_xi relative to xi is a deterministic function of mu plus fluctuations on the order of 1/N.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-24T18:59:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical point",
      "random polynomials",
      "high probability",
      "fixed probability measure mu",
      "mu plus fluctuations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160106417H"
    }
  }
}