{
  "id": "1909.07939",
  "version": "v1",
  "published": "2019-09-17T17:23:29.000Z",
  "updated": "2019-09-17T17:23:29.000Z",
  "title": "Sums of random polynomials with independent roots",
  "authors": [
    "Sean O'Rourke",
    "Tulasi Ram Reddy"
  ],
  "comment": "21 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability measures $\\mu$ and $\\nu$ in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum $p+q$ as $n$ tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of $\\mu$ and $\\nu$. More generally, we consider the sum of $m$ independent degree $n$ random polynomials when $m$ is fixed and $n$ tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-17T17:23:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independent roots",
      "independent random polynomials",
      "logarithmic potential",
      "free probability theory",
      "limiting distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}