{
  "id": "2110.08623",
  "version": "v2",
  "published": "2021-10-16T17:30:53.000Z",
  "updated": "2022-07-14T17:41:06.000Z",
  "title": "Sums of random polynomials with differing degrees",
  "authors": [
    "Isabelle Kraus",
    "Marcus Michelen",
    "Sean O'Rourke"
  ],
  "comment": "30 pages, 2 figures. Major update containing new methods and improved results",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mu$ and $\\nu$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $\\mu$ and $\\nu$, respectively. Under assumptions on the measures $\\mu$ and $\\nu$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as $n \\to \\infty$. In this paper, we generalize and extend this result to the case where $p$ and $q$ have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of $\\mu$ and $\\nu$, scaled by the limiting ratio of the degrees of $p$ and $q$. Additionally, our approach provides a complete description of the limiting distribution for the zeros of $p + q$ for any pair of measures $\\mu$ and $\\nu$, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-07-14T17:41:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "differing degrees",
      "limiting distribution",
      "logarithmic potential",
      "independent random polynomials",
      "third author"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}