{
  "id": "2505.02723",
  "version": "v1",
  "published": "2025-05-05T15:24:24.000Z",
  "updated": "2025-05-05T15:24:24.000Z",
  "title": "Limit law for root separation in random polynomials",
  "authors": [
    "Marcus Michelen",
    "Oren Yakir"
  ],
  "comment": "77 pages",
  "categories": [
    "math.PR",
    "math.CA",
    "math.CV"
  ],
  "abstract": "Let $f_n$ be a random polynomial of degree $n\\ge 2$ whose coefficients are independent and identically distributed random variables. We study the separation distances between roots of $f_n$ and prove that the set of these distances, normalized by $n^{-5/4}$, converges in distribution as $n\\to \\infty$ to a non-homogeneous Poisson point process. As a corollary, we deduce that the minimal separation distance between roots of $f_n$, normalized by $n^{-5/4}$ has a non-trivial limit law. In the course of the proof, we establish a related result which may be of independent interest: a Taylor series with random i.i.d. coefficients almost-surely does not have a double zero anywhere other than the origin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-05T15:24:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random polynomial",
      "root separation",
      "non-trivial limit law",
      "minimal separation distance",
      "non-homogeneous poisson point process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 77,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}