{
  "id": "2212.11867",
  "version": "v1",
  "published": "2022-12-22T17:01:56.000Z",
  "updated": "2022-12-22T17:01:56.000Z",
  "title": "Zeros of a growing number of derivatives of random polynomials with independent roots",
  "authors": [
    "Marcus Michelen",
    "Xuan-Truong Vu"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR",
    "math.CA",
    "math.CV"
  ],
  "abstract": "Let $X_1,X_2,\\ldots$ be independent and identically distributed random variables in $\\mathbb{C}$ chosen from a probability measure $\\mu$ and define the random polynomial $$ P_n(z)=(z-X_1)\\ldots(z-X_n)\\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \\leq \\log n / (5 \\log\\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $\\mu$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy who proved the fixed $k$ case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-22T17:01:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random polynomial",
      "independent roots",
      "growing number",
      "derivative",
      "extends work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}