{
  "id": "2111.10875",
  "version": "v2",
  "published": "2021-11-21T18:49:50.000Z",
  "updated": "2023-02-19T22:18:51.000Z",
  "title": "The number of real zeros of elliptic polynomials",
  "authors": [
    "Nhan D. V. Nguyen"
  ],
  "comment": "33 pages, 2 figures, 1 table, references clarified, minor edits and corrections",
  "categories": [
    "math.PR"
  ],
  "abstract": "The aim of this paper is to further explore the number $N_n(a,b)$ of real zeros of elliptic polynomials of degree $n$ on an arbitrary interval $(a,b)$, where $a$ and $b$ may depend on $n$. We first develop an exact and accessible formula for the variance of $N_n(a,b)$ and exploit it to derive all terms in the large $n$ asymptotic expansion. We then provide sharp estimates for the cumulants and central moments of this quantity. These estimates play an important role in determining sufficient conditions on the interval $(a,b)$ under which $N_n(a,b)$ satisfies a central limit theorem and a strong law of large numbers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-02-19T22:18:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F05",
      "41A60",
      "60G15"
    ],
    "keywords": [
      "real zeros",
      "elliptic polynomials",
      "central limit theorem",
      "large numbers",
      "arbitrary interval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}