{
  "id": "2407.16120",
  "version": "v1",
  "published": "2024-07-23T02:03:28.000Z",
  "updated": "2024-07-23T02:03:28.000Z",
  "title": "Extensions of the Bloch -- Pólya Theorem on the number of real zeros of polynomials (II)",
  "authors": [
    "Tamás Erdélyi"
  ],
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that for every $$a_0,a_1, \\ldots,a_n \\in [1,M]\\,, \\qquad 1 \\leq M \\leq \\exp(c_2n)\\,,$$ there are $$b_0,b_1,\\ldots,b_n \\in \\{-1,0,1\\}$$ such that the polynomial $\\displaystyle{P(z) = \\sum_{j=0}^n{b_ja_jz^j}}$ has at least $$c_1 \\left( \\frac{n}{\\log(4M)} \\right)^{1/2}$$ distinct sign changes in $I_a := (1-2a,1-a)$, where $\\displaystyle{a := \\left( \\frac{\\log(4M)}{n} \\right)^{1/2} \\leq 1/3}$. This improves and extends earlier results of Bloch and P\\'olya and Erd\\'elyi, and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-23T02:03:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10",
      "12D10"
    ],
    "keywords": [
      "real zeros",
      "pólya theorem",
      "polynomial",
      "extensions",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}