{
  "id": "2404.07971",
  "version": "v1",
  "published": "2024-04-11T17:55:57.000Z",
  "updated": "2024-04-11T17:55:57.000Z",
  "title": "The Newman algorithm for constructing polynomials with restricted coefficients and many real roots",
  "authors": [
    "Markus Jacob",
    "Fedor Nazarov"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\\subset\\mathbb{R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\\sum_{k=1}^n\\varepsilon_k x^k$, $\\varepsilon_k\\in E_k$, with at least $c\\sqrt{n}$ distinct roots in $[0,1]$, which matches the classical upper bound up to the value of the constant $c>0$. Our sufficient conditions cover the Littlewood ($E_k=\\{-1,1\\}$) and Newman ($E_k=\\{0,(-1)^k\\}$) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence $E_k$ is periodic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-11T17:55:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real roots",
      "newman algorithm",
      "constructing polynomials",
      "restricted coefficients",
      "natural sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}