{
  "id": "1612.03774",
  "version": "v1",
  "published": "2016-12-12T16:37:43.000Z",
  "updated": "2016-12-12T16:37:43.000Z",
  "title": "Root sets of polynomials and power series with finite choices of coefficients",
  "authors": [
    "Han Yu"
  ],
  "comment": "19 pages, 3 figures",
  "categories": [
    "math.NT",
    "math.CA",
    "math.MG"
  ],
  "abstract": "In this paper we consider the root set of power series with finite choices of coefficients: $\\{z\\in\\mathbb{C}| \\exists a_n\\in H, \\sum_{n=0}^{\\infty} a_nz^n=0\\}$ where $H\\subset\\mathbb{C}$ is a finite subset. For $H=\\{\\pm 1\\}$, the root set of Littlewood series is under consideration for a long while. Here we study the case when $H=\\{e^{i\\frac{2\\pi}{p}k},k=0,1,2,...,p-1\\}$ is the set of all roots of unity with prime order. We show that the correponding root set contains annulus $\\{z| l_p<|z|\\leq 1\\}$ where $l_p\\to \\frac{1}{2}$ for $p\\to\\infty$. Intuitively speaking, power series with more coefficients has significantly larger root set. A corollary of this result is that for large $p$, the root set of polynomials with coefficients in $H$ is dense in $\\{z| l_p<|z|\\leq \\frac{1}{l_q}\\}$ where $l_p\\to \\frac{1}{2}$ for $p\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-12T16:37:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B30"
    ],
    "keywords": [
      "power series",
      "finite choices",
      "coefficients",
      "polynomials",
      "correponding root set contains annulus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}