{
  "id": "1101.4255",
  "version": "v1",
  "published": "2011-01-22T03:02:01.000Z",
  "updated": "2011-01-22T03:02:01.000Z",
  "title": "Maximum Gap in (Inverse) Cyclotomic Polynomial",
  "authors": [
    "Hoon Hong",
    "Eunjeong Lee",
    "Hyang-Sook Lee",
    "Cheol-Min Park"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $\\Phi_n$ denote the $n$-th cyclotomic polynomial and let $\\Psi_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(\\Phi_n)$ and $g(\\Psi_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(\\Phi_{p_1})$, $g(\\Psi_{p_1})$ and $g(\\Psi_{p_1p_2})$. Hence the simplest non-trivial cases are $g(\\Phi_{p_1p_2})$ and $g(\\Psi_{p_1p_2p_3})$. We provide an exact expression for $g(\\Phi_{p_1p_2}).$ We also provide an exact expression for $g(\\Psi_{p_1p_2p_3})$ under a mild condition. The condition is almost always satisfied (only finite exceptions for each $p_1$). We also provide a lower bound and an upper bound for $g(\\Psi_{p_1p_2p_3})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-22T03:02:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum gap",
      "th inverse cyclotomic polynomial",
      "exact expression",
      "simplest non-trivial cases",
      "th cyclotomic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}