{
  "id": "1202.5785",
  "version": "v1",
  "published": "2012-02-26T18:04:23.000Z",
  "updated": "2012-02-26T18:04:23.000Z",
  "title": "On Certain Computations of Pisot Numbers",
  "authors": [
    "Qi Cheng",
    "Jincheng Zhuang"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper presents two algorithms on certain computations about Pisot numbers. Firstly, we develop an algorithm that finds a Pisot number $\\alpha$ such that $\\Q[\\alpha] = \\F$ given a real Galois extension $\\F$ of $\\Q$ by its integral basis. This algorithm is based on the lattice reduction, and it runs in time polynomial in the size of the integral basis. Next, we show that for a fixed Pisot number $\\alpha$, one can compute $ [\\alpha^n] \\pmod{m}$ in time polynomial in $(\\log (m n))^{O(1)}$, where $m$ and $n$ are positive integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-26T18:04:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16"
    ],
    "keywords": [
      "computations",
      "time polynomial",
      "integral basis",
      "real galois extension",
      "fixed pisot number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}