{
  "id": "1601.06867",
  "version": "v1",
  "published": "2016-01-26T02:17:03.000Z",
  "updated": "2016-01-26T02:17:03.000Z",
  "title": "Irreducible polynomials with several prescribed coefficients",
  "authors": [
    "Junsoo Ha"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the number of irreducible polynomials over $\\mathbf{F}_{q}$ with some coefficients prescribed. Using the technique developed by Bourgain, we show that there is an irreducible polynomial of degree $n$ with $r$ coefficients prescribed in any location when $r \\leq \\left[\\left(1/4 - \\epsilon\\right)n \\right]$ for any $\\epsilon>0$ and $q$ is large; and when $r\\leq\\delta n$ for some $\\delta>0$ and for any $q$. The result is improved from the earlier work of Pollack that the similar result holds for $r\\leq\\left[(1-\\epsilon)\\sqrt{n}\\right]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-26T02:17:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "11T23"
    ],
    "keywords": [
      "irreducible polynomial",
      "prescribed coefficients",
      "similar result holds",
      "earlier work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160106867H"
    }
  }
}