{
  "id": "1805.07105",
  "version": "v1",
  "published": "2018-05-18T09:07:52.000Z",
  "updated": "2018-05-18T09:07:52.000Z",
  "title": "Irreducible Polynomials over $\\mathbb{F}_{2^r}$ with Three Prescribed Coefficients",
  "authors": [
    "Ofir Gorodetsky"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For any positive integers $n \\ge 3$ and $r \\ge 1$, we prove that the number of monic irreducible polynomials of degree $n$ over $\\mathbb{F}_{2^r}$ in which the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has period $24$ as a function of $n$, after a suitable normalization. A similar result holds over $\\mathbb{F}_{5^r}$, with the period being $60$. We also show that this is a phenomena unique to characteristics $2$ and $5$. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it answers a question of Katz.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-18T09:07:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "prescribed coefficients",
      "similar result holds",
      "cyclotomic function fields",
      "monic irreducible polynomials",
      "phenomena unique"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}