{
  "id": "1909.13074",
  "version": "v1",
  "published": "2019-09-28T11:32:23.000Z",
  "updated": "2019-09-28T11:32:23.000Z",
  "title": "Primitive values of rational functions at primitive elements of a finite field",
  "authors": [
    "Stephen D. Cohen",
    "Hariom Sharma",
    "Rajendra Sharma"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.AC"
  ],
  "abstract": "Given a prime power $q$ and an integer $n\\geq2$, we establish a sufficient condition for the existence of a primitive pair $(\\alpha,f(\\alpha))$ where $\\alpha \\in \\mathbb{F}_q$ and $f(x) \\in \\mathbb{F}_q(x)$ is a rational function of degree $n$. (Here $f=f_1/f_2$, where $f_1, f_2$ are coprime polynomials of degree $n_1,n_2$, respectively, and $n_1+n_2=n$.) For any $n$, such a pair is guaranteed to exist for sufficiently large $q$. Indeed, when $n=2$, such a pair definitely does {\\em not} exist only for 28 values of $q$ and possibly (but unlikely) only for at most $3911$ other values of $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-28T11:32:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T23"
    ],
    "keywords": [
      "rational function",
      "finite field",
      "primitive values",
      "primitive elements",
      "sufficient condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}