{
  "id": "2010.15657",
  "version": "v1",
  "published": "2020-10-29T15:05:36.000Z",
  "updated": "2020-10-29T15:05:36.000Z",
  "title": "Low-degree permutation rational functions over finite fields",
  "authors": [
    "Zhiguo Ding",
    "Michael E. Zieve"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We degree all degree-4 rational functions f(x) in F_q(X) which permute P^1(F_q), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational functions. We also determine all degree-8 rational functions f(X) in F_q(C) which permute P^1(F_q) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for most other positive integers n<4096, for each sufficiently large q we determine all degree-n rational functions f(X) in F_q(X) which permute P^1(F_q) but which are not compositions of lower-degree rational functions in F_q(X). Some of these results are proved by using a new Galois-theoretic characterization of additive (linearized) polynomials among all rational functions, which is of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-29T15:05:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "11R32"
    ],
    "keywords": [
      "low-degree permutation rational functions",
      "finite fields",
      "lower-degree rational functions",
      "degree-n rational functions",
      "degree-one rational functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}