{
  "id": "1905.04202",
  "version": "v1",
  "published": "2019-05-10T14:57:35.000Z",
  "updated": "2019-05-10T14:57:35.000Z",
  "title": "Permutation polynomials of degree 8 over finite fields of odd characteristic",
  "authors": [
    "Xiang Fan"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper provides an algorithmic generalization of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most $6$ were essentially deduced from manual analysis of these polynomial equations. However, these polynomials, needed for that purpose when $d>6$, are too complicated to solve. Our idea is to make them more solvable by calculating some radicals of ideals generated by them, implemented by a computer algebra system (CAS). Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree $8$ over an arbitrary finite field of odd order $q>8$. The main result is that for an odd prime power $q>8$, a PP $f$ of degree $8$ exists over the finite field of order $q$ if and only if $q\\leqslant 31$ and $q\\not\\equiv 1\\ (\\mathrm{mod}\\ 8)$, and $f$ is explicitly listed up to linear transformations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-10T14:57:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "12Y05"
    ],
    "keywords": [
      "odd characteristic",
      "polynomial equations",
      "personal computer work",
      "arbitrary finite field",
      "computer algebra system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}