{
  "id": "1211.6044",
  "version": "v1",
  "published": "2012-11-23T01:25:49.000Z",
  "updated": "2012-11-23T01:25:49.000Z",
  "title": "Permutation polynomials of finite fields",
  "authors": [
    "Christopher J. Shallue"
  ],
  "comment": "Honours thesis, Monash University",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be the finite field of $q$ elements. Then a \\emph{permutation polynomial} (PP) of $\\mathbb{F}_q$ is a polynomial $f \\in \\mathbb{F}_q[x]$ such that the associated function $c \\mapsto f(c)$ is a permutation of the elements of $\\mathbb{F}_q$. In 1897 Dickson gave what he claimed to be a complete list of PPs of degree at most 6, however there have been suggestions recently that this classification might be incomplete. Unfortunately, Dickson's claim of a full characterisation is not easily verified because his published proof is difficult to follow. This is mainly due to antiquated terminology. In this project we present a full reconstruction of the classification of degree 6 PPs, which combined with a recent paper by Li \\emph{et al.} finally puts to rest the characterisation problem of PPs of degree up to 6. In addition, we give a survey of the major results on PPs since Dickson's 1897 paper. Particular emphasis is placed on the proof of the so-called \\emph{Carlitz Conjecture}, which states that if $q$ is odd and `large' and $n$ is even then there are no PPs of degree $n$. This important result was resolved in the affirmative by research spanning three decades. A generalisation of Carlitz's conjecture due to Mullen proposes that if $q$ is odd and `large' and $n$ is even then no polynomial of degree $n$ is `close' to being a PP. This has remained an unresolved problem in published literature. We provide a counterexample to Mullen's conjecture, and also point out how recent results imply a more general version of this statement (provided one increases what is meant by $q$ being `large').",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-23T01:25:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite field",
      "permutation polynomials",
      "conjecture",
      "dicksons claim",
      "full characterisation"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.6044S"
    }
  }
}