{
  "id": "1609.03662",
  "version": "v1",
  "published": "2016-09-13T02:51:35.000Z",
  "updated": "2016-09-13T02:51:35.000Z",
  "title": "Permutation Polynomials of the form ${\\tt X}^r(a+{\\tt X}^{2(q-1)})$ --- A Nonexistence Result",
  "authors": [
    "Xiang-dong Hou"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $f={\\tt X}^r(a+{\\tt X}^{2(q-1)})\\in{\\Bbb F}_{q^2}[{\\tt X}]$, where $a\\in{\\Bbb F}_{q^2}^*$ and $r\\ge 1$. The parameters $(q,r,a)$ for which $f$ is a permutation polynomial (PP) of ${\\Bbb F}_{q^2}$ have been determined in the following cases: (i) $a^{q+1}=1$; (ii) $r=1$; (iii) $r=3$. These parameters together form three infinite families. For $r>3$ (there is a good reason not to consider $r=2$) and $a^{q+1}\\ne 1$, computer search suggested that $f$ is not a PP of ${\\Bbb F}_{q^2}$ when $q$ is not too small relative to $r$. In the present paper, we prove that this claim is true. In particular, for each $r>3$, there are only finitely many $(q,a)$, where $a^{q+1}\\ne 1$, for which $f$ is a PP of ${\\Bbb F}_{q^2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-13T02:51:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06"
    ],
    "keywords": [
      "permutation polynomial",
      "nonexistence result",
      "parameters",
      "infinite families",
      "computer search"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}