{
  "id": "1006.2610",
  "version": "v2",
  "published": "2010-06-14T06:59:46.000Z",
  "updated": "2012-05-03T09:40:40.000Z",
  "title": "Functions which are PN on infiitely many extensions of Fp, p odd",
  "authors": [
    "Elodie Leducq"
  ],
  "categories": [
    "math.NT",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "Let $p$ be an odd prime number. We prove that for $m\\equiv1\\mod p$, $x^m$ is perfectly nonlinear over $\\mathbb{F}_{p^n}$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\\in\\mathbb{N}$. First, we study singularities of $f(x,y)=\\frac{(x+1)^m-x^m-(y+1)^m+y^m}{x-y}$ and we use Bezout theorem to show that for $m\\neq 1+p^l$, $f(x,y)$ has an absolutely irreducible factor. Then by Weil theorem, f(x,y) has rationnal points such that $x\\neq y$ which means that $x^m$ is not PN.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-03T09:40:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extensions",
      "odd prime number",
      "rationnal points",
      "study singularities",
      "bezout theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.2610L"
    }
  }
}