# Fonctions PN sur une infinité d'extensions de $\mathbb{F}_p$, $p$ impair

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.
Document type :
Preprints, Working Papers, ...
Domain :

Cited literature [1 references]

https://hal.archives-ouvertes.fr/hal-00488098
Contributor : Elodie Leducq <>
Submitted on : Tuesday, June 1, 2010 - 10:53:53 AM
Last modification on : Thursday, April 4, 2019 - 1:29:11 AM
Document(s) archivé(s) le : Friday, September 17, 2010 - 11:31:15 AM

### Files

nombre_exceptionnel.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00488098, version 1
• ARXIV : 1006.2610

### Citation

Elodie Leducq. Fonctions PN sur une infinité d'extensions de $\mathbb{F}_p$, $p$ impair. 2010. ⟨hal-00488098⟩

Record views