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

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

2010-06-14

arxiv.org/abs/1006.2610v1

Mathematics

Number Theory

Scientific paper

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.

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