  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.
