Functions which are PN on infiitely many extensions of Fp, p odd

Let $p$ be an odd prime number. We prove that for $m\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}p$, $x^m$ 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 $m\mathrm{\ne }1+p^l$, $f(x,y)$ has an absolutely irreducible factor. Then by Weil theorem, f(x,y) has rationnal points such that $x\mathrm{\ne }y$ which means that $x^m$ is not PN.