The generalized Catalan equation in positive characteristic

Abstract

Let $K={\mathbb{F}}_p(z_1,\dots ,z_r)$ be a finitely generated field over ${\mathbb{F}}_p$. In this article we study the generalized Catalan equation $a{x}^m+b{y}^n=1$ in $x,y\in K$ and integers $m,n>1$ coprime with $p$. Our main result shows that there are only finitely many solutions up to the action of Frobenius.