The generalized Catalan equation in positive characteristic

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 $ax^m

• by^n = 1$in$x, y \in K$andintegers$m, n > 1$coprimewith$p$. Our main result shows that there are only finitely many solutions up to the action of Frobenius.