The generalized Catalan equation in positive characteristic

TL;DR

This paper investigates the solutions of a generalized Catalan equation over finitely generated fields in positive characteristic, establishing finiteness results for solutions up to Frobenius action.

Contribution

It proves that only finitely many solutions exist for the generalized Catalan equation in positive characteristic fields, considering the action of Frobenius.

Findings

Finitely many solutions exist up to Frobenius action.

Solutions are constrained by the coprimality of exponents and characteristic.

The result extends classical Catalan equation finiteness to positive characteristic fields.

Abstract

Let $K = \mathbb{F}_p(z_1, \ldots, z_r)$ be a finitely generated field over $\mathbb{F}_p$. In this article we study the generalized Catalan equation $ax^m + by^n = 1$ in $x, y \in K$ 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.