Remarks on Catalan's equation over function fields

TL;DR

This paper investigates solutions to Catalan's equation over function fields, establishing conditions under which no non-trivial solutions exist within the ring of functions with poles only at a specified prime.

Contribution

It provides new results on the non-existence of solutions to Catalan's equation over certain function fields under specific conditions.

Findings

No non-constant solutions exist under certain conditions

Results extend understanding of Catalan's equation in function field settings

Conditions identified for the absence of solutions

Abstract

Let $\ell$ be a prime number, $F$ be a global function field of characteristic $\ell$. Assume that there is a prime $P_\infty$ of degree $1$. Let $\mathcal{O}_F$ be the ring of functions in $F$ with no poles outside of $\{P_\infty\}$. We study solutions to Catalan's equation $X^m-Y^n=1$ over $\mathcal{O}_F$ and show that under certain additional conditions, there are no non-constant solutions which lie in $\mathcal{O}_F$, when $m,n>1$.