Remarks on Catalan's equation over function fields

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 $m,n>1$.

arXiv commentVersion 6: Minor changes, accepted for publication in Finite fields and their applications