Diophantine Equations over Global Functions Fields II: R-Integral Solutions of Thue Equations

Abstract

Let {\small $K$} be an algebraic function field over a finite field. Let {\small $L$} be an extension field of {\small $K$} of degree at least 3. Let {\small $R$} be a finite set of valuationsof {\small $K$} and denote by {\small $S$} the set of extensions of valuations of {\small $R$} to {\small $L$}. Denote by {\small $O_{K,R},O_{L,S}$} the ring of {\small $R$}-integers of {\small $K$} and {\small $S$}-integers of {\small $L$}, respectively. Assume that {\small $\alpha\in O_{L,S}$} with {\small $L=K(\alpha)$}, let {\small $0\neq \mu\in O_{K,R}$}, and consider the solutions {\small $(x,y)\in O_{K,R}$} of the Thue equation

{\small \[ N_{L/K}(x-\alpha y)=\mu.\]}

We give an efficient method for calculating the {\small $R$}-integral solutions of the above equation. The method is different from that in our previous paper and is much more efficient in many cases.