A note on Thue's equation over function fields (Q1345295)

Let \(K\) be a function field of dimension 1 and genus \(g\) over the algebraically closed field \(k\) of characteristic zero. Let \(S\) be a finite subset of the valuations of \(K/k\). Denote by \({\mathcal O}_ S\) the ring of \(S\)-integers of \(K\). Let \(F\in K[X, Y]\) be a form of degree \(n\geq 3\) with distinct roots in \({\mathcal O}_ S\), and let \(0\neq \mu\in {\mathcal O}_ S\). The authors show that all solutions of the Thue equation \[ F(x,y)= \mu \qquad \text{in} \quad x,y\in {\mathcal O}_ S \] satisfy \[ H(x, y,1)\leq {\textstyle {14 \over n}} H(F)+ {\textstyle {3\over n}} H(\mu, 1)+ 2g+| S|-1. \] This upper bound improves the bounds of \textit{R. C. Mason} [Diophantine equations over function fields (London Math. Soc. Lect. Note Ser. 96) (Cambridge Univ. Press 1984; Zbl 0533.10012)]. The proof refines Mason's approach by using an averaging argument. Applications of the main result to diophantine approximation of algebraic functions are given, as well.