A note on Thue's equation over function fields
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.
diophantine approximation of algebraic functions
scientific article; zbMATH DE number 729090
scientific article