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A note on Thue’s equation over function fields. (English) Zbl 0820.11019
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 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.
Reviewer: I.Gaál (Debrecen)

##### MSC:
 11D57 Multiplicative and norm form equations 11D75 Diophantine inequalities
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##### References:
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