A note on Thue's equation over function fields
type: Item
instance of: scholarly article
title: A note on Thue's equation over function fields
zbMATH Open document ID: 0820.11019
DOI: 10.1007/BF01301690
author: Roberto Dvornicich
author: Umberto Zannier
published in: Monatshefte für Mathematik
publication date: 1995-09-14T00:00:00Z
full work available at URL: https://eudml.org/doc/178665
review text: 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.
reviewed by: István Gaál
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 729090
zbMATH Keywords: function field
zbMATH Keywords: Thue equation
zbMATH Keywords: upper bounds
zbMATH Keywords: diophantine approximation of algebraic functions
MaRDI profile type: Publication
cites work: Q3315578
cites work: An effective lower bound on the “Diophantine approximation” of algebraic functions by rational functions
cites work: On Osgood's effective thue theorem for algebraic functions
cites work: Q4170806
cites work: Q3856855
OpenAlex ID: W2036795127
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description: scientific article; zbMATH DE number 729090