On a conjecture of Siegel
type: Item
instance of: scholarly article
title: On a conjecture of Siegel
zbMATH Open document ID: 0903.11013
DOI: 10.1007/BF01305344
author: Julia Mueller
author: Enrico Bombieri
published in: Monatshefte für Mathematik
publication date: 1998-12-20T00:00:00Z
full work available at URL: https://eudml.org/doc/178815
review text: Let \(k\) be an algebraically closed field of characteristic 0, \(R=k[t]\). Let \(r\geq 2\) and \(a_1,\ldots,a_r,c\in R\) and consider the equation \[ \sum_{i=1}^r a_i x^{\alpha_{1i}} y^{\alpha_{2i}}=c \] in the variables \(x,y\in R\) subject to the conditions: (A) \(a_i\neq 0\) for \(i=1,\ldots,r\), (B) \(| \alpha_{\nu i}-\alpha_{\nu j}| \geq K\) for \(i\neq j\), (C) \(\alpha_{11}>\ldots>\alpha_{1r}\) and \(\alpha_{21}<\ldots<\alpha_{2r}\). The main theorem of the paper is that the above equation, satisfying (A), (B), (C) with \(K=(2r+1)^{(r+1)(r+2)}\), has at most \((2r+1)^{r-1}\) non-proportional, primitive solutions. This class of equations includes Thue equations over function fields. The proof applies the \(abc...z\) theorem in function fields. Note that a modified form of Siegel's conjecture over \(\mathbb{Z}\) has been proved by \textit{J. Mueller} and \textit{W. M. Schmidt} [Acta Math. 160, 207-247 (1988; Zbl 0655.10016)] for Thue equations, by a completely different method.
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 1156710
zbMATH Keywords: Thue equations
zbMATH Keywords: function fields
zbMATH Keywords: Siegel's conjecture
zbMATH Keywords: number of primitive solutions
Wikidata QID: Q123333482
reviewed by: István Gaál
MaRDI profile type: Publication
cites work: Q4838346
cites work: Vanishing sums in function fields
cites work: Number theory III: Diophantine geometry
cites work: Q3474708
cites work: Thue's equation and a conjecture of Siegel
OpenAlex ID: W2091878593
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description: scientific article; zbMATH DE number 1156710