On Thue's equation
type: Item
instance of: scholarly article
title: On Thue's equation
zbMATH Open document ID: 0614.10018
DOI: 10.1007/BF01405092
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143445
review text: In this very interesting paper the authors consider the number \(N(F,h)\) of solutions of Thue's equation \(F(x,y)=h\) in integers \(x,y\in\mathbb{Z}\) with \((x,y)=1\). Here, \(F\in\mathbb{Z}[x,y]\) is a binary form of degree \(r\ge 3\) and h is fixed. In 1983 J. H. Evertse was the first to give a bound for \(N(F,h)\) depending only on \(r\) and \(h\), but independent of the coefficients of \(F\). The dependence on \(r\) is of the form \(\exp(r^2)\). In the present paper the authors give a vast improvement of Evertse's theorem. They show \(N(F,h)<c_1r^{1+t}\), where \(t\) is the number of distinct prime divisors of \(h\) and \(c_1\) is some computable constant which is \(<215\) if \(r\) is big enough. In case \(h=1\) this bound is optimal in its dependence on \(r\), since \(x^r+(y-x)(y-2x) \cdots (y-rx)=1\) obviously has at least \(r\) solutions. The proof starts with the case \(h=1\) and then, by induction on \(t\), is extended to general \(h\). To bound the number of large solutions the classical Thue-Siegel method is used. For the small solutions the authors first split the original problem in a union of Thue equations with large discriminant and then use some ingenious proximity arguments to finish the proof.
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11D59
zbMATH DE Number: 3993681
zbMATH Keywords: upper bounds
zbMATH Keywords: primitive solutions
zbMATH Keywords: Thue equation
zbMATH Keywords: binary form of higher degree
zbMATH Keywords: large solutions
zbMATH Keywords: small solutions
reviewed by: Frits Beukers
MaRDI profile type: Publication
cites work: Q3663376
cites work: Q3665191
cites work: On the representations of numbers by binary cubic forms
cites work: On the representation of integers by binary forms
cites work: An inequality for the discriminant of a polynomial
cites work: On Thue's theorem.
cites work: Representations of integers by binary forms and the rank of the Mordell- Weil group
author: Enrico Bombieri
author: Wolfgang M. Schmidt
description: scientific article; zbMATH DE number 3993681