On large values of binary forms
type: Item
instance of: scholarly article
title: On large values of binary forms
zbMATH Open document ID: 0876.11011
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-10-29T00:00:00Z
full work available at URL: contents.htm
review text: Let \(F(X,Y)\) be an irreducible binary form of degree \(n\geq 3\) with coefficients in \({\mathbb{Z}}\), and let \(m\) be a positive integer having \(s\) distinct prime factors. The author deals with the Thue equation (*) \(|F(x,y)|=m\) in coprime integers \(x,y\), where solutions \((x,y)\) and \((-x,-y)\) are considered equal. \textit{E. Bombieri} and \textit{W. M. Schmidt} [Invent. Math. 88, 69-81 (1987; Zbl 0614.10018)] showed that (*) has at most \(cn^{1+s}\) solutions, where \(c\) is an absolute constant which for \(n\) sufficiently large can be taken equal to \(215\). The author shows that if \(n\geq 6\) and \(m\) is larger than an effectively computable number \(C\), depending only on the discriminant of \(F\), then (*) has at most \(n\big((n^{4/(n-2)})^s +6\big)\) solutions. In his proof, the author estimates the number of solutions which are ``large'' in terms of \(m\) in the usual way, using the Thue-Siegel principle and a gap principle. The author treats the ``small'' solutions by means of a new idea, which uses a theorem of \textit{K. Györy} and \textit{J.-H. Evertse} [Compos. Math. 79, 169-204 (1991; Zbl 0746.11020)], stating that every irreducible binary form \(F\in {\mathbb{Z}}[X,Y]\) of discriminant \(D\) is SL\(_2({\mathbb{Z}})\)-equivalent to a binary form \(G\) whose coefficients are effectively bounded in terms of \(D\).
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 1026391
zbMATH Keywords: Thue equations
zbMATH Keywords: irreducible binary form
zbMATH Keywords: large solutions
zbMATH Keywords: small solutions
author: Bela Brindza
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
OpenAlex ID: W2069887164
cites work: Q3663376
cites work: On Thue's equation
cites work: Q3665191
cites work: Q3829605
cites work: Q3974142
cites work: Q4059031
cites work: Q3139245
cites work: Diophantine approximations and diophantine equations
cites work: Integer points and the rank of Thue elliptic curves
cites work: On the Number of Solutions of Polynomial Congruences and Thue Equations
DOI: 10.1216/RMJM/1181072025
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description: scientific article; zbMATH DE number 1026391