Cubic Thue inequalities with negative discriminant.
type: Item
instance of: scholarly article
title: Cubic Thue inequalities with negative discriminant.
zbMATH Open document ID: 1042.11021
DOI: 10.1016/S0022-314X(02)00010-0
author: Isao Wakabayashi
published in: Journal of Number Theory
publication date: 2003-04-02T00:00:00Z
review text: Let \(X^3+aX+b\) with \(a,b \in \mathbb Z\) be irreducible over \(\mathbb Z [X]\) with discriminant \(D\), and let \(\theta\) denote the real root (of smallest absolute value, if \(D>0\)) of this polynomial. Theorem 1 gives explicit expressions \(c(a,b), \lambda(a,b) > 0\) such that for all \(p,q \in \mathbb Z\) with \(q>0\) one has \[ \left| \theta- \frac pq \right| > \frac {c(a,b)}{q^{\lambda(a,b)}}, \] where \(\lambda(a,b)<3\) for \(| a| \) sufficiently large and tends to \(2\) for \(| a| \to \infty\). For the proof the author constructs suitable Padé approximations by using the integrals of \textit{J. H. Rickert} [Math. Proc. Camb. Philos. Soc. 113, 461--472 (1993; Zbl 0786.11040)]. This result is applied to give explicit bounds for the solutions of the Thue inequality \[ | x^3+axy^2+by^3| \leq k, \] supposing that \(a\) exceeds some positive bound depending on \(| b| \). As an application, for \(k=a+| b| +1\) the above inequality is completely solved for \(| b| \leq 2\) and arbitrary \(a>0\) as well as for all \((a,b)\) satisfying \(a \geq 360 b^4\). For this purpose, the first 21 terms of the continued fraction expansion of \(\theta\) are explicitly given.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 1891440
zbMATH Keywords: Padé approximation
zbMATH Keywords: measure of irrationality
zbMATH Keywords: continued fraction
zbMATH Keywords: Thue equation
zbMATH Keywords: Thue inequality
zbMATH Keywords: Legendre's theorem
describes a project that uses: KANT/KASH
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891440