On the resolution of index form equations in biquadratic number fields. I
type: Item
instance of: scholarly article
title: On the resolution of index form equations in biquadratic number fields. I
zbMATH Open document ID: 0726.11022
DOI: 10.1016/0022-314X(91)90090-X
published in: Journal of Number Theory
publication date: 1991-01-01T00:00:00Z
review text: For every algebraic number field K of degree n with integral basis \(\{1,w_ 2,...,w_ n\}\) one has \[ D_{K/{\mathbb{Q}}}(w_ 2X_ 2+...+w_ nX_ n)=(I(X_ 2,...,X_ n))^ 2D_{K/{\mathbb{Q}}}, \] where \(I(X_ 2,...,X_ n)\) is a homogeneous polynomial of degree n(n-1)/2 with coefficients in \({\mathbb{Z}}\), the so-called index form of the basis \(\{1,w_ 2,...,w_ n\}\). The authors consider the index form equation \(I(x_ 2,x_ 3,x_ 4)=\pm 1\) in \(x_ 2,x_ 3,x_ 4\in {\mathbb{Z}}\) in case that K is a totally real biquadratic number field. They present a method to find all small solutions of this equation, i.e. with \(| x_ i| \leq 10^ 7\) for \(i=2,3,4\). Their idea is to reduce the index form equation to an equation of the form (1) \(G_ n=y^ 2-d\), where \(\{G_ n\}\) is a binary recurrence sequence with rational integral terms and d is a constant, and to check for all small n whether (1) holds for some \(y\in {\mathbb{Z}}\). In this way, the authors find all small solutions of index form equations as above for all real quadratic extensions of \({\mathbb{Q}}(\sqrt 5)\) of discriminant 63000 and of \({\mathbb{Q}}(\sqrt{2})\) of discriminant 39000. [For part II, see the following review.]
Mathematics Subject Classification ID: 11D99
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 4198081
zbMATH Keywords: index form equation
zbMATH Keywords: biquadratic number field
zbMATH Keywords: small solutions
zbMATH Keywords: binary recurrence sequence
author: István Gaál
author: Attila Pethő
author: Michael E. Pohst
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
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cites work: Q4050663
cites work: The diophantine equation \(y^2+k=x^3\).
cites work: Computing All Power Integral Bases of Cubic Fields
cites work: Q4124912
cites work: Q3932877
cites work: Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains
cites work: Q3872539
cites work: Q5582375
cites work: Q3342642
cites work: Q3342641
cites work: On the resolution of Thue inequalities
cites work: Q3804775
cites work: Berechnung unabhängiger Einheiten und Klassenzahlen in total reellen biquadratischen Zahlkörpern
cites work: Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper.
cites work: On the Diophantine equation $ax^{2t}+bx^ty+cy^2=d$ and pure powers in recurrence sequences.
cites work: On Mordell's Equation y 2 - k = x 3 : A Problem of Stolarsky
cites work: Q4137269
full work available at URL: 0022-314x(91)90090-x
OpenAlex ID: W2029775181
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description: scientific article; zbMATH DE number 4198081