On Siegel's lemma
type: Item
instance of: scholarly article
title: On Siegel's lemma
zbMATH Open document ID: 0533.10030
DOI: 10.1007/BF01393823
published in: Inventiones Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143033
review text: The following problem is fundamental in effecting auxiliary constructions in transcendental number theory and diophantine approximation: one has \(M<N\) linear equations \(\sum^{N}_{n=1}a_{mn}x_ n=0\); \(m=1,2,...,M\) with rational integer coefficients \(a_{mn}\) not all zero, to be solved for non-zero N-tuples \(x=(x_ 1,...,x_ N)\) of rational integers with a good upper bound for the \(| x_ n|\). For more refined applications the \(a_{mn}\) may be elements of a number field K, whilst the \(x_ n\) are restricted to integers of a subfield \(k\subseteq K\); if \([K:k]=r\) we then need \(N>Mr.\) The present paper solves the above problem once-for-all. The authors find and define a canonical height for the given linear system. They prove the existence of N-Mr linearly independent solutions \(x_{\ell}= (x_{1\ell},x_{2\ell},... ,x_{N\ell})\) of N-tuples of integers of k so that the product of the heights of these solutions is economically bounded. The results obtained depend on \textit{J. D. Vaaler}'s cube-slicing inequality [Pac. J. Math. 83, 543-553 (1979; Zbl 0465.52011)] and an adèlic generalization of Minkowski's theorem on successive minima [see \textit{R. B. McFeat}, Diss. Math. 88 (1971; Zbl 0229.10014)] proved here independently. These methods are considerably more sophisticated than in Dirichlet's box principle which was traditionally applied in this context. This paper will influence the future of transcendental number theory and of diophantine approximation.
Mathematics Subject Classification ID: 11J99
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 11R56
zbMATH DE Number: 3845680
zbMATH Keywords: geometry of numbers over adeles
zbMATH Keywords: auxiliary functions in transcendence theory
zbMATH Keywords: linear equations
zbMATH Keywords: canonical height
zbMATH Keywords: cube-slicing inequality
Wikidata QID: Q55899117
author: Jeffrey D. Vaaler
author: Enrico Bombieri
MaRDI profile type: Publication
cites work: On the Thue-Siegel-Dyson theorem
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cites work: Note on a Theorem of Minkowski
cites work: A geometric inequality with applications to linear forms
cites work: Bounds for solutions of systems of linear equations
cites work: Arithmetic on algebraic varieties
cites work: Q5685192
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description: scientific article; zbMATH DE number 3845680