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On Siegel’s lemma. (English) Zbl 0533.10030
Invent. Math. 73, 11-32 (1983); addendum ibid. 75, 377 (1984).
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 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 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.

MSC:
11J99 Diophantine approximation, transcendental number theory
11J81 Transcendence (general theory)
11H50 Minima of forms
11R56 Adèle rings and groups
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References:
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