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A remark on the heights of subspaces. (English) Zbl 0714.11022

A tribute to Paul Erdős, 359-360 (1990).
In this brief paper, the author recalls the definition of the (multiplicative) height \(H(S)\) of a linear subspace \(S\subseteq K^ n\), where \(K\) is a number field. He then proves the inequality \(H(S+T)H(S\cap T)\leq H(S)H(T)\) for subspaces \(S,T\subseteq K^ n\). As the author notes, the proposition has also been proved in T. Struppeck and J. D. Vaaler, Inequalities for heights of algebraic subspaces and the Thue-Siegel principle, Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL (USA) 1989, Prog. Math. 85, 493–528 (1990; Zbl 0722.11033).
[For the entire collection see Zbl 0706.00007.]

MSC:

11D57 Multiplicative and norm form equations
11J87 Schmidt Subspace Theorem and applications
11H99 Geometry of numbers
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