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Siegel’s Lemma with additional conditions. (English) Zbl 1192.11018
Summary: Let \(K\) be a number field, and let \(W\) be a subspace of \(K^N\), \(N\geq 1\). Let \(V_1,\dots,V_M\) be subspaces of \(K^N\) of dimension less than dimension of \(W\). We prove the existence of a point of small height in \(W\backslash \bigcup_{i=1}^m V_i\), providing an explicit upper bound on the height of such a point in terms of heights of \(W\) and \(V_1,\dots,V_M\). Our main tool is a counting estimate we proved for the number of points of a subspace of \(K^N\) inside of an adelic cube [Monatsh. Math. 147, No. 1, 25–41 (2006; Zbl 1091.11024)].
As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.

MSC:
11D04 Linear Diophantine equations
11H06 Lattices and convex bodies (number-theoretic aspects)
11H46 Products of linear forms
11J13 Simultaneous homogeneous approximation, linear forms
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