Siegel's lemma with additional conditions

Details Siegel's lemma with additional conditions Siegel's lemma with additional conditions

2004-09-21

arxiv.org/abs/math/0409375v3

J. Number Theory 120 (2006), no. 1, 13--25

Mathematics

Number Theory

12 pages, revised version, to appear in Journal of Number Theory

Scientific paper

Let $K$ be a number field, and let $W$ be a subspace of $K^N$, $N \geq 1$. Let $V_1,...,V_M$ be subspaces of $K^N$ of dimension less than dimension of $W$. We prove the existence of a point of small height in $W \setminus \bigcup_{i=1}^M V_i$, providing an explicit upper bound on the height of such a point in terms of heights of $W$ and $V_1,...,V_M$. Our main tool is a counting estimate we prove for the number of points of a subspace of $K^N$ inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a non-vanishing point for a decomposable form and an effective subspace extension lemma.

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