Siegel's lemma with additional conditions

Abstract

Let $K$ be a number field, and let $W$ be a subspace of $K^N$, $N\ge 1$. 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.

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