Algebraic Points of Small Height Missing a Union of Varieties

Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN , N ≥ 2. Let ZK be a union of varieties defined over K such that V * ZK . We prove the existence of a point of small height in V \ZK , providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of a hypersurface containing ZK , where dependence on both is optimal. This generalizes and improves upon the results of [6] and [7]. As a part of our argument, we provide a basic extension of the function field version of Siegel’s lemma [21] to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field.