Algebraic points of small height missing a union of varieties

TextLet 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 KNKN, N⩾2N⩾2. Let ZKZK be a union of varieties defined over K   such that V⊈ZKV⊈ZK. We prove the existence of a point of small height in V∖ZKV∖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 ZKZK, where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) [6] and [7]. As a part of our argument, we provide a basic extension of the function field version of Siegel's lemma (Thunder, 1995) [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.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=R-o6lr8s0Go.