Algebraic points of small height missing a union of varieties

Abstract: Text: Let K be a number field, , or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of , . Let be a union of varieties defined over K such that . We prove the existence of a point of small height in , 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 , where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) . As a part of our argument, we provide a basic extension of the function field version of Siegel''s lemma (Thunder, 1995) 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. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=R-o6lr8s0Go. [Copyright &y& Elsevier]