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 where 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 hypersurface containing ZK, where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) [6,7]. As 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.
© 2010 Elsevier Inc.
Fukshansky, Lenny. "Algebraic Points of Small Height Missing a Union of Varieties." Journal of Number Theory 130.10 (2010): 2099-2118. Web. 3 Apr. 2012.