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 of genus 0 or 1 over a perfect field, and let V be a subspace of KN, N > 1. Let ZK be a union of varieties defined over K such that V is not contained in ZK. We prove the existence of a point of small height in V outside of 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. Our method is based on some counting lattice points in slices of a cube, a version of combinatorial nullstellensatz, and a version of Siegel's lemma 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. (Received February 24, 2009)
© 2009 Lenny Fukshansky
Fukshansky, Lenny. "Points of Small Height Missing a Union of Varieties." AMS Special Session: Algebra and Number Theory with Polyhedra, AMS Spring Western Section Meeting, San Francisco, CA. 26 April 2009.