Lenny Fukshansky, Claremont McKenna

November 3, 4:00 - 5:00 pm in MS 5148

Siegel's lemma outside of a union of varieties

Let K be a number field, Q-bar, 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 K^N, N>1. Let Z_K be a union of varieties defined over K such that V is not contained in Z_K. We prove the existence of a point of small height in V outside of Z_K, 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 Z_K, where dependence on both is optimal. A key tool required in the function field case is 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.