On Siegel's Lemma Outside of 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. 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.
© 2010 Lenny Fukshansky
Fukshansky, Lenny. "On Siegel's Lemma Outside of a Union of Varieties." Oberseminar des Institutes für Algebra und Geometrie, University of Magdeburg, Magdeburg, Germany. 9 November 2010.