Points of Small Height Missing a Union of Varieties
Document Type
Lecture
Department
Mathematics (CMC)
Publication Date
4-26-2009
Abstract
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)
Rights Information
© 2009 Lenny Fukshansky
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Recommended Citation
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.
Comments
This lecture is related to Lenny Fukshansky's article "Algebraic points of small height missing a union of varieties" from the Journal of Number Theory, vol. 130 no. 10, pg. 2099-2118.
It is also related to his lecture "Algebraic points of small height missing a union of varieties" from the Western Number Theory Conference at Fort Collins, CO, in December 2008.