# 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 *K ^{N}*,

*N*> 1. Let

*Z*be a union of varieties defined over

_{K}*K*such that

*V*is not contained in

*Z*. We prove the existence of a point of small height in

_{K}*V*outside of

*Z*, providing an explicit upper bound on the height of such a point in terms of the height of

_{K}*V*and the degree of a hypersurface containing

*Z*, 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)

_{K}## Rights Information

© 2009 Lenny Fukshansky

## Terms of Use & License Information

## 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.