#### Title

Algebraic Points of Small Height Missing a Union of Varieties

#### Document Type

Article

#### Department

Mathematics (CMC)

#### Publication Date

10-2010

#### Abstract

Let *K* be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let *V* be a subspace of *K ^{N}* where

*N*≥ 2. Let

*Z*be a union of varieties defined over

_{K}*K*such that

*V*⊈

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

_{K}*V*\

*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 hypersurface containing

*Z*, where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) [6,7]. As part of our argument, we provide a basic extension of the function field version of Siegel's lemma (Thunder, 1995) [21] to an inequality 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.

_{K}#### Rights Information

© 2010 Elsevier Inc.

#### Terms of Use & License Information

#### Recommended Citation

Fukshansky, Lenny. "Algebraic Points of Small Height Missing a Union of Varieties." Journal of Number Theory 130.10 (2010): 2099-2118. Web. 3 Apr. 2012.

## Comments

This article is related to a lecture by the author: "Algebraic points of small height missing a union of varieties," given during the Western Number Theory Conference in Fort Collins, CO, December 2008.