#### Title

On Siegel's Lemma Outside of a Union of Varieties

#### Document Type

Lecture

#### Department

Mathematics (CMC)

#### Publication Date

11-9-2010

#### 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. 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.

_{K}#### Rights Information

© 2010 Lenny Fukshansky

#### Terms of Use & License Information

#### Recommended Citation

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.

## Comments

Versions of this talk, called "Siegel's lemma outside of a union of varieties," were also given during the AMS Special Session: Number Theory, AMS Fall Eastern Section Meeting at Middletown, CT in October 2008 and during the Number Theory Seminar at UCLA in November 2008.