On Siegel's Lemma Outside of a Union of Varieties

Authors

Lenny Fukshansky, Claremont McKenna CollegeFollow

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_K be a union of varieties defined over K such that V is not contained in Z_K. We prove the existence of a point of small height in V outside of Z_K, 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 Z_K, 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.

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.

Rights Information

© 2010 Lenny Fukshansky

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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.