On Siegel's Lemma Outside of a Union of Varieties

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Mathematics (CMC)

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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. 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.


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.

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© 2010 Lenny Fukshansky

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