Points of Small Height Missing 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. 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)


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.

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© 2009 Lenny Fukshansky

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