Algebraic points of small height missing a union of varieties

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Let K be a number field, , or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN, N?2. Let ZK be a union of varieties defined over K such that V?ZK. We prove the existence of a point of small height in V?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. This generalizes and improves upon the results of Fukshansky (2006) 6] and 7]. As a 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.

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