Algebraic points of small height missing a union of varieties

Abstract

Let $K$ be a number field, $\overline{\mathbb{Q}}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N\ge 2$. Let $Z_K$ be a union of varieties defined over $K$ such that $V⊈Z_K$. We prove the existence of a point of small height in $V\setminus Z_K$, 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. This generalizes and improves upon the previous results of the author. As a part of our argument, we provide a basic extension of the function field version of Siegel's lemma of J. Thunder 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.

20 pages; revised and improved version -- in particular, the main result now stands over function fields of any genus; to appear in the Journal of Number Theory -- an earlier version also appeared in the MPIM preprint series