Algebraic points of small height missing a union of varieties

TL;DR

This paper proves the existence of small-height points outside a union of varieties in a vector space over various fields, providing explicit bounds and extending Siegel's lemma to inhomogeneous heights.

Contribution

It introduces a new explicit bound for small-height points outside varieties, generalizing previous results and extending Siegel's lemma to inhomogeneous heights.

Findings

Existence of small-height points outside given varieties with explicit bounds

Extension of Siegel's lemma to inhomogeneous heights in function fields

Lower bounds for algebraic integers of bounded height in number fields

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 $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such that $V \nsubseteq Z_K$. 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.