Diophantine Approximation on varieties III: Approximation of non-algebraic points by algebraic points

TL;DR

This paper proves that non-algebraic points on varieties can be approximated by algebraic points with bounded height and degree, leading to new criteria and estimates in transcendence and algebraic independence theory.

Contribution

It establishes optimal approximation results for non-algebraic points on varieties, enhancing tools for algebraic independence and transcendence theory.

Findings

Approximation of non-algebraic points by algebraic points is essentially optimal.

New algebraic independence criteria are derived.

Provides estimates for algebraic points of bounded height and degree.

Abstract

For $\theta$ a non-algebraic point on a quasi projective variety over a number field, I prove that $\theta$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result will include a proof of a slightly strengthened version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theory on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields.