Diophantine Approximation on varieties V: Algebraic independence criteria

TL;DR

This paper develops new criteria for algebraic independence of complex numbers based on their approximation properties in projective space, extending previous methods with derivative evaluations and broader applicability.

Contribution

It introduces a new proof of an algebraic independence criterion and a novel criterion with wider applicability, building on approximation techniques.

Findings

Provides a new proof of Laurent and Roy's algebraic independence criterion.

Introduces a new algebraic independence criterion with broader applicability.

Extends approximation techniques by incorporating derivatives of global sections.

Abstract

For a tuple $(\theta_1,..,\theta_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by $\theta_1, ..., \theta_M$ over the field of rational numbers from the approximability of the point $\theta=(1,\theta_1,...,\theta_M)$ in projective space by hypersurfaces. The first given result is an new proof of an algebraic independence criterion, that was formerly proved by Laurent and Roy, and generalizes the Philippon criterion by introducing also evaluations of derivatives of global sections. The second result is a new kind of algebraic independence criteria that has a wider range of applicabilty than the first one.