On the criteria for linear independence of Nesterenko, Fischler and Zudilin

TL;DR

This paper reviews and simplifies criteria for linear and algebraic independence of numbers, highlighting recent refinements that extend Nesterenko's original work with more straightforward proofs.

Contribution

The authors provide a simplified proof of Fischler and Zudilin's refined criterion, extending Nesterenko's criterion to algebraic independence with a new, unified approach.

Findings

Simplified proof of Fischler and Zudilin's criterion

Extension of Nesterenko's criterion to algebraic independence

Unified approach to linear and algebraic independence criteria

Abstract

In 1985, Yu. V. Nesterenko produced a criterion for linear independence, which is a variant of Siegel's. While Siegel uses upper bounds on full systems of forms, Nesterenko uses upper and lower bounds on sufficiently dense sequences of individual forms. The proof of Nesterenko's criterion was simplified by F. Amoroso and P. Colmez in 2003. More recently, S. Fischler and W. Zudilin produced a refinement, together with a much simpler proof. This new proof rests on a simple argument which we expand here. We get a new result, which contains Nesterenko's criterion, as well as criteria for algebraic independence.