Une version effective du th\'eor\`eme de Lindemann-Weierstrass par des m\'ethodes d'ind\'ependance alg\'ebrique

TL;DR

This paper introduces a new, simpler effective proof of the Lindemann-Weierstrass theorem using algebraic independence methods, improving previous estimates and making the proof more accessible.

Contribution

It provides a novel, simplified effective proof of the Lindemann-Weierstrass theorem based on algebraic independence, with improved bounds over prior methods.

Findings

Simpler construction of auxiliary functions

Effective bounds slightly weaker than Sert's estimate

Improved bounds over Ably's estimate using algebraic independence

Abstract

We present a new completely effective proof of the Lindemann-Weierstrass theorem based on algebraic independence methods. Although it is slightly weaker than the best known estimate due to A. Sert, it improves the best estimate due to M. Ably obtained by such methods. The novelty of the proof lies in the simplicity of the construction of auxiliary functions, a fact that we exploit to introduce the non-specialist to methods of algebraic independence.