Minorations simultan\'ees de formes lin\'eaires de logarithmes de nombres alg\'ebriques

TL;DR

This paper advances the theory of linear forms in logarithms of algebraic numbers by providing new lower bounds in both archimedean and p-adic contexts, integrating modern adelic slope tools and a novel Siegel's lemma.

Contribution

It introduces a new approach by combining classical methods with modern adelic slope theory and a new small values Siegel's lemma for lower bounds.

Findings

Established lower bounds for simultaneous linear forms in logarithms.

Unified treatment of archimedean and p-adic cases.

Integrated modern adelic tools into classical proof techniques.

Abstract

This work falls within the theory of linear forms in logarithms over a commutative linear group defined over a number field. We give lower bounds for simultaneous linear forms in logarithms of algebraic numbers, treating both the archimedean and $p$-adic cases. The proof includes Baker's method, Hirata's reduction, Chudnovsky's process of variable change. The novelty is that we integrated into the proof the modern tools of adelic slope theory, using also a new small values Siegel's lemma.