Simultaneous lower bounds for linear forms in logarithms of algebraic numbers (Q2925392)

Let \(\Lambda_i=\beta_{i0}+\beta_{i1}u_1+\cdots+ \beta_{in}u_n\) (\(i=1,\dots,t\)) be linear combinations with algebraic coefficients \(\beta_{ij}\) of logarithms of algebraic numbers \(\alpha_j\), with \(\exp(u_j)=\alpha_j\) (\(j=1,\dots,n\)). The main result of this paper is a new, explicit and sharp lower bound for \(\max_{1\leq i\leq t} |\Lambda_i|_v\), where the absolute value \(v\) is either Archimedean or ultrametric. This estimate improves and generalizes several previous lower bounds obtained by different authors. The proof combines the main tools which have already been introduced in this context, from Baker's method, to Hirata's reduction, to Chudnovsky's process of variable change. The new strategy, the auxiliary section method, combines the idea of auxiliary function with the adelic approach involving slope theory. The author introduces a new tool, of independent interest, namely an absolute small value Siegel's lemma.