An explicit lower bound for a~homogeneous rational linear form in logarithms of algebraic numbers

E. M. Matveev

Izvestiya. Mathematics , Tome 62 (1998) no. 4, pp. 723-772

Résumé

In this paper we study linear forms $\Lambda=b_1\ln\alpha_1+\dots+b_n\ln\alpha_n$ with rational integer coefficients $b_j$ ($b_n\ne 0$, $n\geqslant 2$), where the $\alpha_j$ are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an explicit estimate of the form $$ |\Lambda|>\exp\bigl(-C^nD^{n+2}\Omega\ln\bigl(C^nD^{n+2}\Omega'\bigr)\ln(eB)\bigr). $$ Its novel feature is that it contains no factors of the form $n^n$.