Geodesic
An explicit lower bound for a~homogeneous rational linear form in logarithms of algebraic numbers
Izvestiya. Mathematics , Tome 62 (1998) no. 4, pp. 723-772

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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 $n^n$. 
@article{IM2_1998_62_4_a3,
     author = {E. M. Matveev},
     title = {An explicit lower bound for a~homogeneous rational linear form in logarithms of algebraic numbers},
     journal = {Izvestiya. Mathematics },
     pages = {723--772},
     publisher = {mathdoc},
     volume = {62},
     number = {4},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1998_62_4_a3/}
}
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E. M. Matveev. An explicit lower bound for a~homogeneous rational linear form in logarithms of algebraic numbers. Izvestiya. Mathematics , Tome 62 (1998) no. 4, pp. 723-772. http://geodesic.mathdoc.fr/item/IM2_1998_62_4_a3/