Geodesic
The Hilbert polynomial and linear forms in the logarithms of algebraic numbers
Izvestiya. Mathematics , Tome 72 (2008) no. 6, pp. 1063-1110

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We prove a new estimate for homogeneous linear forms with integer coefficients in the logarithms of algebraic numbers. We obtain a qualitative improvement of the estimate depending on the coefficients of the linear form and the best value of the constant in the estimate in the case when the number of logarithms is not too large. 
@article{IM2_2008_72_6_a1,
     author = {Yu. M. Aleksentsev},
     title = {The {Hilbert} polynomial and linear forms in the logarithms of algebraic numbers},
     journal = {Izvestiya. Mathematics },
     pages = {1063--1110},
     publisher = {mathdoc},
     volume = {72},
     number = {6},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_6_a1/}
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Yu. M. Aleksentsev. The Hilbert polynomial and linear forms in the logarithms of algebraic numbers. Izvestiya. Mathematics , Tome 72 (2008) no. 6, pp. 1063-1110. http://geodesic.mathdoc.fr/item/IM2_2008_72_6_a1/