| Résume | Lower bounds for linear forms\Lambda=b_1\log\alpha_1+...+b_n\log\alpha_n, (b_j,\alpha_j\in\overline Q),have numerous applications to different problems of Number Theory.Usually these bounds for |\Lambda| are expressed in terms of theheights h(\alpha_j), in b_j and of the degree of the field Kgenerated by the numbers \alpha_j. After a long line of resultsfollowing A.O.Gelfond (1935-1949, n=2) and A.Baker (1966, n\geq 3)the dependence of the bounds in these parameters is practicallyestablished. In a recent article (1998) E.Matveev has improved thedependence in terms of n. In the special case where\deg_K K(\alpha_1^{1/2},...,\alpha_n^{1/2})=2^n {1}he reduced a factor n^n in the bound to C^n, where C is anabsolute constant. Now he proved such result in the general casewithout the extra condition {1}. This proof is a subject of the talk. |
