Linear Forms in Logarithms of Rational Numbers

Abstract

The history of the theory of linear forms in logarithms is well known. We shall briefly sketch only some of the moments connected with new technical progress and important for our article. This theory was originated by pioneer works of A.O. Gelfond (see, for example, [5, 6]); with the help of the ideas which arose in connection with the solution of 7-th Hilbert problem (construction of auxiliary functions, extrapolation of zeros and small values), the bounds for the homogenous linear forms in two logarithms were proved. In the middle of the sixties A. Baker, [1], using auxiliary functions in several complex variables, for the first time obtained bounds for linear forms in any number of logarithms, both in the homogenous and non-homogenous cases. All further development of the theory is connected with improvements of these bounds. So N.I. Feldman introduced the so-called binomial polynomials in the construction of the auxiliary function, and due to this the dependence of the bounds on the coefficients of the linear forms was improved. The Kummer theory was used by A. Baker and H. Stark, [2], to improve the dependence of the estimates on the parameters related to the algebraic numbers appearing as arguments of the logarithms. The further improvements of this dependence are connected with the introduction of bounds for the number of zeros of polynomials on algebraic groups in works of G. Wüstholz. [14], P. Philippon and M. Waldschmidt, [11], A. Baker and G. Wüstholz, [3].

This research was partially supported by INTAS-RFBR grant No 97–1904.