On the linear independence of numbers over number fields

In the present paper, the problem of a lower bound for the measure of linear independence of a given collection of numbers₁, ,_n is considered under the assumption that, for a sequence of polynomials whose coefficients are algebraic integers, upper and lower estimates at the point (₁, ,_n) are known. We use a method that generalizes the Nesterenko method to the case of an arbitrary algebraic number field.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 506–517, October, 1998.The author wishes to thank Professor Yu. V. Nesterenko for setting the problem and valuable advice and Professor D. Bertrand for fruitful discussions.This research was partially supported by the International Science Foundation (Soros Foundation) under grant No. 507_s.