JOURNAL OF NUMBER THEORY 5, 80-94 (1973) An Auxiliary Result in the Theory of Transcendental Numbers* R. TIJDEMAN’ School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 Communicated by K. Mahler Received March 25, 1971 This is an improvement on some estimates of exponential polynomials proved by Gelfond, Mahler, and Baker. This type of estimate is useful in the theory of transcendental numbers. In several proofs in the theory of transcendental numbers one finds functions of the type n-1 m,-1 n-1 E(z) = C C AwVzUe’+, E(z) + 0, m = C m, , v=o Leo “=O (0.0 where the numbers A,, and 01,are complex constants, such that on a large set of points E(z) and some of its derivatives assume small absolute values, say I JfwB,)l d E for 0 < p < r, - 1, 0 < u < s - 1. (0.2) This situation occurs for example in the proofs of some well known transcendence measures for P(ab) and P(log u/log b), where a and b are algebraic numbers and P(z) is a polynomial with integral coefficients, due to Gelfond [6, p. 164, formula (257) and p. 171, formula (294)]. More recently, Baker dealt with a similar situation in his famous papers, “Linear forms in the logarithms of algebraic numbers,” and some other papers [l-4; in particular 2, p. 104, formula (6)]. The function E(z) cannot vanish at “too many points.” It has been proved in [9, p. 58; IO], that the number of zeros of E(z) in a disk of * This work was supported by Air Force Office of Scientific Research grant AF- AFOSR-69-1712. +Author’s current address: Mathematisch Instituut, Wassenaarseweg 80, Leiden, The Netherlands. 80 Copyright 0 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.