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Linear independence measures for logarithms of algebraic numbers. (English) Zbl 1093.11054
Amoroso, Francesco (ed.) et al., Diophantine approximation. Lectures given at the C. I. M. E. summer school, Cetraro, Italy, June 28–July 6, 2000. Berlin: Springer (ISBN 3-540-40392-2/pbk). Lect. Notes Math. 1819, 249-344 (2003).
Let $$\alpha_1,\dots,\alpha_n$$ be non zero (complex) algebraic numbers and let $$\lambda_1,\dots,\lambda_n$$ be complex numbers such that $$e^{\lambda_i}=\alpha_i$$ for all $$i=1,\dots,n$$. A celebrated theorem of A. Baker [“Linear forms in the logarithms of algebraic numbers: I, II, III, IV”. Mathematika Lond. 13, 204–216 (1966; Zbl 0161.05201); ibid. 14, 102–107 (1967; Zbl 0161.05202); ibid. 14, 220–228 (1967; Zbl 0161.05301); ibid. 15, 204-216 (1968; Zbl 0169.37802)] asserts that if $$\lambda_1,\dots,\lambda_n$$ are linearly independent over the rationals, then they are linearly independent over the field of algebraic numbers. More important, Baker’s method provides a non-trivial lower bound for the quantity $$| b_0+b_1\lambda_1+\dots+b_n\lambda_n|$$, where $$b_0,\dots,b_n$$ are algebraic numbers. Much of what is known about effective solutions of Diophantine equations ultimately depends on such lower bounds. For this reason the subject has been much developed in the last forty years.
The paper under review consists of detailed notes of six lectures delivered by the author on this theory.
The first result (Theorem 1.1) states that for every positive integer $$n$$ the real number $$e^n$$ is not an integer. The author shows two proofs of this fact and leaves the open question of proving the lower bound $$| m-e^n| >n^{-c}$$ for a positive constant $$c$$ and all integers $$m,n\geq 2$$. Starting from this basic result, he proceeds to prove the newest sophisticated lower bounds for linear forms in logarithms. Many results are proved by using the method of interpolation determinant, introduced by M. Laurent in this context [“Sur quelques résultats récents de transcendence. Journées arithmétiques”, (Luminy 1989). Astérisque 198–200, 209–230(1991; Zbl 0762.11027)].
The outcome is a self-contained text presenting both classical results in transcendence theory and some of the most recent advances in the theory of linear forms in logarithms, carefully written by a leading contributor to this theory.
For the entire collection see [Zbl 1015.00016].

##### MSC:
 11J82 Measures of irrationality and of transcendence 11-02 Research exposition (monographs, survey articles) pertaining to number theory
##### Keywords:
linear forms in logarithms; Baker’s method; lower bounds