An effective version of the Lindemann-Weierstrass theorem for interpolation determinants
The best known quantitative version of the Lindemann-Weierstrass theorem was given by \textit{M. Ably} [Acta Arith. 67, 29-45 (1994; Zbl 0813.11042)] by using Philippon's criterion for algebraic independence. Here the author uses Laurent's interpolation determinants to obtain a lower bound of the same type for \(|P(e^{\alpha_1},\dots, e^{\alpha_p})|\) where the \(\alpha_j\) are complex algebraic numbers, linearly independent over \(\mathbb{Q}\), and \(P\) is a polynomial with algebraic coefficients. In comparison with Ably's result (where \(P\) has rational coefficients) this bound is a little better for the dependence on \(P\) and especially entirely explicit for the dependence on the \(\alpha_j\). This paper is a very nice piece of virtuosity.
scientific article; zbMATH DE number 1310086