Michel Waldschmidt

Linear forms in two logarithms and Schneider's method. II

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Linear Independence of Logarithms of Algebraic Numbers

In Chap. 4, we proved Baker’s homogeneous Theorem 1.5: if logarithms of algebraic numbers are linearly independent over ℚ, then they are linearly independent over \( \overline {\Bbb Q} \). The proof was an extension of Gel’fond’s solution to Hilbert’s seventh problem. Here we give a second proof of the same theorem, using an extension of Schneider’s method. The two main tools are an upper bound for the absolute value of an alternant in several variables (Proposition 6.6) and the zero estimate (namely Theorem 5.1).

Simultaneous Approximation and Algebraic Independence

AbstractWe establish several new measures of simultaneous algebraic approximations for families of complex numbers

Diophantine approximation by conjugate algebraic integers

(\theta _1 ,....,\theta _n )

Recent advances in Diophantine approximation

related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the field

Density measure of rational points on Abelian varieties

\mathbb{Q}(\theta _1 ,....,\theta _n )