By Michel Waldschmidt
The idea of transcendental numbers is heavily with regards to the examine of diophantine approximation. This booklet bargains with values of the standard exponential functionality e^z. A significant open challenge is the conjecture on algebraic independence of logarithms of algebraic numbers. This e-book contains proofs of the most easy effects (theorems of Hermite-Lindemann, Gelfond-Schneider, 6 exponentials theorem), an advent to peak capabilities with a dialogue of Lehmer's challenge, numerous proofs of Baker's theorem in addition to specific measures of linear independence of logarithms. An unique characteristic is that proofs make systematic use of Laurent's interpolation determinants. the main basic result's the so-called Theorem of the Linear Subgroup, an efficient model of that is additionally incorporated. It yields new result of simultaneous approximation and of algebraic independence. 2 chapters written through D. Roy supply whole and while simplified proofs of 0 estimates (due to P. Philippon) on linear algebraic groups.