Logarithmic forms and Diophantine geometry
“The quantity of recent results quoted in this book reveals the intense vitality of the subject,”summarizes Michel Waldschmidt his extensive review of Alan Baker's and Gisbert Wüstholz's “Logarithmic forms and Diophantine geometry” [New Mathematical Monographs 9. Cambridge: Cambridge University Press. (2007; Zbl 1145.11004)].
There is no doubt that the area has undergone a remarkable development during the last decades, and there is a lot of edge-cutting research going on. To quote the reviewer:“During the last 40 years, substantial progress has been made in transcendental number theory and its applications. New transcendence results have been obtained, a number of open problems have been solved, many applications have been developed, in particular in arithmetic algebraic geometry. One of the main source of the revival of the theory was the solution in 1966 by the first author of a problem raised by A. O. Gel'fond on proving linear independence measures for logarithms of algebraic numbers. This pioneering contribution of A. Baker to the theory has been extended in several directions by a number of specialists. Among them is the second author, and the book contributes to highlighting the fundamental works of each of the two authors. The main emphasis is therefore on the study of linear independence of logarithms of algebraic points on commutative algebraic groups, including linear independence measures as well as applications. The applications which are discussed in this book are essentially those related with Diophantine Geometry, where effective results are obtained thanks to linear independence estimates arising from transcendental number theory.”