Introduction to Algebraic Independence Theory - 1752 (2001 edition.)

In the last five years there has been very significant progress in the development of transcendence theory.

A new approach to the arithmetic properties of values of modular forms and theta-functions was found.

The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough.

The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.