Introduction to Algebraic Independence Theory [Paperback]

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.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.PrefaceList of ContributorsChapter 1. PHI (tau,z) and Transcendence1. Differential rings and modular forms2. Explicit differential equations3. Singular values4. Transcendence on phi and zChapter 2. Mahler's conjecture and other transcendence results1. Introduction2. A proof of Mahler's conjecture3. K. Barr?'s work on modular functions4. Conjectures about modular and exponential functionsChapter 3. Algebraic independence for values of Ramanujan functions1. Main theorem and consequences2. How it can be proved?3. Constructions of the sequence of polynomials4. Algebraic fundamentals5. Another proof of Theorem 1.1Chapter 4. Some remarks on proofs of algebral#q