Introduction to Algebraic Independence Theory

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.

?(?, z) and Transcendence.- Mahler’s conjecture and other transcendence Results.- Algebraic independence for values of Ramanujan Functions.- Some remarks on proofs of algebraic independence.- Elimination multihomogene.- Diophantine geometry.- Géométrie diophantienne multiprojective.- Criteria for algebraic independence.- Upper bounds for (geometric) Hilbert functions.- Multiplicity estimates for solutions of algebraic differential equations.- Zero Estimates on Commutative Algebraic Groups.- Measures of algebraic independence for Mahler functions.- Algebraic Independence in Algebraic Groups. Part 1: Small Transcendence Degrees.- Algebraic Independence in Algebraic Groups. Part II: Large Transcendence Degrees.- Some metric results in Transcendental Numbers Theory.- The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence.