Introduction to Algebraic Independence Theory

Yuri V. Nesterenko, Patrice Philippon (eds.)9783540414964, 3540414967

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.

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Table of contents :
Θ(τ, z ) and Transcendence….Pages 1-11
Mahler’s conjecture and other transcendence Results….Pages 13-26
Algebraic independence for values of Ramanujan Functions….Pages 27-46
Some remarks on proofs of algebraic independence….Pages 47-51
Elimination multihomogene….Pages 53-81
Diophantine geometry….Pages 83-94
Géométrie diophantienne multiprojective….Pages 95-131
Criteria for algebraic independence….Pages 133-141
Upper bounds for (geometric) Hilbert functions….Pages 143-148
Multiplicity estimates for solutions of algebraic differential equations….Pages 149-165
Zero Estimates on Commutative Algebraic Groups….Pages 167-185
Measures of algebraic independence for Mahler functions….Pages 187-197
Algebraic Independence in Algebraic Groups. Part 1: Small Transcendence Degrees….Pages 199-211
Algebraic Independence in Algebraic Groups. Part II: Large Transcendence Degrees….Pages 213-225
Some metric results in Transcendental Numbers Theory….Pages 227-237
The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence….Pages 239-248