TY  - BOOK
AU  - Nesterenko,Yuri V.
AU  - Philippon,Patrice
ED  - SpringerLink (Online service)
TI  - Introduction to Algebraic Independence Theory
T2  - Lecture Notes in Mathematics,
SN  - 9783540445500
AV  - QA241-247.5 
U1  - 512.7 23
PY  - 2001///
CY  - Berlin, Heidelberg
PB  - Springer Berlin Heidelberg
KW  - Mathematics
KW  - Geometry, algebraic
KW  - Number theory
KW  - Number Theory
KW  - Algebraic Geometry
N1  - ?(?, 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
N2  - 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
UR  - http://dx.doi.org/10.1007/b76882
ER  -