Introduction to algebraic independence theory / Yuri V. Nesterenko, Patrice Philippon (eds.) ; with contributions from F. Amoroso [and others].

Contributor(s): Nesterenko, I︠U︡riĭ Valentinovich | Philippon, Patrice, 1954-
Material type: TextTextSeries: Lecture notes in mathematics (Springer-Verlag): 1752.Publisher: Berlin ; New York : Springer, ©2001Description: 1 online resource (xiii, 256 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783540445500; 3540445501Subject(s): Algebraic independence | Algebraic independenceGenre/Form: Electronic books. Additional physical formats: Print version:: Introduction to algebraic independence theory.DDC classification: 510 s | 512/.73 LOC classification: QA3 | .L28 no. 1752Online resources: Click here to access online
Contents:
PHI(tau, z) and Transcendence -- Mahler's conjecture and other transcendence results -- Algebraic independence for values of Ramanujan functions -- Some remarks in proofs of algebraic independence -- limination multihomogne -- Diophantine geometry -- Gomtrie 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 2: Large Transcendence Degrees -- Some metric results in Transcendental Numbers Theory -- The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence.
Summary: 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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Includes bibliographical references and index.

PHI(tau, z) and Transcendence -- Mahler's conjecture and other transcendence results -- Algebraic independence for values of Ramanujan functions -- Some remarks in proofs of algebraic independence -- limination multihomogne -- Diophantine geometry -- Gomtrie 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 2: Large Transcendence Degrees -- Some metric results in Transcendental Numbers Theory -- The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence.

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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