Book contents
- Frontmatter
- Contents
- Preface
- Contributors
- 1 On effective approximations to cubic irrationals
- 2 Applications of measure theory and Hausdorff dimension to the theory of Diophantine approximation
- 3 Galois representations and transcendental numbers
- 4 Some new results on algebraic independence of E-functions
- 5 Algebraic values of hypergeometric functions
- 6 Some new applications of an inequality of Mason
- 7 Aspects of the Hilbert Nullstellensatz
- 8 On the irrationality of certain series: problems and results
- 9 S-unit equations and their applications
- 10 Decomposable form equations
- 11 On Gelfond's method
- 12 On effective bounds for certain linear forms
- 13 Automata and transcendence
- 14 The study of Diophantine equations over function fields
- 15 Linear relations on algebraic groups
- 16 Estimates for the number of zeros of certain functions
- 17 An application of the S-unit theorem to modular forms on Γ0(N)
- 18 Lower bounds for linear forms in logarithms
- 19 Reducibility of lacunary polynomials, IX
- 20 The number of solutions of Thue equations
- 21 On arithmetic properties of the values of E-functions
- 22 Some exponential Diophantine equations
- 23 Arithmetic specializations theory
- 24 On the transcendence methods of Gelfond and Schneider in several variables
- 25 A new approach to Baker's theorem on linear forms in logarithms III
- 26 Linear forms in logarithms in the p-adic case
24 - On the transcendence methods of Gelfond and Schneider in several variables
Published online by Cambridge University Press: 05 January 2012
- Frontmatter
- Contents
- Preface
- Contributors
- 1 On effective approximations to cubic irrationals
- 2 Applications of measure theory and Hausdorff dimension to the theory of Diophantine approximation
- 3 Galois representations and transcendental numbers
- 4 Some new results on algebraic independence of E-functions
- 5 Algebraic values of hypergeometric functions
- 6 Some new applications of an inequality of Mason
- 7 Aspects of the Hilbert Nullstellensatz
- 8 On the irrationality of certain series: problems and results
- 9 S-unit equations and their applications
- 10 Decomposable form equations
- 11 On Gelfond's method
- 12 On effective bounds for certain linear forms
- 13 Automata and transcendence
- 14 The study of Diophantine equations over function fields
- 15 Linear relations on algebraic groups
- 16 Estimates for the number of zeros of certain functions
- 17 An application of the S-unit theorem to modular forms on Γ0(N)
- 18 Lower bounds for linear forms in logarithms
- 19 Reducibility of lacunary polynomials, IX
- 20 The number of solutions of Thue equations
- 21 On arithmetic properties of the values of E-functions
- 22 Some exponential Diophantine equations
- 23 Arithmetic specializations theory
- 24 On the transcendence methods of Gelfond and Schneider in several variables
- 25 A new approach to Baker's theorem on linear forms in logarithms III
- 26 Linear forms in logarithms in the p-adic case
Summary
Introduction
The methods we consider here were introduced by Gelfond and Schneider in their solutions of Hilbert's seventh problem on the transcendence of αβ (for algebraic α and β). Gelfond's proof [5] involved the two functions ez and eβz, with their derivatives, at the multiples of log α, while Schneider's proof [12] involved the two functions z and αz, evaluated at the points Z + Z.β (without derivatives).
Both methods have been extensively developed later. In his Bourbaki lecture [2], D. Bertrand pointed out a similarity between two of the most recent results which have been obtained, one by the method of Gelfond - Baker [16], and the other by Schneider's method [15].
The purpose of this paper is to prove a theorem which contains the two above-mentioned results, by combining the methods of Gelfond and Schneider.
Here is a corollary of our main result. Let G be a commutative algebraic group of dimension d ≥ 1 which is defined over the field of algebraic numbers. We denote by TG(ℂ) the tangent space of G at the origin, and by expG : TG(ℂ) → G(ℂ) the exponential map of the Lie group G(ℂ). Let do (resp. d1) be the dimension of the maximal unipotent (resp. multiplicative) factor of G, so that, where G2 is of dimension d2 = d - d0 - d1.
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- Information
- New Advances in Transcendence Theory , pp. 375 - 398Publisher: Cambridge University PressPrint publication year: 1988
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