Transcendental Numbers

Andrei B. Shidlovskii, Neal Koblitz

• Publisher: De Gruyter
• Year: 2011
• Print ISBN: 9783110115680
• eBook ISBN: 9783110889055
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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Table of contents

• Foreword
• Preface to the English edition
• Preface
• Notation
• Introduction
• § 1. Approximation of algebraic numbers
• § 2. The classical method of Hermite-Lindemann
• § 3. Methods arising from the solution of Hilbert’s Seventh Problem, and their subsequent development
• § 4. Siegel’s method and its further development
• Chapter 1. Approximation of real and algebraic numbers
• § 1. Approximation of real numbers by algebraic numbers
• § 2. Simultaneous approximation
• § 3. Approximation of algebraic numbers by rational numbers
• § 4. Approximation of algebraic numbers by algebraic numbers
• § 5. Further refinements and generalizations of Liouville’s Theorem
• Remarks
• Chapter 2. Arithmetic properties of the values of the exponential function at algebraic points
• § 1. Transcendence of e
• § 2. Transcendence of π
• § 3. Transcendence of the values of the exponential function at algebraic points
• § 4. Approximation of ez by rational functions
• § 5. Linear approximating forms for eρ1z,..., eρmz
• § 6. A set of linear approximating forms
• § 7. Lindemann’s Theorem
• § 8. Linear approximating forms and the Newton interpolation series for the exponential function
• Remarks
• Chapter 3. Transcendence and algebraic independence of the values of Ε-functions which are not connected by algebraic equations over the field of rational functions
• § 1. E-functions
• § 2. The First Fundamental Theorem
• § 3. Some properties of linear and fractional-linear forms
• § 4. Properties of linear forms in functions which satisfy a system of homogeneous linear differential equations
• § 5. Order of zero of a linear form at z = 0
• § 6. The determinant of a set of linear forms
• § 7. Passing to linearly independent numerical linear forms
• § 8. Auxiliary lemmas on solutions of systems of homogeneous linear equations
• § 9. Functional linear approximating forms
• § 10. Numerical linear approximating forms
• § 11. Rank of the m-tuple f1(ξ),..., fm(ξ)
• § 12. Proof of the First Fundamental Theorem
• § 13. Consequences of the First Fundamental Theorem
• Remarks
• Chapter 4. Transcendence and algebraic independence of the values of Ε-functions which are connected by algebraic equations over the field of rational functions
• § 1. Rank of the m-tuple f1(ξ),..., fm(ξ)
• § 2. Some lemmas
• § 3. Estimate for the dimension of a vector space spanned by monomials in elements of a field extension
• § 4. The Third Fundamental Theorem
• § 5. Transcendence of the values of Ε-functions connected by arbitrary algebraic equations over C(z)
• § 6. Algebraic independence of the values of E-functions which are connected by arbitrary algebraic equations over C(z)
• § 7. Ε-functions connected by special types of equations
• § 8. Ε-functions connected by algebraic equations with constant coefficients
• § 9. Ε-functions which are connected by a single algebraic equation over C(z)
• § 10. Minimal equations
• § 11. Dimension of the vector spaces spanned by monomials in the elements of a field extension
• § 12. Algebraic independence of the values of IE-functions
• § 13. Algebraic independence of the values of KE-functions
• Remarks
• Chapter 5. Transcendence and algebraic independence of the values of E-functions which satisfy first order linear differential equations
• § 1. Hypergeometric E-functions
• § 2. The simplest hypergeometric E-functions
• § 3. Sets of solutions of first order linear differential equations
• § 4. Some lemmas
• § 5. Proof of the theorems
• Remarks
• Chapter 6. Algebraic independence of the values of E-functions which satisfy second order linear differential equations
• § 1. A general theorem on algebraic independence of the values of an Ε-function and its derivative
• § 2. The functions Κλ(z) associated to Bessel functions
• § 3. The functions Κλ(z) and
• § 4. Kummer functions
• § 5. Solutions of non-homogeneous linear differential equations
• Remarks
• Chapter 7. Solutions of certain linear differential equations of arbitrary order
• § 1. Solutions of non-homogeneous differential equations
• § 2. Solutions of homogeneous differential equations
• § 3. Corollaries of Theorems 1 and 2
• Remarks
• Chapter 8. Arithmetic methods applied to solutions of linear differential equations of arbitrary order
• § 1. Statement of the theorems
• § 2. Auxiliary lemmas
• § 3. Proof of Theorems 1–5
• § 4. Proof of Theorems 6 and 7
• § 5. Further results
• Chapter 9. Siegel’s Theorem
• § 1. Statement of the theorem and some basic auxiliary results
• § 2. Some lemmas
• § 3. Some properties of solutions of second order homogeneous linear differential equations
• § 4. Algebraic independence of solutions of a set of second order homogeneous linear differential equations
• § 5. Proof of Siegel’s Theorem
• § 6. Solutions of non-homogeneous linear differential equations
• § 7. Generalizations of Siegel’s Theorem
• Chapter 10. Solutions of linear differential equations of prime order p
• § 1. Statement of the basic results
• § 2. The homogeneous ideal I
• § 3. Algebraic functions of several variables
• §4. The differential operator G
• § 5. The differential operators S and δ
• § 6. A lemma on linear approximation
• § 7. End of the proof of Theorem 7
• § 8. Linear reducibility
• § 9. Proof of Theorems 6 and 5
• Remarks
• Chapter 11. The algebraic independence measure of values of IE-functions
• § 1. Definition of the measures
• § 2. The linear independence measure of values of IE-functions
• § 3. The algebraic independence measure of values of IE-functions which are not connected by algebraic equations over C(z)
• § 4. Auxiliary results
• § 5. The algebraic independence measure of values of IE-functions which are connected by algebraic equations over C(z)
• § 6. Some applications of the general theorems
• Remarks
• Chapter 12. The algebraic independence measure of values of KE-functions
• § 1. The fundamental lemma
• § 2. Bounds for the measures of the values of E-functions which are not connected by algebraic equations over C(z)
• § 3. Bounds for the measures of the values of E-functions which are connected by a single algebraic equation over C(z)
• § 4. Bounds for the measures of the values of E-functions which are connected by arbitrary algebraic equations over C(z)
• § 5. Algebraic independence of the values of E-functions in conjugate fields
• § 6. An auxiliary theorem
• § 7. Consequences of the auxiliary theorem
• § 8. Some applications of the general theorems
• Remarks
• Chapter 13. Effective bounds for measures
• § 1. Definitions and notation
• § 2. Refinement of the fundamental lemmas
• § 3. Bounds for linear independence measures
• § 4. Bounds for algebraic independence measures
• § 5. Some applications of the general theorems
• Remarks
• Concluding remarks
• Supplementary remarks on recent work for the English edition
• Bibliography