Linear Independence Measures for Logarithms of Algebraic Numbers

Second 2000 C.I.M.E. Session on Diophantine Approximation, International Mathematical Year
Cetraro (Cosenza), from June 28 to July 6, 2000.

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Table of content:

§1 Introduction to Transcendence Proofs (p.5-19)

§1.1 Sketch of Proof
§1.2 Tools for the Auxiliary Function
§1.3 Proof with an Auxiliary Function and without Zero Estimate
§1.4 Tools for the Interpolation Determinant Method
§1.5 Proof with an Interpolation Determinant and a Zero Estimate
§1.6 Remarks

§2. Extrapolation with Interpolation Determinants (p.20-29)

§2.1 Upper Bound for a Determinant in a Single Variable
§2.2 Proof of Hermite-Lindemann's Theorem with an Interpolation Determinant and without Zero Estimate

§3. Linear Independence of Logarithms of Algebraic Numbers (p.30-51)

§3.1 Introduction to Baker's Method
§3.2 Proof of Baker's Theorem
§3.3 Further Extrapolation with the Auxiliary Function
§3.4 Upper Bound for a Determinant in Several Variables
§3.5 Extrapolation with an Interpolation Determinant

§4. Introduction to Diophantine Approximation (p.52-66)

§4.1 On a Conjecture of Mahler
§4.2 Fel'dman's Polynomials
§4.3 Output of the Transcendence Argument
§4.4 From Polynomial Approximation to Algebraic Approximation
§4.5 Proof of Theorem 4.5

§5. Measures of Linear Independence of Logarithms of Algebraic Numbers (p.67-81)

§5.1 Introduction
§5.2 Baker's Method with an Auxiliary Function
Appendix : From 2 to n Logarithms

§6. Matveev's Theorem with Interpolation Determinants (p.85-90)

§6.1 First Extrapolation
§6.2 Using Kummer's Condition
§6.3 Second Extrapolation
§6.4 An Approximate Schwarz Lemma for Interpolation Determinants

References (p.91-92)