9783540403920

The C.I.M.E. session in Diophantine Approximation, held in Cetraro (Italy) June 28 - July 6, 2000 focused on height theory, linear independence and transcendence in group varieties, Baker's method, approximations to algebraic numbers and applications to polynomial-exponential diophantine equations and to diophantine theory of linear recurrences. Very fine lectures by D. Masser, Y. Nesterenko, H.-P. Schlickewei, W.M. Schmidt and M. Waldschmidt have resulted giving a good overview of these topics, and describing central results, both classical and recent, emphasizing the new methods and ideas of the proofs rather than the details. They are addressed to a wide audience and do not require any prior specific knowledge.

Heights, Transcendence, and Linear Independence on Commutative Group Varieties

1

(52)

David Masser

First lecture. Introduction and basic techniques

1

(7)

Second lecture. More on heights

8

(9)

Third lecture. Elliptic functions and elliptic curves

17

(7)

Fourth lecture. Linear forms in elliptic logarithms

24

(8)

Fifth lecture. Abelian varieties

32

(7)

Sixth Lecture. Commutative group varieties

39

(14)

References

47

(6)

Linear Forms in Logarithms of Rational Numbers

53

(54)

Yuri Nesterenko

Introduction

53

(1)

Main result and induction assumption

54

(5)

Construction of auxiliary function

59

(20)

Binomial polynomial

59

(3)

Siegel's lemma with weights

62

(4)

Some topics from the geometry of numbers

66

(3)

Upper bound for an index

69

(2)

Construction

71

(8)

Extrapolation of zeros

79

(16)

Interpolation formula

81

(2)

Extrapolation of zeros in Q

83

(7)

Extrapolation with Kummer descent

90

(5)

Zero estimates and the end of the proof of Theorem 2.1

95

(12)

Zero estimates on linear algebraic groups

97

(1)

Construction of the sublattice Φ from Proposition 2.6

98

(8)

References

106

(1)

Approximation of Algebraic Numbers

107

(64)

Hans Peter Schlickewei

Results

107

(5)

Roth's proof of theorem 1.1

112

(6)

Vanishing

113

(3)

Non-Vanishing

116

(1)

Conclusion

117

(1)

Schmidt's proof of theorem 1.2

118

(12)

Parallelepipeds

118

(2)

The approximation part

120

(7)

The geometry part

127

(3)

The proof of theorem 1.3

130

(14)

Parallelepipeds

131

(4)

The approximation part

135

(2)

The geometry part

137

(7)

Generalization of theorem 1.4

144

(17)

Parallelepipeds

146

(4)

The approximation part

150

(11)

Gap principles

161

(10)

Vanishing determinants

161

(5)

Application of Minkowski's Theorem

166

(4)

References

170

(1)

Linear Recurrence Sequences

171

(78)

Wolfgang M. Schmidt

Introduction

171

(2)

Functions of Polynomial-Exponential Type

173

(6)

Generating Functions

179

(1)

Factorization of Polynomial-Exponential Functions

180

(5)

Gourin's Theorem

185

(5)

Hadamard Products, Quotients and Roots

190

(2)

The Zero-Multiplicity, and Polynomial-Exponential Equations

192

(3)

Proof of Laurent's Theorem in the Number Field Case

195

(5)

A Specialization Argument

200

(2)

A Method of Zannier Using Derivations

202

(5)

Applications to Linear Recurrences

207

(6)

Bounds for the Number of Solutions of Polynomial-Exponential Equations

213

(5)

The Bavencoffe-Bezivin Sequence

218

(5)

Proof of Evertse's Theorem on Roots of Unity

223

(3)

Reductions for Theorem 12.3

226

(3)

Special Solutions

229

(2)

Properties of Special Solutions

231

(3)

Large Solutions

234

(1)

Small Solutions, and the end of the proof of Theorem 12.3

235

(1)

Linear Recurrence Sequences Again

236

(7)

Final Remarks

243

(6)

References

245

(4)

Linear Independence Measures for Logarithms of Algebraic Numbers

249

Michel Waldschmidt

First Lecture. Introduction to Transcendence Proofs

252

Sketch of Proof

252

Tools for the Auxiliary Function

253

Proof with an Auxiliary Function and without Zero Estimate

255

Tools for the Interpolation Determinant Method

260

Proof with an Interpolation Determinant and a Zero Estimate

261

Remarks

262

Second Lecture. Extrapolation with Interpolation Determinants

267

Upper Bound for a Determinant in a Single Variable

267

Proof of Hermite-Lindemann's Theorem with an Interpolation Determinant and without Zero Estimate

273

Third Lecture. Linear Independence of Logarithms of Algebraic Numbers

277

Introduction to Baker's Method

278

Proof of Baker's Theorem

283

Further Extrapolation with the Auxiliary Function

289

Upper Bound for a Determinant in Several Variables

291

Extrapolation with an Interpolation Determinant

297

Fourth Lecture. Introduction to Diophantine Approximation

300

On a Conjecture of Mahler

300

Fel'dman's Polynomials

306

Output of the Transcendence Argument

307

From Polynomial Approximation to Algebraic Approximation

312

Proof of Theorem 4.2

315

Fifth Lecture. Measures of Linear Independence of Logarithms of Algebraic Numbers

316

Introduction

316

Baker's Method with an Auxiliary Function

318

Sixth Lecture. Matveev's Theorem with Interpolation Determinants

336

First Extrapolation

337

Using Kummer's Condition

338

Second Extrapolation

340

An Approximate Schwarz Lemma for Interpolation Determinants

341

References

342

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Diophantine Approximation : Lectures Given at the C. I. M. E. Summer School Held in Cetraro, Italy, June 28 - July 6 2000

3540403922

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