Transcendental Number Theory

Alan Baker's systematic account of transcendental number theory, with a new introduction and afterword explaining recent developments.

Alan Baker (Author), David Masser (Foreword by)

9781009229944, Cambridge University Press

Paperback / softback, published 9 June 2022

190 pages
22.8 x 15.1 x 1.1 cm, 0.29 kg

'Baker's book is the book on transcendental numbers. He covers a majority of those areas that have reached definitive results, presents most of the proofs in a complete yet far more compact form than hitherto available, and covers historical and bibliographical matters with great thoroughness and impeccable scholarship. As literature, it compares well with the finest works of Landau, Rademacher, and Titchmarsh.' Kenneth B. Stolarsky, Bulletin of the American Mathematical Society

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is, the theory of those numbers that cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory, which continues to see rapid progress today. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalization of the Thue–Siegel–Roth theorem, of Shidlovsky's work on Siegel's E-functions and of Sprindžuk's solution to the Mahler conjecture. This edition includes an introduction written by David Masser describing Baker's achievement, surveying the content of each chapter and explaining the main argument of Baker's method in broad strokes. A new afterword lists recent developments related to Baker's work.

Introduction David Masser
Preface
1. The origins
2. Linear forms in logarithms
3. Lower bounds for linear forms
4. Diophantine equations
5. Class numbers of imaginary quadratic fields
6. Elliptic functions
7. Rational approximations to algebraic numbers
8. Mahler's classification
9. Metrical theory
10. The exponential function
11. The Shiegel–Shidlovsky theorems
12. Algebraic independence
Bibliography
Original papers
Further publications
New developments
Some Developments since 1990 David Masser
Index.

Subject Areas: Algorithms & data structures [UMB], Number theory [PBH]