Librería Portfolio

This unified account of various aspects of a powerful classical method, easy to understand in its simplest forms, is illustrated by applications in several areas of number theory. As well as including diophantine approximation and transcendence, which were mainly responsible for its invention, the author places the method in a broader context by exploring its application in other areas, such as exponential sums and counting problems in both finite fields and the field of rationals. Throughout the book, the method is explained in a ´molecular´ fashion, where key ideas are introduced independently. Each application is the most elementary significant example of its kind and appears with detailed references to subsequent developments, making it accessible to advanced undergraduates as well as postgraduate students in number theory or related areas. It provides over 700 exercises both guiding and challenging, while the broad array of applications should interest professionals in fields from number theory to algebraic geometry.

The method is placed in a broad context through the inclusion of applications outside diophantine approximation and transcendence
Key ideas are introduced independently, along with the motivation for each one
Includes over 700 exercises ranging from simple to challenging



Table of Contents
Introduction
1. Prologue
2. Irrationality I
3. Irrationality II - Mahler´s method
4. Diophantine equations - Runge´s method
5. Irreducibility
6. Elliptic curves - Stepanov´s method
7. Exponential sums
8. Irrationality measures I - Mahler
9. Integer-valued entire functions I - Pólya
10. Integer-valued entire functions II - Gramain
11. Transcendence I - Mahler
12. Irrationality measures II - Thue
13. Transcendence II - Hermite-Lindemann
14. Heights
15. Equidistribution - Bilu
16. Height lower bounds - Dobrowolski
17. Height upper bounds
18. Counting - Bombieri-Pila
19. Transcendence III - Gelfond-Schneider-Lang
20. Elliptic functions
21. Modular functions
22. Algebraic independence
Appendix: Néron´s square root
References
Index.