Auxiliary Polynomials in Number Theory

Description

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 Content: 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 - Polya 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: Neron's square root References Index.