Analytic Methods for Diophantine Equations and Diophantine Inequalities (Cambridge Mathematical Library)
Cambridge University Press
Edition: 2nd Revised ed., 3/3/2005
EAN 9780521605830, ISBN10: 0521605830
Paperback, 160 pages, 22.8 x 15.2 x 0.9 cm
Language: English
Originally published in English
Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities. It provides an excellent introduction to a timeless area of number theory that is still as widely researched today as it was when the book originally appeared. The three main themes of the book are Waring's problem and the representation of integers by diagonal forms, the solubility in integers of systems of forms in many variables, and the solubility in integers of diagonal inequalities. For the second edition of the book a comprehensive foreword has been added in which three prominent authorities describe the modern context and recent developments. A thorough bibliography has also been added.
Preface
Foreword
1. Introduction
2. Waring's problem
history
3. Weyl's inequality and Hua's inequality
4. Waring's problem
the asymptotic formula
5. Waring's problem
the singular series
6. The singular series continued
7. The equation C1xk1 +…+ Csxks=N
8. The equation C1xk1 +…+ Csxks=0
9. Waring's problem
the number G (k)
10. The equation C1xk1 +…+ Csxks=0 again
11. General homoogeneous equations
Birch's theorem
12. The geometry of numbers
13. cubic forms
14. Cubic forms
bilinear equations
15. Cubic forms
minor arcs and major arcs
16. Cubic forms
the singular integral
17. Cubic forms
the singular series
18. Cubic forms
the p-adic problem
19. Homogeneous equations of higher degree
20. A Diophantine inequality
Bibliography
Index.