Analytic methods for Diophantine equations and Diophantine inequalities

✍ Scribed by Harold Davenport, Tim Browning

• Publisher: Cambridge University Press
• Year: 2005
• Tongue: English
• Leaves: 162
• Series: mathematical library
• Edition: 2
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Waring’s problem: Chapters 1–10......Page 9
Forms in many variables: Chapters 11–19......Page 13
Diophantine inequalities: Chapter 20......Page 17
Editorial preface......Page 21
1 Introduction......Page 23
2 Waring’s problem: history......Page 25
3 Weyl’s inequality and Hua’s inequality......Page 29
4 Waring’s problem: the asymptotic formula......Page 37
5 Waring’s problem: the singular series......Page 46
6 The singular series continued......Page 55
7 The equation…......Page 61
8 The equation…......Page 67
9 Waring’s problem: the number G(k)......Page 73
10 The equation…again......Page 85
11 General homogeneous equations: Birch’s theorem......Page 89
12 The geometry of numbers......Page 97
13 Cubic forms......Page 107
14 Cubic forms: bilinear equations......Page 114
15 Cubic forms: minor arcs and major arcs......Page 121
16 Cubic forms: the singular integral......Page 126
17 Cubic forms: the singular series......Page 129
18 Cubic forms: the p-adic problem......Page 133
19 Homogeneous equations of higher degree......Page 142
20 A Diophantine inequality......Page 147
References......Page 156
Index......Page 161