Book contents
- Frontmatter
- Contents
- Foreword
- Editorial preface
- 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 homogeneous 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
- References
- Index
Foreword
Published online by Cambridge University Press: 08 January 2010
- Frontmatter
- Contents
- Foreword
- Editorial preface
- 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 homogeneous 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
- References
- Index
Summary
Waring's problem: Chapters 1–10
When Davenport produced these lecture notes there had been very little progress on Waring's problem since important work by Davenport and Vinogradov something like a quarter of a century earlier, and the main interest was to report on the more recent work on forms as described in the later chapters. Indeed there was a generally held view, with regard to Waring's problem at least, that they had extracted everything that could be obtained reasonably by the Hardy–Littlewood method and that the method was largely played out. Moreover, the material on Waring's problem was not intended, in general, to be state of the art, but rather simply an introduction to the Hardy–Littlewood method, with a minimum of fuss by a masterly expositor, which could then be developed as necessary for use in the study of the representation of zero by general integral forms, especially cubic forms, in the later chapters. There is no account of Davenport's own fundamental work on Waring's problem, namely G(4) = 16 (Davenport), G(5) ≤ 23, G(6) ≤ 36 (Davenport), nor of Vinogradov's G(k) ≤ 2k log k + o(k log k) for large k or Davenport's proof that almost all natural numbers are the sum of four positive cubes.
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- Chapter
- Information
- Analytic Methods for Diophantine Equations and Diophantine Inequalities , pp. vii - xviiiPublisher: Cambridge University PressPrint publication year: 2005