Geometry of Numbers - 1st Edition - ISBN: 9780720421088, 9781483259277

Geometry of Numbers

1st Edition

Authors: C. G. Lekkerkerker
Editors: N. G. De Bruijn J. De Groot A. C. Zaanen
eBook ISBN: 9781483259277
Imprint: North Holland
Published Date: 1st January 1969
Page Count: 520
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Description

Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume VIII: Geometry of Numbers focuses on bodies and lattices in the n-dimensional euclidean space.

The text first discusses convex bodies and lattice points and the covering constant and inhomogeneous determinant of a set. Topics include the inhomogeneous determinant of a set, covering constant of a set, theorem of Minkowski-Hlawka, packing of convex bodies, successive minima and determinant of a set, successive minima of a convex body, extremal bodies, and polar reciprocal convex bodies. The publication ponders on star bodies, as well as points of critical lattices on the boundary, reducible, and irreducible star bodies and reduction of automorphic star bodies.

The manuscript reviews homogeneous and inhomogeneous s forms and some methods. Discussions focus on asymmetric inequalities, inhomogeneous forms in more variables, indefinite binary quadratic forms, diophantine approximation, sums of powers of linear forms, spheres and quadratic forms, and a method of Blichfeldt and Mordell.

The text is a dependable reference for researchers and mathematicians interested in bodies and lattices in the n-dimensional euclidean space.

Table of Contents


Chapter 1. Preliminaries

1. Notations. Convex Bodies

2. Ray Sets and Star Bodies

3. Lattices

4. Algebraic Number Fields

Chapter 2. Convex Bodies and Lattice Points

5. The Fundamental Theorem of Minkowski

6. Generalizations of the Theorem of Blichfeldt

7. Generalizations of the Theorem of Minkowski

8. A theorem of Rédei and Hlawka

9. Successive Minima of a Convex Body

10. Reduction Theory

11. Successive Minima of Non-Convex Sets

12. Extremal Bodies

13. The Inhomogeneous Minimum

14. Polar Reciprocal Convex Bodies

15. Compound Convex Bodies

16. Convex Bodies and Arbitrary Lattices

Chapter 3. The Critical Determinant, The Covering Constant and the Inhomogeneous Determinant of a Set

17. Mahler's Selection Theorem. Critical Determinant and Absolute Homogeneous Minimum. Critical Lattices

18. The Successive Minima and the Determinant of a Set

19. The Theorem of Minkowski-Hlawka

20. Packing of Convex Bodies

21. Covering Constant of a Set. Covering by Sets

22. Packings and Coverings in the Plane

23. Inhomogeneous Determinant of a Set

24. A Theorem of Mordell-Siegel-Hlawka-Rogers

Chapter 4. Star Bodies

25. The Functional Δ(S), Γ(S),f(Λ), g(Λ)

26. Points of Critical Lattices on the Boundary. Automorphic Star Bodies

27. Reducible and Irreducible Star Bodies

28. Reduction of Automorphic Star Bodies

Chapter 5. Some Methods

29. The Critical Determinant of a Two-Dimensional Star Body. Methods of Mahler and Mordell

30. Some Special Two-Dimensional Domains

31. The Critical Determinant of an n-Dimensional Domain

32. Some Special Domains

33. A Method of Blichfeldt. Density Functions

34. A Method of Blichfeldt and Mordell

35. A Theorem of Macbeath

36. Comparison of Star Bodies in Spaces of Unequal Dimensions

Chapter 6. Homogeneous Forms

37. Homogeneous Forms, Absolute Minima, Extreme Forms

38. Spheres and Quadratic Forms

39. Extreme Positive Definite Quadratic Forms

40. Sums of Powers of Linear Forms

41. Products of Linear Forms

42. Other Homogeneous Forms

43. Extreme Forms. Isolated Minima

44. Asymmetric and One-Sided Inequalities

45. Diophantine Approximation

Chapter 7. Inhomogeneous Forms

46. Inhomogeneous Minima of Forms

47. Indefinite Binary Quadratic Forms

48. Delone's Algorithm. Lower Bounds for μ (Q, Y)

49. Inhomogeneous Forms in More Variables

50. Asymmetric Inequalities

51. Inequalities with Infinitely Many Solutions

Bibliography

Index

Details

No. of pages:
520
Language:
English
Copyright:
© North Holland 1969
Published:
Imprint:
North Holland
eBook ISBN:
9781483259277

About the Author

C. G. Lekkerkerker

About the Editor

N. G. De Bruijn

J. De Groot

A. C. Zaanen