Geometry of Numbers

Geometry of Numbers

1st Edition - January 1, 1969

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  • Author: C. G. Lekkerkerker
  • eBook ISBN: 9781483259277

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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



Product details

  • No. of pages: 520
  • Language: English
  • Copyright: © North Holland 1969
  • Published: January 1, 1969
  • Imprint: North Holland
  • eBook ISBN: 9781483259277

About the Author

C. G. Lekkerkerker

About the Editors

N. G. De Bruijn

J. De Groot

A. C. Zaanen

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