Geometry of Numbers

1st Edition - January 1, 1969
Author: C. G. Lekkerkerker
Editors: N. G. De Bruijn, J. De Groot, A. C. Zaanen
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 5 9 2 7 - 7

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… Read more

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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.

Chapter 1. Preliminaries 1. Notations. Convex Bodies 2. Ray Sets and Star Bodies 3. Lattices 4. Algebraic Number FieldsChapter 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 LatticesChapter 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-RogersChapter 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 BodiesChapter 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 DimensionsChapter 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 ApproximationChapter 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 SolutionsBibliographyIndex