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.
Inhalt
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 (S), G(S),f( ), g( )
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