The Geometry of Numbers /

The Geometry of Numbers presents a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice-points on lines, circles, and inside simple polygons in the plane. Little mathematical expertise is required beyond an acquaintance with those objects a...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Otros Autores: Olds, C. D., Lax, Anneli, Davidoff, Giuliana P.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge : Cambridge University Press, 2012.
Colección:Anneli Lax new mathematical library.
Temas:
Geometry of numbers.
Number theory.
Géométrie des nombres.
Théorie des nombres.
Geometry of numbers
Number theory
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Title Page
  • Contents
  • Preface
  • Part I Lattice Points and Number Theory
  • 1 Lattice Points and Straight Lines
  • 1.1 The Fundamental Lattice
  • 1.2 Lines in Lattice Systems
  • 1.3 Lines with Rational Slope
  • 1.4 Lines with Irrational Slope
  • 1.5 Broadest Paths without Lattice Points
  • 1.6 Rectangles on Paths without Lattice Points
  • Problem Set for Chapter 1
  • References
  • 2 Counting Lattice Points
  • 2.1 The Greatest Integer Function, [x]
  • Problem Set for Section 2.1
  • 2.2 Positive Integral Solutions of ax + by = n
  • Problem Set for Section 2.22.3 Lattice Points inside a Triangle
  • Problem Set for Section 2.3
  • References
  • 3 Lattice Points and the Area of Polygons
  • 3.1 Points and Polygons
  • 3.2 Pick's Theorem
  • Problem Set for Section 3.2
  • 3.3 A Lattice Point Covering Theorem for Rectangles
  • Problem Set for Section 3.3
  • References
  • 4 Lattice Points in Circles
  • 4.1 How Many Lattice Points Are There?
  • 4.2 Sums of Two Squares
  • 4.3 Numbers Representable as a Sum of Two Squares
  • Problem Set for Section 4.3
  • 4.4 Representations of Prime Numbers as Sums of TwoSquares4.5 A Formula for R(n)
  • Problem Set for Section 4.5
  • References
  • Part II An Introduction to the Geometry of Numbers
  • 5 Minkowski's Fundamental Theorem
  • 5.1 Minkowski's Geometric Approach
  • Problem Set for Section 5.1
  • 5.2 Minkowski M-Sets
  • Problem Set for Section 5.2
  • 5.3 Minkowski's Fundamental Theorem
  • Problem Set for Section 5.3
  • 5.4 (Optional) Minkowski's Theorem in n Dimensions
  • References
  • 6 Applications of Minkowski's Theorems
  • 6.1 Approximating Real Numbers
  • 6.2 Minkowski's First TheoremProblem Set for Section 6.2
  • 6.3 Minkowski's Second Theorem
  • Problem for Section 6.3
  • 6.4 Approximating Irrational Numbers
  • 6.5 Minkowski's Third Theorem
  • 6.6 Simultaneous Diophantine Approximations
  • Reading Assignment for Chapter 6
  • References
  • 7 Linear Transformations and Integral Lattices
  • 7.1 Linear Transformations
  • Problem Set for Section 7.1
  • 7.2 The General Lattice
  • 7.3 Properties of the Fundamental Lattice
  • Problem Set for Section 7.3
  • 7.4 Visible Points
  • 8 Geometric Interpretations of Quadratic Forms8.1 Quadratic Representation
  • 8.2 An Upper Bound for the Minimum Positive Value
  • 8.3 An Improved Upper Bound
  • 8.4 (Optional) Bounds for the Minima of Quadratic Formsin More Than Two Variables
  • 8.5 Approximating by Rational Numbers
  • 8.6 Sums of Four Squares
  • References
  • 9 A New Principle in the Geometry of Numbers
  • 9.1 Blichfeldt's Theorem
  • 9.2 Proof of Blichfeldt's Theorem
  • 9.3 A Generalization of Blichfeldt's Theorem
  • 9.4 A Return to Minkowski's Theorem

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