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Computational Geometry of Positive Definite Quadratic Forms: Polyhedral Reduction Theories, Algorithms, and Applications
 
Achill Schürmann Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany
Computational Geometry of Positive Definite Quadratic Forms
Softcover ISBN:  978-0-8218-4735-0
Product Code:  ULECT/48
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-1643-0
Product Code:  ULECT/48.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-0-8218-4735-0
eBook: ISBN:  978-1-4704-1643-0
Product Code:  ULECT/48.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
Computational Geometry of Positive Definite Quadratic Forms
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Computational Geometry of Positive Definite Quadratic Forms: Polyhedral Reduction Theories, Algorithms, and Applications
Achill Schürmann Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany
Softcover ISBN:  978-0-8218-4735-0
Product Code:  ULECT/48
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-1643-0
Product Code:  ULECT/48.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-0-8218-4735-0
eBook ISBN:  978-1-4704-1643-0
Product Code:  ULECT/48.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
  • Book Details
     
     
    University Lecture Series
    Volume: 482008; 147 pp
    MSC: Primary 11; 52; 90; Secondary 20;

    Starting from classical arithmetical questions on quadratic forms, this book takes the reader step by step through the connections with lattice sphere packing and covering problems. As a model for polyhedral reduction theories of positive definite quadratic forms, Minkowski's classical theory is presented, including an application to multidimensional continued fraction expansions. The reduction theories of Voronoi are described in great detail, including full proofs, new views, and generalizations that cannot be found elsewhere. Based on Voronoi's second reduction theory, the local analysis of sphere coverings and several of its applications are presented. These include the classification of totally real thin number fields, connections to the Minkowski conjecture, and the discovery of new, sometimes surprising, properties of exceptional structures such as the Leech lattice or the root lattices.

    Throughout this book, special attention is paid to algorithms and computability, allowing computer-assisted treatments. Although dealing with relatively classical topics that have been worked on extensively by numerous authors, this book is exemplary in showing how computers may help to gain new insights.

    Readership

    Graduate students and research mathematicians interested in the geometry of numbers, discrete geometry, and computational mathematics.

  • Table of Contents
     
     
    • Chapters
    • Chapter 1. From quadratic forms to sphere packings and coverings
    • Chapter 2. Minkowski reduction
    • Chapter 3. Voronoi I
    • Chapter 4. Voronoi II
    • Chapter 5. Local analysis of coverings and applications
    • Appendix A. Polyhedral representation conversion under symmetries
    • Appendix B. Possible future projects
  • Reviews
     
     
    • The book is a valuable contribution to the existing literature, filling a big gap.

      Mathematical Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 482008; 147 pp
MSC: Primary 11; 52; 90; Secondary 20;

Starting from classical arithmetical questions on quadratic forms, this book takes the reader step by step through the connections with lattice sphere packing and covering problems. As a model for polyhedral reduction theories of positive definite quadratic forms, Minkowski's classical theory is presented, including an application to multidimensional continued fraction expansions. The reduction theories of Voronoi are described in great detail, including full proofs, new views, and generalizations that cannot be found elsewhere. Based on Voronoi's second reduction theory, the local analysis of sphere coverings and several of its applications are presented. These include the classification of totally real thin number fields, connections to the Minkowski conjecture, and the discovery of new, sometimes surprising, properties of exceptional structures such as the Leech lattice or the root lattices.

Throughout this book, special attention is paid to algorithms and computability, allowing computer-assisted treatments. Although dealing with relatively classical topics that have been worked on extensively by numerous authors, this book is exemplary in showing how computers may help to gain new insights.

Readership

Graduate students and research mathematicians interested in the geometry of numbers, discrete geometry, and computational mathematics.

  • Chapters
  • Chapter 1. From quadratic forms to sphere packings and coverings
  • Chapter 2. Minkowski reduction
  • Chapter 3. Voronoi I
  • Chapter 4. Voronoi II
  • Chapter 5. Local analysis of coverings and applications
  • Appendix A. Polyhedral representation conversion under symmetries
  • Appendix B. Possible future projects
  • The book is a valuable contribution to the existing literature, filling a big gap.

    Mathematical Reviews
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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