Computational Geometry of Positive Definite Quadratic Forms: Polyhedral Reduction Theories, Algorithms, and Applications
About this Title
Achill Schürmann, Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany
Publication: University Lecture Series
Publication Year 2008: Volume 48
ISBNs: 978-0-8218-4735-0 (print); 978-1-4704-1643-0 (online)
MathSciNet review: MR2466406
MSC: Primary 11H55; Secondary 05B40, 11J70, 20B25, 52-02, 52B15, 52B55
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.
Graduate students and research mathematicians interested in the geometry of numbers, discrete geometry, and computational mathematics.
Table of Contents
- 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