Numerically Solving Polynomial Systems with Bertini - Bates, Daniel J

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Résumé :
This book is a guide to concepts and practice in numerical algebraic geometry - the solution of systems of polynomial equations by numerical methods. Through numerous examples, the authors show how to apply the well-received and widely used open-source Bertini software package to compute solutions, including a detailed manual on syntax and usage options. The authors also maintain a complementary web page where readers can find supplementary materials and Bertini input files.

Numerically Solving Polynomial Systems with Bertini approaches numerical algebraic geometry from a user's point of view with numerous examples of how Bertini is applicable to polynomial systems. It treats the fundamental task of solving a given polynomial system and describes the latest advances in the field, including algorithms for intersecting and projecting algebraic sets, methods for treating singular sets, the nascent field of real numerical algebraic geometry, and applications to large polynomial systems arising from differential equations.

Those who wish to solve polynomial systems can start gently by finding isolated solutions to small systems, advance rapidly to using algorithms for finding positive-dimensional solution sets (curves, surfaces, etc.), and learn how to use parallel computers on large problems. These techniques are of interest to engineers and scientists in fields where polynomial equations arise, including robotics, control theory, economics, physics, numerical PDEs, and computational chemistry....

Biographie:
Daniel Bates was a Postdoctoral Fellow at the Institute for Mathematics and its Applications (IMA) before starting as an Assistant Professor of Mathematics at Colorado State University in 2008. Professor Bates is a member of the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM) and is an active member of the SIAM Activity Group on Algebraic Geometry.

Sommaire:
List of figures; Conventions; Preface; Part I. Isolated Systems: 1. Polynomial systems; 2. Basic polynomial continuation; 3. Adaptive precision and endgames; 4. Projective space; 5. Types of homotopies; 6. Parameter homotopies; 7. Advanced topics about isolated solutions; Part II. Positive-Dimensional Solution Sets: 8. Positive-dimensional components; 9. Computing witness supersets; 10. The numerical irreducible decomposition; 11. Advanced topics about positive-dimensional solution sets; Part III. Further Algorithms and Applications: 12. Intersection; 13. Singular sets; 14. Real solutions; 15. Applications to algebraic geometry; 16. Projections of algebraic sets; 17. Big polynomial systems arising from differential equations; Part IV. Bertini User's Manual: Appendix A. Bertini quick start guide; Appendix B. Input format; Appendix C. Calling options; Appendix D. Output files; Appendix E. Configuration settings; Appendix F. Tips and tricks; Appendix G. Parallel computing; Appendix H. Related software; Bibliography; Software index; Subject index.