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Invitation to Nonlinear Algebra
About this book
Nonlinear algebra provides modern mathematical tools to address challenges arising in the sciences and engineering. It is useful everywhere, where polynomials appear: in particular, data and computational sciences, statistics, physics, optimization. The book offers an invitation to this broad and fast-developing area. It is not an extensive encyclopedia of known results, but rather a first introduction to the subject, allowing the reader to enter into more advanced topics. It was designed as the next step after linear algebra and well before abstract algebraic geometry. The book presents both classical topics—like the Nullstellensatz and primary decomposition—and more modern ones—like tropical geometry and semidefinite programming. The focus lies on interactions and applications. Each of the thirteen chapters introduces fundamental concepts. The book may be used for a one-semester course, and the over 200 exercises will help the readers to deepen their understanding of the subject.
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Information
Table of contents
- Publisher’s Notice
- Preface
- Acknowledgments
- Chapter 1. Polynomial Rings
- Chapter 2. Varieties
- Chapter 3. Solving and Decomposing
- Chapter 4. Mapping and Projecting
- Chapter 5. Linear Spaces and Grassmannians
- Chapter 6. Nullstellensätze
- Chapter 7. Tropical Algebra
- Chapter 8. Toric Varieties
- Chapter 9. Tensors
- Chapter 10. Representation Theory
- Chapter 11. Invariant Theory
- Chapter 12. Semidefinite Programming
- Chapter 13. Combinatorics
- Bibliography