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.
Author(s): Mateusz Michałek, Bernd Sturmfels
Series: Graduate Studies in Mathematics 211
Edition: 1
Publisher: American Mathematical Society
Year: 2021
Language: English
Pages: 226
Tags: Nonlinear, Algebra
Cover 1
Title page 4
Preface 12
Acknowledgments 14
Chapter 1. Polynomial Rings 16
1.1. Ideals 16
1.2. Gröbner Bases 21
1.3. Dimension and Degree 27
Exercises 32
Chapter 2. Varieties 34
2.1. Affine Varieties 34
2.2. Projective Varieties 42
2.3. Geometry in Low Dimensions 46
Exercises 51
Chapter 3. Solving and Decomposing 54
3.1. 0-Dimensional Ideals 54
3.2. Primary Decomposition 58
3.3. Linear PDEs with Constant Coefficients 64
Exercises 70
Chapter 4. Mapping and Projecting 72
4.1. Elimination 72
4.2. Implicitization 76
4.3. The Image of a Polynomial Map 81
Exercises 86
Chapter 5. Linear Spaces and Grassmannians 88
5.1. Coordinates for Linear Spaces 88
5.2. Plücker Relations 91
5.3. Schubert Calculus 94
Exercises 99
Chapter 6. Nullstellensätze 102
6.1. Certificates for Infeasibility 102
6.2. Hilbert’s Nullstellensatz 105
6.3. Let’s Get Real 107
Exercises 111
Chapter 7. Tropical Algebra 114
7.1. Arithmetic and Valuations 114
7.2. Linear Algebra 119
7.3. Tropical Varieties 124
Exercises 128
Chapter 8. Toric Varieties 130
8.1. The Affine Story 130
8.2. Varieties from Polytopes 136
8.3. The World Is Toric 141
Exercises 145
Chapter 9. Tensors 148
9.1. Eigenvectors 148
9.2. Tensor Rank 155
9.3. Matrix Multiplication 161
Exercises 164
Chapter 10. Representation Theory 166
10.1. Groups, Representations and Characters 166
10.2. Invertible Matrices and Permutations 173
10.3. Exploiting Symmetry 179
Exercises 184
Chapter 11. Invariant Theory 186
11.1. Finite Groups 186
11.2. Classical Invariant Theory 191
11.3. Geometric Invariant Theory 194
Exercises 199
Chapter 12. Semidefinite Programming 202
12.1. Spectrahedra 202
12.2. Optimization and Duality 206
12.3. Sums of Squares 211
Exercises 215
Chapter 13. Combinatorics 216
13.1. Matroids 216
13.2. Lattice Polytopes 221
13.3. Generating Functions 226
Exercises 231
Bibliography 234
Index 238
