Invitation to Nonlinear Algebra (Graduate Studies in Mathematics, 211)

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. 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 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. This book offers an invitation to this broad and fast-developing area. "The pdf contains a draft title page, draft copyright page and a draft manuscript"-- Provided by publisher