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Invitation to Nonlinear Algebra: Invitation to Nonlinear Algebra
: Mateusz Michalek, Bernd Sturmfels
: American Mathematical Society
: 2021
: 242
:
: pdf (true)
: 10.1 MB

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.

Linear algebra is the foundation of much of mathematics, particularly applied mathematics. Numerical linear algebra is the basis of scientific computing, and its importance for the sciences and engineering can hardly be overestimated. The ubiquity of linear algebra has overshadowed the fairly recent growth in the use of nonlinear models across the mathematical sciences. There has been a proliferation of methods based on systems of multivariate polynomial equations and inequalities. This expansion is fueled by recent theoretical advances, development of efficient software, and an increased awareness of these tools. At the heart of this growing area lies algebraic geometry, but there are links to many other branches of mathematics, such as combinatorics, algebraic topology, commutative algebra, convex and discrete geometry, tensors and multilinear algebra, number theory, representation theory, and symbolic and numerical computation. Application areas include optimization, statistics, and complexity theory, among many others.

Nonlinear algebra is not simply a rebranding of algebraic geometry. It represents a recognition that a focus on computation and applications, and the theoretical underpinnings that this requires, results in a body of inquiry that is complementary to the existing curriculum. The term nonlinear algebra is intended to capture these trends, and to be more friendly to applied scientists.














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