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Invitation to nonlinear algebra. (English) Zbl 1477.14001

Graduate Studies in Mathematics 211. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-6551-3/pbk; 978-1-4704-6308-3/ebook). xiii, 226 p. (2021).
The book under review is an introduction to several areas of basic and advanced algebraic geometry, combinatorics, convex geometry, multilinear algebra, representation theory and optimization. It is organized in \(13\) chapters, each followed by an exercises section. The authors assume no previous knowledge of the topics and give references of additional sources, for further study.
The first six chapters consist of an introduction to algebraic geometry. The topics covered include Gröbner bases, Hilbert’s basis theorem, Hilbert function and series, the Hilbert polynomial, dimension, degree, affine and projective varieties, Zariski topology, irreducibility, primary decomposition (with an application to linear PDEs), elimination and implicitization, resultants, Chevalley’s Theorem, Grassmannians, Plücker relations, Schubert calculus, the Nullstellensätze and the Positivstellensatz. For proofs of the real Nullstellenstaz and the Positivstellensatz, the authors refer to [M. Marshall, Positive polynomials and sums of squares. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1169.13001)].
Chapter \(7\) sees to the introduction of the concepts of the tropical semiring and tropical varieties. For much of the details of the general theory the authors refer to [D. Maclagan and B. Sturmfels, Introduction to tropical geometry. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1321.14048)]. This chapter also contains a section dedicated to the applications of tropical linear algebra, making reference to [P. Butkovič, Max-linear systems. Theory and algorithms. London: Springer (2010; Zbl 1202.15032)]. In chapter \(8\), affine and projective toric varieties and ideals are described, as well as their connection to rational convex polyhedral cones and lattice polytopes. Chapter \(9\) is an introduction to the spectral theory of tensors, tensor rank and border rank, with a section dedicated to the tensor of matrix multiplication. Chapter \(10\) gives an introduction to representation theory, from basic concepts to the representation theory of the general and special linear groups, with a section dedicated to applications to lower bounds of the border rank of tensors. Chapter \(11\) contains topics of geometric invariant theory, including the proof of Hilbert’s finiteness theorem and Noether’s degree bound, for the invariant ring of a finite group action on a polynomial ring over a field of characteristic zero. It also includes Derksen’s algorithm for the computation of the invariant ring of a reductive algebraic group acting polynomially on a vector space. Chapter \(12\) introduces semidefinite programming, including the notion of spectrahedra, the theory of duality and applications to polynomial optimization. The final chapter of the book gives an introduction to matroids, matroid representability and the connection to toric varieties.

MSC:

14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
05-XX Combinatorics
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
14Txx Tropical geometry
20Cxx Representation theory of groups
52-XX Convex and discrete geometry
90C22 Semidefinite programming
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