Invitation to Nonlinear Algebra

While just about everybody reading this column already knows what linear algebra is, and what topics are associated with that subject, there may not be such broad understanding of just what nonlinear algebra, the titular subject of this book, is concerned with. In fact, one reason I accepted the editor’s invitation to review this book was to see for myself just what the subject was about. 

 

Because some of the most basic nonlinear objects in algebra are polynomials, I had assumed that algebraic geometry played a big role in nonlinear algebra, and that assumption proved to be accurate. The first six chapters of this book, almost half the text in total length, comprise a summary of classical algebraic geometry, slanted towards computation and applications: polynomial rings, Grobner bases, affine and projective varieties, Grassmanians, and the Nullstellensatz (both for algebraically closed fields of characteristic 0 and the real field). However, as the authors note in the Preface to the text, nonlinear algebra “is not simply a rebranding of algebraic geometry” but instead “links to many other branches of mathematics, such as combinatorics, algebraic topology, commutative algebra, convex and discrete geometry, linear and multilinear algebra, number theory, representation theory, and symbolic and numerical computation.” The authors use the term “nonlinear algebra” to “capture these trends, and to be more friendly to applied scientists.” 

 

So, for example, after the first six chapters of the text, there are individual chapters on tropical algebra, toric geometry, tensors, group representation theory, invariant theory, semidefinite programming and combinatorics. There are a total of 13 chapters, each one corresponding to a week’s worth of lectures in a graduate course.  In fact, each chapter is divided into three sections, presumably reflecting the fact that courses frequently meet three times a week.  

 

Consistent with its title, and its use by the authors as a text for graduate courses in this country and in Europe, this book is not intended as a research monograph or scholarly reference work. It is very definitely a text, with a student reader in mind. The authors write in a readable, interesting and accessible way, providing in each chapter a good introduction to the subject matter. There are lots of examples and a good number of exercises at the end of each chapter.  Based on a quick perusal, without actually taking pencil to paper, it seemed to me that these exercises were appropriate to a graduate student audience—few were trivial, but most seemed reasonably accessible. 

 

Although the authors do not specify precisely what the prerequisites for the book are, I would say a firm background in abstract and linear algebra is essential, and a good overall knowledge of undergraduate mathematics is very valuable as well.  For example, the phrase “Zariski topology” is used on page 20, for example, obviously under the assumption that the reader knows what a topological space is. The authors have taken pains to define certain concepts (like “ideal”) that play a big role in the text. 

 

To summarize and conclude: this book will not make the reader an expert in any of the topics listed in the third paragraph of this review, but it will provide him or her with a quick, clear and accessible introduction to them, and will certainly facilitate further reading. The authors state in the preface to this book that they think that student readers “will enjoy our presentation”. I think they will too.

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Mark Hunacek (mhunacek@iastate.edu) is a Teaching Professor Emeritus at Iowa State University.