This book offers an interesting approach, combining numerical analysis and polynomial algebra. The emphasis is on multivariate polynomial systems arising in problems from scientific computing, and their solution with methods adopted from numerical analysis, instead of methods from computer algebra. One of the principal objectives is the use of polynomials with empirical data, and possibilities for their meaningful numerical treatment. The material is presented in four parts, with a total of 11 chapters.

Part 1, “Polynomials and Numerical Analysis,” contains four chapters: two introductory chapters, “Polynomials” and “Representation of Polynomial Ideals,” and two chapters on the basics of numerical analysis in the context of polynomial systems, “Polynomials with Coefficients of Limited Accuracy” and “Approximate Numerical Computation.” Basic definitions in both areas are presented precisely, together with many examples illustrating the terms defined. The material in Part 1 in particular covers fundamentals from polynomials and numerical analysis, which will be needed by readers as a common basis for later chapters about the combined approach. As the author argues, many readers might be familiar with only one of the subjects, and the text has the goal of providing knowledge in the lesser known subject. For readers with expertise in numerical analysis in particular, the book offers an intuitive introduction to polynomials, by relating them to facts from linear algebra.

Part 2 covers “Univariate Polynomial Problems,” in two chapters. Chapter 5, “Univariate Polynomials,” presents the well-known field of univariate polynomials with exact coefficients, so-called intrinsic polynomials. Chapter 6, “Various Tasks with Empirical Univariate Polynomials,” discusses new approaches, and provides material on approximation methods for the solution of univariate polynomials whose coefficients are not exact.

The largest part, Part 3, addresses “Multivariate Polynomial Problems,” and the last, smaller part, Part 4, discusses “Positive-Dimensional Polynomial Systems.” These parts are less definitive in character, and are meant to exhibit questions and difficulties arising in connection with multivariate numerical algebraic problems. Part 3 contains chapter 7, “One Multivariate Polynomial,” chapter 8, “Zero Dimensional Systems of Multivariate Polynomials,” chapter 9, “Systems of Empirical Multivariate Polynomials,” and chapter 10, “Numerical Basis Computations.” Chapter 7 is concerned with meaningful computational problems posed for one multivariate polynomial, with approximate data and approximate computation. Chapter 8 discusses aspects of regular polynomial systems that are fundamental for their numerical solution, and chapter 9 discusses numerical aspects. Chapter 10 is devoted to the computation of a Groebner basis, for which software is commonly available, but which is not part of the main focus of this book. The short chapter 11, of Part 4, offers some basic insight into “Matrix Eigenproblems for Positive-Dimensional Systems.”

This book is written as a textbook rather than a monograph, so it could easily be used by graduate students, or for courses on the subject. The material is self-contained, thanks to the introductory material in the first part and the many important definitions, accompanied by detailed examples in all of the chapters. All of the parts, and most of the chapters, start with a mathematically intuitive motivation. At the end of each chapter, historical and bibliographical notes cover the course of development, from the beginning to recent results, and cite milestones and relevant literature for further reading. Exercises are also provided for each of the chapters. The text seems most appropriate for readers with some expertise in one of the subjects, and especially for numerical analysts. Newly introduced terms and concepts are, for example, motivated by well-known facts from numerical linear algebra or functional analysis. Standard linear algebra notation is used throughout the book.

In summary, the author meets his goals to write a book for two different communities--numerical analysis and algebra--which usually have different backgrounds, and do not interact much. The material is presented so that readers from either community can get insight into the other field, in a precise, yet intuitive way; they might even get a different perspective on their own field. Most importantly, the book presents problems, difficulties, and working approaches to attacking algebraic approaches by numerical methods, in a concise, yet entertaining and intuitive way. This might motivate people to look into this scientific field, between two well-established mathematical disciplines.