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Reviewer: Joel Cohen

This is the second edition of a comprehensive textbook on the mathematical and algorithmic foundations of computer algebra. This edition is similar to the first edition [1], with corrections and some changes to chapters 3, 15, and 22. The authors' stated goal is to provide an introduction to the subject that includes the mathematical underpinnings of the field, the asymptotic analysis of algorithms, and a description of asymptotic fast methods for symbolic problems. The book provides this, and much more. The book is organized into five sections: Euclid, Newton, Gauss, Fermat, and Hilbert. These (roughly) reflect five areas of mathematical concepts, algorithmic concepts, and approaches in computer algebra. The Euclid section is roughly organized around, and motivated by, the concept of a Euclidean algorithm for integers and polynomials. It includes a description of the basic algebraic concepts that are used to describe and analyze the algorithms, as well as classical and modern algorithms for basic arithmetic, classical greatest common divisor (gcd) computation, modular approaches for gcd computation, the Chinese remainder problem, determinant computation, and partial fraction decomposition. Most of the material in this section is similar to what is found in the first edition, although chapter 3 ("The Euclidean Algorithm") has been reorganized. The Newton section is roughly motivated by, and organized around, Newton's root finding algorithm, in a symbolic rather than numeric setting. It includes algorithms for fast multiplication of integers and polynomials, and the fast Fourier transform, a Newton-type iteration algorithm for polynomial division, a p -adic expansion algorithm for polynomials, fast polynomial evaluation and interpolation algorithms, a fast Euclidean algorithm, and fast algorithms in linear algebra. The chapters in this section are similar to those in the first edition. The Gauss section, which takes its name from the early work of Gauss on polynomial factorization, is concerned with polynomial factorization algorithms, including the Berlekamp/Hensel algorithm, and the polynomial time factorization algorithm of Lenstra, Lenstra, and Lovász. Most of the material in this section is similar to what is found in the first edition, although parts of chapter 15 ("Hensel Lifting and Factoring Polynomials") have been reorganized. The Fermat section, which is roughly motivated by, and organized around, Fermat's little theorem, is concerned with various approaches to primality testing, one integer factorization, and its applications to RSA encryption. The chapters in this section are similar to those in the first edition. The Hilbert section is primarily organized and motivated by Hilbert's two well-known algebraic theorems-the Nullstellensatz theorem and the irreducibility theorem-which provide a theoretical basis for some topics. It includes a description of Gröbner bases and Buchberger's algorithm, and a brief introduction to symbolic integration and symbolic summation. Most of the material in this section is similar to what is found in the first edition, although chapter 22 ("Symbolic Integration") has been expanded to include the integration of hyperexponential functions. This book differs from others in computer algebra by its large variety of topics, and connections to other areas of mathematics and computer science. The authors have an encyclopedic knowledge of computer algebra, its historical background, some related mathematical fields, and important application areas. The book is full of historical anecdotes, insights, and surprising connections to other mathematical areas, such as fractals, coding theory, and Diophantine equations, and application areas, such as audio and image compression, the analysis of musical scales, robotics, automated theorem proving in geometry, and general proof systems. It also includes suggestions for research problems. All this makes for great reading and reviewing. The authors have used the text for one- and two-semester courses in computer algebra at both the undergraduate and graduate levels. Although the stated prerequisites for the book include linear algebra, and a "certain level of mathematical maturity," it will also be very useful to have some background in abstract algebra, discrete mathematics, probability theory, number theory, and algorithm analysis. (An appendix briefly reviews some of these topics.) In any event, instructors will have to carefully pick and choose topics that address the time constraints of their courses, and the backgrounds of their students. The book's Web site, http://www-math.upb.de/mca, contains answers to some exercises, as well as errata and other reviews. Online Computing Reviews Service