et al: 585 pp., 拢84.00, ISBN 0 7923 9259 0 (Kluwer Academic Press, 1992). This book is devoted to a mathematical treatment of some of the most fundamental algorithms that are employed by computer algebra systems such as DERIVE, MAPLE and MATHEMATICA. It does not contain any details or comparative information on these systems themselves. The bulk of the book is concerned with algorithms for polynomial manipulation, which undoubtedly form the central core of all such systems. The topic of indefinite integration is also handled in some detail. However, a number of important subjects such as differential equations and advanced linear algebra are not covered, mainly for reasons of space.Given the increasing use of computer algebra systems across the whole range of the scientific community, this book is welcome as the first systematic treatment of the underlying mathematics that is being employed. As a textbook, it is aimed at advanced undergraduate and postgraduate students, and it contains a large number of exercises of varying difficulty. It is, of course, also useful as a reference book. For undergraduates it would need to be used with care and some selectivity because, although the level is appropriate, there is a vast amount of material, and it is covered in considerable detail. Most of the algorithms have many possible minor variations which work better or worse depending on the circumstances, and each of these requires its own mathematical analysis and justification. The algebraic concepts of rings, homomorphisms and ideals play a significant role. For those students who have already taken a course in ring theory, this would provide an excellent demonstration of how the abstract concepts of pure mathematics can be applied to produce tools that are proving invaluable to the whole scientific community. On the other hand, for well-motivated students who have not studied ring theory already, the book does provide enough theoretical background to fill in the gaps in their knowledge. Individual algorithms are presented using a PASCAL-like notation, and numerous examples are worked through in detail, some involving quite large numbers and complicated expressions. Indeed, this is essential for the demonstration of the necessity and effectiveness of some of the techniques that are being employed.The central themes of the book are the computation of greatest common divisors (GCDs), and the factorisation of polynomials in one or more commuting variables, over various fields and integral domains, including the integers, rational numbers and their algebraic extensions, and finite fields. Remember that computer algebra is essentially about exact computation rather than numerical computation, and so floating point arithmetic does not play such a large role. Of course, we all know how to compute GCDs using the Euclidean Algorithm, but it turns out that when we are working over the integers or rational numbers, this process can rapidly lead to the phenomenon of coefficient explosion. In other words, the input and output data ma...
