The objective of this text is to lay out the modern mathematical and algorithm analysis foundations for constructive exact mathematical computation (computer algebra). The authors emphasize much of the now well-understood mathematical foundations and other topics that particularly interest them (often asymptotically fast algorithms in computational number theory). As an extension to this material, they pick up on the relatively easy analysis of the complexity of operations on polynomials in a single variable over a finite field (modular arithmetic). This ground is covered in such detail that a second polynomial variable is first introduced on page 471 of this 750-page book!

There are a few offbeat, amusing applications, including audio and video compression, cryptographic systems, the geometry of molecules, and music. Each chapter has a useful collection of bibliographic and historical notes; to their credit, the authors have made significant efforts to find older references. The bibliography contains some 600 references. Each chapter has exercises, most of which extend the text along the lines of additional mathematical proofs or analyses of running time.

Factoring of integers, an interesting subject in its own right with many distinguished results, occupies some 50 pages, although its status as a fundamental building block outside of cryptosystems is uncertain. By comparison, only 11 pages sketch symbolic integration, and the discussion only covers rational functions.

Given the book’s primarily theoretical and foundational orientation, readers interested in how computer algebra systems such as Mathematica or Maple are constructed should look elsewhere. This book mostly ignores data structures. Thus the important computational results attributed to sparse algorithms and representation are deferred, and readers are directed to the references. Perhaps the authors view their practical efficiency as insufficient compensation for their resistance to neat asymptotic analyses. In any case, a more systems-building approach can be found in Geddes et al. [1].

How might this text be most useful? I see it as a supplement to an advanced mathematics course on modern algebra with applications. For computer scientists, this book may be most useful as a view into historical foundations as well as a systematic presentation of recent advances on discrete algorithms and their computational complexity. As such, it may also form the basis for a nontraditional survey and introduction to the literature for students looking for interesting research problems.

The authors’ erudition and attention to detail are refreshing, as are the occasional touches of color that enliven the figures. The publisher has set a reasonable price for this large and well-produced volume.