From Computing Reviews, http://www.reviews.com/ =============================================== Fundamental problems of algorithmic algebra by Yap C. Oxford University Press, Inc., New York, NY, 2000. Review by Richard Fateman (May 2000) Subject Classification: Algorithms (I.1.2); Nonnumerical Algorithms And Problems (F.2.2); Expressions And Their Representation (I.1.1); Computations On Discrete Structures (F.2.2...) Theory, Algorithms =============================================== The objective of this text is to elaborate on the themes of three fundamental problems in the modern mathematical and algorithm-analysis foundations for constructive exact mathematical computation (computer algebra). These problems are finding polynomial zeros, solving systems of polynomial equations (algebraic geometry), and the ideal membership problem. Readers interested in how computer algebra systems such as Mathematica or Maple are constructed should look elsewhere. While readers might easily get the impression that the current state of the art in system building consists of implementing certain asymptotically fast algorithms, in practice the dominant considerations in building such systems are different from the optimality of algorithms on worst-case inputs. Given its primarily theoretical and foundational orientation, then, how might this text be most useful? I see it as a sequel to a course in modern algebra for students interested in complexity aspects of constructive mathematics. For graduate students and researchers, it is an excellent review of the chosen topics with contemporary references. This text can be compared to the recently published Modern computer algebra, by von zur Gathen and Gerhard  or Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, by Cox et al. . It consists of a selection of topics treated authoritatively, with a breadth and depth that are appropriate for bringing a student or researcher up to date on a spectrum of optimal-complexity algorithms. After some preliminary material, the tasks discussed range from fast integer multiplication using the fast Fourier transform, through greatest common divisor, subresultants, polynomial root isolation, factoring, linear and nonlinear elimination (Grbner bases), and polynomial ideal theory, to continued fractions. Compared to the competition, I give the author high marks for the good taste of his selections, as well as for providing continuity (and some new material) in evolving areas. REFERENCES  von zur Gathen, J. and Gerhard, J. Modern computer algebra. Cambridge University Press, New York, 1999.  Cox, D. A.; OShea, D.; and Little, J. B. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer, New York, 1992.