```From Computing Reviews, http://www.reviews.com/
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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
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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 [1] or Ideals, varieties, and
algorithms: an introduction to computational algebraic geometry and
commutative algebra, by Cox et al. [2].

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
[1] von zur Gathen, J. and Gerhard, J. Modern computer
algebra. Cambridge University Press, New York, 1999.
[2] 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.

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