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Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. (English) Zbl 0756.13017
Undergraduate Texts in Mathematics. New York: Springer-Verlag. xi, 513 p. (1992).
An important base for computations in commutative algebra and algebraic geometry is the algorithm of Buchberger. Using this algorithm one can compute a Gröbner base of an ideal in a polynomial ring (or more general of a submodule of a free module over a polynomial ring). This algorithm is implemented in many computer algebra systems which allow to perform effectively many constructions in algebra and algebraic geometry (e.g. syzygies, Hilbert polynomials, primary decomposition etc.). — The book gives a good introduction into these problems.
On the base of an introduction to algebraic geometry and the relationship between algebra and algebraic geometry the algorithm of Buchberger is explained. First applications are solutions of the ideal membership problem, solving polynomial equations followed by a chapter about elimination theory. — The book contains also applications concerning robotics, automatic geometry theorem proving, invariant theory of finite groups.
The computational question is always related with basic topics of algebraic geometry (Hilbert basis theorem, the Nullstellensatz, invariant theory, projective geometry, dimension theory etc.).
In an appendix several computer algebra systems (Maple, Mathematica, Reduce etc.) are introduced and discussed. — The book contains a lot of exercises. It is a good introduction for students of algebraic geometry taking care of the growing importance of computational techniques.
Reviewer: G.Pfister (Berlin)

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14Q99 Computational aspects in algebraic geometry
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra