Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 2nd ed. (English) Zbl 0861.13012

Undergraduate Texts in Mathematics. New York, NY: Springer. xiii, 536 p. (1996).
Buchberger’s algorithm is an important tool for computations in commutative algebra and algebraic geometry. Using this algorithm one can compute a Gröbner base of an ideal in a polynomial ring (or, more generally, of a submodule of a free module over a polynomial ring). This algorithm is implemented in many computer algebra systems, which allow the effective performance of 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, Buchberger’s algorithm is explained. First applications are solutions of the ideal membership problem, solving polynomial equations followed by the 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 to 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.
Compared to the first edition, some proofs have been improved, some chapters have been rewritten, respectively added (for instance, one chapter with Bezout’s theorem).


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