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Zbl 0861.13012
Cox, David; Little, John; O'Shea, Donal
Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 2nd ed. (English)
[B] Undergraduate Texts in Mathematics. New York, NY: Springer. xiii, 536 p. DM 68.00; öS 496.40; sFr 60.00 (1996). ISBN 0-387-94680-2

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.\par 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.\par 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.\par 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).
[G.Pfister (Kaiserslautern)]

MSC 2000:: *13P10 Polynomial ideals, Groebner bases
14Q99 Computational aspects of algebraic geometry
13F20 Polynomial rings
14-02 Research monographs (algebraic geometry)
13-02 Research monographs (commutative rings and algebras)

Keywords: computations in algebraic geometry; ideal membership problem; solving polynomial equations; elimination theory; robotics; automatic geometry theorem proving; computations in commutative algebra; Gröbner base; Buchberger's algorithm

Cited in: Zbl 1222.93055 Zbl 1118.13001 Zbl 1095.14001 Zbl 0994.13011 Zbl 0956.13008 Zbl 0952.14001 Zbl 0930.14030 Zbl 0896.13021

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