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**Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra.
3rd ed.**
*(English)*
Zbl 1118.13001

Undergraduate Texts in Mathematics. New York, NY: Springer (ISBN 978-0-387-35650-1/hbk; 978-0-387-35651-8/ebook). xv, 551 p. (2007).

Around 1980 two new directions in science and technique came together. One was Buchberger’s algorithms in order to handle Gröbner bases in an effective way for solving polynomial equations. The second one was the development of the personal computers. This was the starting point of a computational perspective in commutative algebra and algebraic geometry. In 1991 the three authors invented the first edition of their book as an introduction for undergraduates to some interesting ideas in commutative algebra and algebraic geometry with a strong perspective to practical and computational aspects [“Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra.” (1992; Zbl 0756.13017)]. A second revised edition appeared in 1996 [“Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra”. 2nd ed. (1996; Zbl 0861.13012)]. That means from the very beginning the book provides a bridge for the new, computational aspects in the field of commutative algebra and algebraic geometry.

To be more precise, the book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations, and elimination theory. Some more spectacular applications are about robotics, automatic geometric theorem proving, and invariants of finite groups. It seems to the reviewer to carry coals to Newcastle for estimating the importance and usefulness of the book. It should be of some interest to ask how many undergraduates have been introduced to algorithmic aspects of commutative algebra and algebraic geometry following the line of the book. The reviewer will be sure that this will continue in the future too.

What are the changes to the previous editions? There is a significant shorter proof of the Extension Theorem, see 3.6 in Chapter 3, suggested by A.H.M. Levelt. A major update has been done in Appendix C “Computer Algebra Systems”. This concerns in the main the section about MAPLE. Some minor updated information concern the use of AXIOM, CoCoA, Macaulay2, Magma, Mathematica, and SINGULAR. This reflects about the recent developments in Computer Algebra Systems. It encourages an interested reader to more practical exercises. The authors have made changes on over 200 pages to enhance clarity and correctness. Many individuals have reported typographical errors and gave the authors feedback on the earlier editions. The book is well-written. The reviewer guesses that it will become more and more difficult to earn 1 $ (sponsored by the authors) for every new typographical error as it was the case also with the first and second edition. The reviewer is sure that it will be a excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry. – For further complements to algebraic geometry see also the authors’ book [“Using Algebraic Geometry” (1998; Zbl 0920.13026)].

To be more precise, the book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations, and elimination theory. Some more spectacular applications are about robotics, automatic geometric theorem proving, and invariants of finite groups. It seems to the reviewer to carry coals to Newcastle for estimating the importance and usefulness of the book. It should be of some interest to ask how many undergraduates have been introduced to algorithmic aspects of commutative algebra and algebraic geometry following the line of the book. The reviewer will be sure that this will continue in the future too.

What are the changes to the previous editions? There is a significant shorter proof of the Extension Theorem, see 3.6 in Chapter 3, suggested by A.H.M. Levelt. A major update has been done in Appendix C “Computer Algebra Systems”. This concerns in the main the section about MAPLE. Some minor updated information concern the use of AXIOM, CoCoA, Macaulay2, Magma, Mathematica, and SINGULAR. This reflects about the recent developments in Computer Algebra Systems. It encourages an interested reader to more practical exercises. The authors have made changes on over 200 pages to enhance clarity and correctness. Many individuals have reported typographical errors and gave the authors feedback on the earlier editions. The book is well-written. The reviewer guesses that it will become more and more difficult to earn 1 $ (sponsored by the authors) for every new typographical error as it was the case also with the first and second edition. The reviewer is sure that it will be a excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry. – For further complements to algebraic geometry see also the authors’ book [“Using Algebraic Geometry” (1998; Zbl 0920.13026)].

Reviewer: Peter Schenzel (Halle)

### MSC:

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

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 |