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**Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra.
4th revised ed.**
*(English)*
Zbl 1335.13001

Undergraduate Texts in Mathematics. Cham: Springer (ISBN 978-3-319-16720-6/hbk; 978-3-319-16721-3/ebook). xvi, 646 p. (2015).

In the 80th of the last century, B. Buchberger invented the study of Gröbner bases into the field of commutative algebra and algebraic geometry. With the development of the personal computers it became a powerful technique for computational aspects in various computer algebra systems. It was the intention of the authors’ to provide a comprehensive easily accessible introduction to important concepts of computational commutative algebra in the first edition of their book see [Zbl 0756.13017]. It became a well accepted source for students and researchers as an introduction to the subject (see reviews about the second [Zbl 0861.13012] and third edition [Zbl 1118.13001].

In each of the new editions the authors’ were interested to incorporate new developments, simplifications of arguments as well as further applications. Thanks to the authors’ this is also the case in the present fourth edition. In fact the authors’ provide the following substantial changes:

(1) They define standard representations and lcm representations in Chapter 2. (2) They give two new proofs of the extension theorem, one based on resultants and one with Gröbner bases (inspired by P. Schauenburg [J. Symb. Comput. 42, No. 9, 859–870 (2007; Zbl 1137.13018)]). (3) In Chapter 4, they present a proof of the weak Nullstellensatz by the use of Gröbner bases (see also [L. Glebsky, “A proof of Hilbert’s Nullstellensatz based on Groebner bases”, arXiv:1204.3128]), they include saturations in addition to ideal quotients and prove the closure theorem by Gröbner bases techniques (see also Schauenburg [loc.cit]). (4) In Chapter 5, they add a section about Noether normalization, in particular for the application in dimension theory. (5) The results on Gröbner bases under specializations in Chapter 6 have been supplemented by the concept of Gröbner covers (see A. Montes and M. Wibmer [J. Symb. Comput. 45, No. 12, 1391–1425 (2010; Zbl 1207.13018)]). (6) There is a new Chapter 10 about the progress over 25 years development about methods for computing Gröbner bases. This includes Traverso’s Hilbert driven Buchberger algorithm, Faugère’s \(F_4\)-algorithm and an introduction to the signature-based family of algorithms. This, in particular illustrates the close bridge between theory and practice in computational commutative algebra. (7) Following the previous editions, the authors’ discuss software for computational aspects (Maple, Mathematica, Sage, CoCoa, Macaulay 2, and Singular) and several other systems that can be used in courses based on the text in the book. (8) In an Appendix, one may find 14 students projects substantially updated with new ideas. (9) The bibliography has been completed and expanded to reflect some of the new developments of the subject.

Thanks to the continuously updating the textbook will remain an excellent source for the computational commutative algebra for students as well as for researchers interested in learning the subject.

In each of the new editions the authors’ were interested to incorporate new developments, simplifications of arguments as well as further applications. Thanks to the authors’ this is also the case in the present fourth edition. In fact the authors’ provide the following substantial changes:

(1) They define standard representations and lcm representations in Chapter 2. (2) They give two new proofs of the extension theorem, one based on resultants and one with Gröbner bases (inspired by P. Schauenburg [J. Symb. Comput. 42, No. 9, 859–870 (2007; Zbl 1137.13018)]). (3) In Chapter 4, they present a proof of the weak Nullstellensatz by the use of Gröbner bases (see also [L. Glebsky, “A proof of Hilbert’s Nullstellensatz based on Groebner bases”, arXiv:1204.3128]), they include saturations in addition to ideal quotients and prove the closure theorem by Gröbner bases techniques (see also Schauenburg [loc.cit]). (4) In Chapter 5, they add a section about Noether normalization, in particular for the application in dimension theory. (5) The results on Gröbner bases under specializations in Chapter 6 have been supplemented by the concept of Gröbner covers (see A. Montes and M. Wibmer [J. Symb. Comput. 45, No. 12, 1391–1425 (2010; Zbl 1207.13018)]). (6) There is a new Chapter 10 about the progress over 25 years development about methods for computing Gröbner bases. This includes Traverso’s Hilbert driven Buchberger algorithm, Faugère’s \(F_4\)-algorithm and an introduction to the signature-based family of algorithms. This, in particular illustrates the close bridge between theory and practice in computational commutative algebra. (7) Following the previous editions, the authors’ discuss software for computational aspects (Maple, Mathematica, Sage, CoCoa, Macaulay 2, and Singular) and several other systems that can be used in courses based on the text in the book. (8) In an Appendix, one may find 14 students projects substantially updated with new ideas. (9) The bibliography has been completed and expanded to reflect some of the new developments of the subject.

Thanks to the continuously updating the textbook will remain an excellent source for the computational commutative algebra for students as well as for researchers interested in learning the subject.

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) |

14Qxx | Computational aspects in algebraic geometry |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |