Using algebraic geometry.

*(English)*Zbl 0920.13026
Graduate Texts in Mathematics. 185. New York, NY: Springer. xii, 499 p. (1998).

The book introduces the reader in a computational manner into a series of classical and modern topics of commutative algebra. These topics are in close connection with fundamental geometric questions, which motivates the title of the book. The algorithms presented are exclusively based on rewriting (Gröbner and standard basis) and linear algebra techniques, using dense or sparse encoding of polynomials and formal power series as fundamental data structures.

Throughout the book (symbolic) computational aspects are treated in the sense of effectivity and universality not in the sense of efficiency. No systematic complexity considerations are made and the computational aspects of commutative algebra are treated in a purely theoretical way. Performant algorithms for real world problems are out of the scope of the book. Nevertheless the rich tresory of (small) examples and the computational presentation of central topics of classical and modern commmutative algebra give the reader the feeling that the underlying abstract theory deals with concrete mathematical objects and in this sense the book constitutes a refreshing contribution to the mostly boring literature on commutative algebra. However the book fails to establish a bridge between modern computer science and classical algebraic geometry. It finishes with a series of promising applications of symbolic computation to concrete problems. The criteria for the selection of the bibliography at the end of the book are unclear for me.

The first six chapters of the book treat classical topics as: polynomials and ideals and their connection to Gröbner bases, algebraic varieties, polynomial equation solving (with emphasis on the zero-dimensional case including the semialgebraic aspect), resultants, standard bases in local rings, Gröbner and standard bases for modules, syzygy theory and Hilbert polynomials. The last three chapters deal with modern topics: sparse polynomial equation solving and resultants, Bernstein’s theorem, integer programming by means of polynomial rewriting techniques, multivariate polynomial splines, algebraic coding theory.

Throughout the book (symbolic) computational aspects are treated in the sense of effectivity and universality not in the sense of efficiency. No systematic complexity considerations are made and the computational aspects of commutative algebra are treated in a purely theoretical way. Performant algorithms for real world problems are out of the scope of the book. Nevertheless the rich tresory of (small) examples and the computational presentation of central topics of classical and modern commmutative algebra give the reader the feeling that the underlying abstract theory deals with concrete mathematical objects and in this sense the book constitutes a refreshing contribution to the mostly boring literature on commutative algebra. However the book fails to establish a bridge between modern computer science and classical algebraic geometry. It finishes with a series of promising applications of symbolic computation to concrete problems. The criteria for the selection of the bibliography at the end of the book are unclear for me.

The first six chapters of the book treat classical topics as: polynomials and ideals and their connection to Gröbner bases, algebraic varieties, polynomial equation solving (with emphasis on the zero-dimensional case including the semialgebraic aspect), resultants, standard bases in local rings, Gröbner and standard bases for modules, syzygy theory and Hilbert polynomials. The last three chapters deal with modern topics: sparse polynomial equation solving and resultants, Bernstein’s theorem, integer programming by means of polynomial rewriting techniques, multivariate polynomial splines, algebraic coding theory.

Reviewer: Joos Heintz (Santander)

##### MSC:

13Pxx | Computational aspects and applications of commutative rings |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

14Q20 | Effectivity, complexity and computational aspects of algebraic geometry |