Four years ago the authors published the first volume of a projected series about computational commutative algebra which "took three years of intense work just to fill three centimeters of your bookshelf". Now they have gifted us with the second volume, and they say that "the completion of this volume took four years and it is about four centimeters thick. Thus we have a confirmed invariant which governs our writing: our velocity is one centimeter per year." These quotations from the foreword of the book give a clue of how amusing it is, and also how mathematically solid it is.

In this second volume the authors continue with the same style as the
first, an almost humorous one. I suggest the reading of the first
volume's review [M. Kreuzer and L. Robbiano, *
Computational commutative algebra. 1*, Springer, Berlin, 2000;
MR1790326 (2001j:13027)] to see more details.

The first volume introduces the theory of Gröbner basis from the beginning, for both rings and modules, along with the necessary basic results of commutative algebra, such as the Hilbert Basis Theorem, Hilbert Nullstellensatz, general grading of rings and modules, etc. The second volume is focused on the study of nine questions posed by the authors in the introduction, which are quoted below:

Question 1. How can we equip $P=K[x_1,\dots, x_n]$ with gradings which are useful for computational commutative algebra?

Question 2. How can we relate an arbitrary ideal or module to a homogeneous ideal or a graded module?

Question 3. How can we exploit homogeneity to compute Gröbner bases and perform elementary operations on ideals and modules more efficiently?

Question 4. How can we compute minimal homogeneous systems of generators, minimal homogeneous presentations, and minimal graded free resolutions of finitely generated graded modules?

Question 5. How can we compute the Hilbert function and the Hilbert series of a finitely generated graded module? What are their basic properties? How fast do Hilbert functions grow?

Question 6. How can we compute the Hilbert polynomial of a graded module $M$? Can we derive other invariants of $M$ from it?

Question 7. What are the algebraic and geometric interpretations of the dimension of a graded ring?

Question 8. Can mathematicians be replaced by computers?

Question 9. What is there beyond Gröbner bases?

The first four questions are answered in Chapter 4 (yes, the first volume stopped at Chapter 3 and this one starts at Chapter 4), where the notion of grading polynomial rings and modules by matrices is introduced. This special case of grading is well suited for computational purposes, since, for actual computations, gradings given by arbitrary monoids are too general.

Questions 5 to 7 are answered in Chapter 5, the "heart" of the book. The introduction of the chapter is an amusing dialog between an amateur mathematician, a gardener and a chess player. Besides the computational questions about Hilbert functions and series, several characterizations of dimension (Krull dimension, combinatorial dimension, transcendence degree, etc.) are nicely studied and related one with another. The authors also present the theorems of Macaulay and Green, with the characterization of Hilbert functions. The last section is about general Hilbert functions, that is, multigraded Hilbert function and multivariate Hilbert series. Some tutorials of this chapter are a travel through "magical applications", such as chess puzzles, photogrammetry, and a kid's toy.

Chapter 6 starts with a continuation of the three friend's dialog, and is a mixture of themes: toric ideals and Hilbert bases, finite sets of points, border bases, SAGBI bases, filtrations and automatic theorem proving. Among others, a curious application is given: counting the numbers of magic squares with given properties.

It was a pleasure to review this nice book but, as always, the equilibrium of the world depends on good and bad things, and the "bad thing" is the authors' decision to not write a third volume. In their own words: "Alas, we have to inform you that this is absolutely and definitively the second and last volume of the trilogy." Fortunately, these two volumes collect so large an amount of information and inspiration that we can say the authors actually fulfilled our expectations.