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Computational commutative algebra. II. (English) Zbl 1090.13021
Berlin: Springer (ISBN 3-540-25527-3/hbk). x, 586 p. (2005).
The book under review is the second volume of the authors’ “Computational commutative algebra”. The first part is published in 2000 (Zbl 0956.13008). In the review of the first part, it was written that the second volume will appear soon. Now it turns out that soon means five years. While it took three years to complete the three centimeter thick first volume the authors continued with the five centimeter of the second volume with the same continuity of speed and collection of material as they write in the introduction. In fact, the second volume of their “Computational commutative algebra” follows the same line of spirit with an ingenious selection of the topics and with a lot of fun for the reader and – as it seems – also for the authors.
Besides of an introduction and several appendices these five centimeters of text are splitted in three main chapters (4. The homogeneous case, 5. Hilbert functions, and 6. Further applications) by continuing with the counting of the main chapters of the first volume.
In the 4th section the authors present a detailed study about gradings. While in the first volume there is a general approach of gradings in order to provide the basics for Gröbner bases of modules it turns out that gradings on polynomial rings by arbitrary commutative monoids are too general. The gradings on \(P = K[x_0,\ldots,x_n]\), \(K\) a field, investigated here are \(\mathbb Z^m\)-gradings defined by a matrix \(M \in \text{ Mat}_{m,n}(\mathbb Z),\) where the gradings of the variables are given by the columns of \(W.\) More restrictive the authors consider gradings of positive type, i.e. those defined by matrices \(W\) such that some linear combination of its rows with integer coefficients have all entries positive. By a grading of positive type the vector space dimension of all elements of any given degree has finite \(K\)-dimension. Moreover the graded version of Nakayama’s lemma holds for such gradings. This is extended to a slight variation of the homogeneous Buchberger algorithm for getting a minimal homogeneous system of generators contained in the original system of generators. This is iterated in order to receive a minimal graded free resolution. This investigation results out of the authors’ consequent point of view of comptutational aspects of graded modules. Various procedures for passing from a non-homogeneous situation to a homogeneous one are another feature of this chapter. More precisely they consider submodules of a finitely generated graded free \(P\)-module and they introduce the concept of Macaulay bases. To be more precise, a positive grading induces an ordering on the set of terms. By mimicking the developments of volume 1 the lexicographically largest degree form of a polynomial is called the degree form. A set of vectors is called a Macaulay basis of a submodule of a free module if their degree forms generate the degree form module of the module. This corresponds to term orderings in Gröbner basis theory. Often the algorithmic approaches developed can be improved in case the input polynomials are homogeneous. A large part of the chapter is devoted to homogenization, in particular effective methods for computing the homogenization and the behaviour of Gröbner bases of ideals under dehomogenization.
Section 5 is devoted to the study of Hilbert functions. It is a well-known fact that the computation of the Hilbert function as well as the Hilbert series might be done by the reduction to the case of monomial ideals. This makes the computation fast and in fact independent of the minimal free resolution of a module. In this chapter there is a thorough investigation of the subject starting from the basics, illustrating the importance of the invariants derived of the Hilbert functions as dimension, regularity index. The authors continue with the bounds for Hilbert functions culminating with the theorems of Macaulay and Green. Following their intention on homogenization they introduce the affine Hilbert function. Consequently they prove that the dimension of an affine algebra defined via the degree of its affine Hilbert function is equal to its Krull dimension. As a consequence of the undertaking there are subsections on primary decompositions, dimension theory and Noether normalization. While most of the part of the section is written for the case of usually (i.e. \(\mathbb Z\)) graded rings and modules they include a subsection on the multigraded situation as it is needed for instance in the study of toric varieties. So, there is an investigation on multivariante Hilbert series. To do this seriously the authors have to introduce the \(\sigma\)-Laurent series ring for a monoid ordering \(\sigma\) on \(\mathbb Z^m.\) As in the case of \(\mathbb Z\)-grading the authors present a Hilbert driven Gröbner basis computation also in the multivariate case. Among the tutorials of the chapter there are those on Veronese subrings, Rees rings and Segre products, from the computational point of view available for the first time in a textbook.
In the final section 6 “Further Applications” there is a summary of separate subjects related to computational commutative algebra and of some interest in itself. The first of them is devoted to toric ideals and Hilbert bases. The approach is based on integer matrices and leads to the computation of lattice ideals and their saturation. As an application there is a tutorial on magic squares. In order to study vanishing ideals of projective point sets (subsection 6.3) by liftings of monomial ideals the subsection 6.2 provides liftings of ideals and distractions. Here an ideal \(J\) is called a lifting of \(I\) whenever \(x_0\) is a non-zero divisor on \(R/I\) and \(J\) is obtained by substituting \(x_0 = 0\) in \(I.\) Recently there has been a deep interest in the study of vanishing ideals of points in projective \(n\)-space. Here the reader might found a computational aspect about them regarding its Hilbert function, the Cayley-Bacharach property, and the minimal resolution conjecture. Further subsections are centered around the border bases of zero-dimensional ideals, filtrations, SAGBI bases, and automatic theorem proving. In particular there is a study of singularities, the proof of Molien’s theorem, and applications in elementary geometry. In fact the book alltogether covers on its 586 pages a wealth of interesting material with several unexpected applications.
As in the first part there is a strong recommendation to an interested reader to experiment with the computer algebra system CoCoA. To this end there is a chapter on the ABC of CoCoA introducing new features and functions that have been added since the first volume appeared. Moreover there is a chapter with suggestions for further reading for each of the sections. Alltogether now there are 99 tutorials including chess playing, photogrammetry, and error-correcting codes.
The book is by the same an encyclopedia on computational commutative algebra, a source for lectures on the subject as well as an inspiration for seminars. The text is recommanded for all those who want to learn and enjoy an algebraic tool that becomes more and more relevant to different fields of applications.

13Pxx Computational aspects and applications of commutative rings
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation
13A02 Graded rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14Qxx Computational aspects in algebraic geometry