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Computational methods of commutative algebra and algebraic geometry. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. (English) Zbl 0896.13021

Algorithms and Computation in Mathematics 2. Berlin: Springer. xi, 394 p. (1998).
The development of computers and recent programming systems have a strong influence in various aspects of pure mathematics. On the other hand the efforts for high performance algorithms related to computational problems in several fields of mathematics lead to research problems for fast algorithms in classical branches of pure mathematics. The new series ‘Algorithms and computation in mathematics’ by Springer-Verlag is devoted to those recent developments. The book under review, the second volume of this new series, is concerned with the computational aspects in commutative algebra, algebraic geometry and homological algebra. This is a panorama in which the processing of small sized problems are expected to lie on the brink of combinatorial explosion. That complicated examples do go through, in particular when the data have an interesting mathematical background, is often surprising. During the past two decades there was a wealth of theoretical and practical approaches in order to handle calculations on a personal computer that takes on a scratch pad days and weeks. Algebraic geometry has its roots in the problem of solving polynomial equations in several variables. So it is closely related to computational considerations on zero sets of polynomials in several variables.
The classical papers to the constructive approach in algebra [G. Hermann, Math. Ann. 95, 736-788 (1926; JFM 52.0127.01) and A. Seidenberg, Trans. Am. Math. Soc. 197, 273-313 (1974; Zbl 0356.13007)] are based on elimination theory, which tend to view each problem under a worst case scenario.
A landmark for an algorithmic approach to problems in ideal theory and algebraic geometry is B. Buchberger’s paper “Gröbner bases: An algorithmic method in polynomial ideal theory” in: Multidimensional systems theory. Progress, directions and open problems, Math. Appl., D. Reidel Publ. Co. 16, 184-232 (1985; Zbl 0587.13009). In fact, this was the initialization for computer algebra systems in commutative algebra and algebraic geometry. The more advanced systems on this field are Cocoa, Macaulay and Singular. The main intention of the present book is to write a theoretical and practical issue for the basic constructions in algebraic geometry and commutative algebra, and how in turn they may affect implementation by symbolic computation programs. The range of the methods lies between Gröbner bases and factorization algorithms. Because of the existence of some well-written textbooks about the rôle of Gröbner bases [see e.g. D. Cox, J. Little, and D. O’Shea, “Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra” (1992; Zbl 0756.13017); see also the 2nd ed. (1996; Zbl 0861.13012)] there is only a short introduction of them. This leads the author to syzygies and to finite free resolution of ideals and modules over polynomial rings. Having this in mind there is a straightforward investigation on homologically determined techniques like operations on modules, Cohen-Macaulay modules, Hilbert functions, cohomology, local duality, etc. This is pushed forward by the author to the computation of sheaf cohomology of algebraic varieties and to the study of vector bundles.
Another main stream of the author’s investigations are centered around factorization methods. This focuses in algorithmic aspects of primary decomposition, primality testing, integral closure, radicals of ideals, Nullstellensätze etc. At several points the author serves explicit examples of an illustrative character. This part of his book covers also root finders for polynomial equations and factorization methods for plynomials.
Chapter 1 is a review of orderings and Gröbner bases. Chapter 2 is concerned with various operations in commutative algebra, among them Noether normalization, testing of flatness and of Cohen-Macaulayness. Chapter 3 deals with techniques related to primary decomposition of ideals. The first part is concerned with methods to decompose an ideal into an intersection of ideals of different codimension, based on homological algebra. The second part, finding the primary decomposition of equidimensional ideals, is much harder. It is based on the reduction to the zero-dimensional case but steer away from factorization methods. The chapters 4 and 5 are dedicated to various methods in order to determine the radical of an ideal. While chapter 4 is dealing with decomposition in Artin algebras and providing an interface between symbolic and numerical solvers, in chapter 5 there is a systematic investigation between an ideal and its attached Jacobian ideals. Among others, the question of extracting the isolated zeros of a system of polynomial equations is considered. The main aspects of the integral closure are treated in chapter 6. Chapter 7 is an account on the computation of ideal transforms. Besides of theoretical attention to several details its more direct methods discuss the progress of subrings, particularly of rings of invariants. Chapter 8, written by D. Eisenbud, details his approach to the computation of the cohomology of projective schemes. The final chapter 9 has a more theoretical flavor and consists in an examination of various complexities of an algebra in comparison to a polynomial ring as Castelnuovo-Mumford regularity, various degress, and exponents of the Nullstellensatz.
The book is completed by three appendices. The first one – “A primer on commutative algebra” – grows out of the author’s intention to present the commutative algebra as background in addition to a textbook at the level of the book by M. F. Atiyah and I. G. Macdonald: “Introduction to commutative algebra” (1969; Zbl 0175.03601). It covers also basic homological algebra. Mostly full proofs are given. This makes the author’s presentations well readable, even when he claims to be ‘aware of some unbalance in this book, with topics ranging from freshman calculus to local cohomology, without enough detail of each and jumping over many topics in between!’. For the reviewer’s point of view the author’s bright view for applications and ‘sophisticated’ theoretical approaches makes the book a cornerstone for computational methods. For the reviewer’s point of view another good preparation for this book is D. Eisenbud’s textbook: “Commutative algebra with a view toward algebraic geometry” (1995; Zbl 0819.13001).
The second appendix – “Hilbert functions”, written by J. Herzog – provides an introduction to graded rings and modules. Based on Macaulay’s and Green’s theorems [the complete proofs may be found in the book by W. Bruns and J. Herzog, “Cohen-Macaulay rings” (1993; Zbl 0788.13005)] there are proofs of Gotzmann’s regularity and persistence theorems.
The third appendix – “Using Macaulay”, by D. Eisenbud, D. R. Grayson, and M. Stillman – presents an introduction to the software Macaulay.
By a short line of intrinsic examples illustrating some of the concepts presented in this book anybody is invited to begin exploring algebras, modules and many structures they support. By the aid of the author’s guides the book is also addressed to advanced students who are interested in explicit computations in algebra and geometry. They will get accesses by examples which was impossible one decade ago. Many parts of it can be read by anyone with a basic abstract algebra course. It seems to the reviewer that – among others – it was one of the author’s intention to equip students who are interested in computational problems with the necessary algebraic background in pure mathematics and to encourage them to further research in commutative algebra and algebraic geometry. But also researchers will benefit from this exposition. They will found an up to date description of the related research. An encyclopaedic list of 218 references summarizes the literature about computational methods.
The reviewer recommands the book to anybody who is interested in commutative algebra and algebraic geometry and its computational aspects.
Reviewer: P.Schenzel (Halle)

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14Qxx Computational aspects in algebraic geometry
13Pxx Computational aspects and applications of commutative rings
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H15 Multiplicity theory and related topics
12F20 Transcendental field extensions
14A05 Relevant commutative algebra
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