Buchberger's results, which are dated 1965 (his Ph.D. thesis) and 1970 (his journal publication), became known within the computer algebra community around 1976; at the same time David A. Spear was implementing, in MACSYMA, a package allowing the solution of ideal (and subring) theoretical problems within commutative rings.

This package was already ahead of most of the recent commonly used, specialized software in commutative algebra, covering classes of rings which are even only partially available in modern software: the classes of rings available covered at least quotient rings of a polynomial ring over any field represented in the Kronecker Model!

While the report of this package contains no documentation, fortunately many of the ideas embedded there soon became available within the research community.

In particular, Zacharias' results (Section 26.1) hint that Spear's notion of admissible rings required at least algorithms for syzygy computation, membership test and membership representation, the tool for lifting such algorithms from R to R [X1,…, Xn] being essentially Gröbner technology.

The relation between Buchberger's result and Spear's own is presented by Spear in his report as follows: The solution to each of the problems described above depends on a fundamental algorithm for expressing ideals in a canonical form.

The Primbasissätze presented in the previous chapter give the tools needed in order to devise algorithms for computing Lasker–Noether decompositions and related concepts.

In Section 35.1 I introduce the computational problems related to Lasker–Noether decomposition which will be discussed throughout the chapter.

Section 35.2 presents the Gianni–Trager–Zacharias solution (mainly a direct application of the Primbasissätze) for a zero-dimensional ideal.

Section 35.3 contains the result (Theorem 35.3.4) allowing us to reduce the general case to the zero-dimensional one, and presents both their approach (GTZ-scheme) and a suggested improvement (ARGH-scheme): the GTZ-scheme has the disadvantage that the algorithms produce many redundant spurious components which must be tested and removed; the ARGH-scheme avoids such production of spurious components but at the price of also removing embedded components which must be recovered later.

Section 35.4 gives the solution, by means of the GTZ- and ARGH-schemes, of the decomposition problems.

If the ideal to be decomposed is in allgemeine position, the best shape of the bases allows us to strongly improve the algorithms (Section 35.5); however, the price of being in allgemeine position is full-density of all the data; this requires techniques allowing the computation of an allgemeine coordinate preserving sparsity as much as possible (Section 35.6); in connection with this problem, there have also been proposals to apply Möller's algorithm (Section 35.7) in order to avoid density when performing the ARGH-scheme.

Buchberger completed his thesis in 1965 and published his results in 1970. The next year, Gröbner quoted them in his notes of a course held by him in Turin and Milan in April–May 1971. There, in a section devoted to the determination of the primary components in the Lasker–Noether decomposition of an ideal, he concluded with the following remark:

OSSERVAZIONE: Riguardo ai calcoli che occorre eseguire per risolvere i problemi della teoria degli ideali negli anelli di polinomi, giova notare che, in linea di principio, tutti i calcoli si possono ridurre alla risoluzione di sistemi di equazioni lineari. Infatti basta risolvere il problema dato nei singoli spazi vettoriali P(t)…In questo procedimento è lecito fermarsi ad un certo grado (finito) T che corresponde al grado massimo attinto dai polinomi che formano la base dell'ideale cercato.

Un criterio per determinare tale numero T è s tato indagato da B. BUCHBERGER (Aequationes mathematicae, Vol. 4, Fasc. 3, 1970, S. 377–388)

REMARK: With regard to the calculations needed to solve the problems in the theory of ideals of polynomial rings, it is helpful to remark that, in principle, all computations can be reduced to the resolution of systems of linear equations. In fact it is sufficient to solve the given problem in the single vector spacesP(t) [the set of all polynomials of degree bounded by t] In this procedure it is sufficient to terminate at a fixed (finite) degree T corresponding to the maximal degree reached by the polynomials which are a basis of the required ideal.

Many of the notions introduced in Section 29.3 in order to describe and apply the linear-algebra structure of the vector-space k[N<(I)] = Spank(N<(I)) ≅ P/I, where I ⊂ P, stemmed on the one hand from a deeper analysis of the Möller algorithm, on the other hand from a reconsideration of Gröbner's description of Macaulay's results within ideal duality.

The aim of this chapter is to survey that result by Macaulay: after presenting Macaulay's computational assumptions and terminology (Section 30.1), we discuss his notation and the basic properties of his inverse systems (Section 30.2).

Section 30.3 is devoted to his linear-algebra algorithms which compute the inverse system of homogeneous and affine ideals.

Macaulay then concentrated his consideration to m-primary ideals and m-closed ideals I, seen as the ‘limit’ of m-primaries – I = ⋂d I + md. For them (Section 30.4) he

introduced the notion of Noetherian equations,

gave algorithms to compute their Noetherian equations, and their P-module structure,

already hinted at the notion of canonical forms, linear representation, and Gröbner representation which he is able to read directly from the Noetherian equations.

His next step generalized this result from zero-dimensional primaries to the higher-dimensional case by means of extension/contraction; in order to avoid the risk of failing to explain his results, I quote in Section 30.5 that chapter of his book, limiting myself to supporting the reader by following Macaulay's argument on a non-trivial example.

If you HOPE that this second SPES volume preserves the style of the previous volume, you will not be disappointed: in fact it maintains a self-contained approach using only undergraduate mathematics in this introduction to elementary commutative ideal theory and to its computational aspects, while my horror vacui compelled me to report nearly all the relevant results in computational algebraic geometry that I know about.

When the commutative algebra community was exposed, in 1979, to Buchberger's theory and algorithm (dated 1965) of Gröbner bases, the more alert researchers, mainly Schreyer and Bayer, immediately realized that this injection of Gröbner technology was all one needed to make effective Macaulay's paradigm for reducing computational problems for ideals either to the corresponding combinatorial problem for monomials or to a more elementary linear algebraic computation.4 This realization gave to researchers a straightforward approach which led them, within more or less fifteen years, to completely effectivize commutative ideal theory.

This second volume of SPES is an eyewitness report on this successful introduction of effective methods to algebraic geometry.

Part three, Gauss, Euclid, Buchberger: Elementary Gröbner Bases, introduces at the same time Buchberger's theory of Gröbner bases, his algorithm for computing them and Macaulay's paradigm.

While I will discuss in depth both of the classical main approaches to the introduction of Gröbner bases – their relation with rewriting rules and the Knuth–Bendix Algorithm, and their connection with Macaulay's H-bases and Hironaka's standard bases as tools for lifting properities to a polynomial algebra from its graded algebra – my presentation stresses the relation of both the notion and the algorithm to elementary linear algebra and Gaussian reduction; an added bonus of this approach is the ability to link Buchberger's algorithm with the most recent alternative linear algebra approach proposed by Faugère.