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.