56 - Macaulay V

Summary

The aim of this section is to discuss some research which is behind the solution of Macaulay's Problem 23.3.3, which he stated as follows.

The object of this note is to discover the limiting relations which must exist between the terms of the series D₀, D₁, …, D_l, … where D_l is the number of linearly independent homogeneous polynomials of degree l (or of degree less than or equal to l in the case of non-homogeneous polynomials) belonging to some actual […] polynomial-ideal.

[…]

The converse and more important question is that of finding the actual values of D₀,D₁, D₂, … for a given polynomial-ideal, that is, an ideal defined by the stated conditions which its members individually and collectively have to satisfy.

After having reformulated the problem, set Macaulay's notation (Section 56.1) and recalled elementary formulas relating the problem with Hilbert functions (Section 56.2), I report Sperner's description (Section 56.3) of Macaulay's solution.

What is odd (see Historical Remark 56.8.5) is that Macaulay's result, with a different proof, already appeared (in Russian) in 1913 and was quoted by Janet in 1927; the author of the proof is Gunther who, at the same time, anticipated some results recently proved as extensions of Macaulay's result.

Here I give a partial résumé of his result: I begin with his illuminating results on Borel's relation and (Delassus's notion of) the generic escalier and term orderings (Section 56.4). Then, after discussing his Macaulay-like formula for Borel sets, which is essentially an anticipation of Cartan's formula (Section 56.5), I report his proof of Macaulay's result (Section 56.6). Next I discuss how Gunther applies the lex segment as a tool for computing Hilbert function and his description of the growth of the revlex segment (Section 56.7) and his views and results regarding the Riquier– Janet procedure (or, if you like, Buchberger's algorithm) (Section 56.8).