The most effective approaches to ‘solving’ proposed at the beginning of the twentieth century were both deeply inspired by the work of Hilbert and are behind the two most efficient alternatives to Buchberger's Algorithm.

1. • The procedure suggested by Riquier and reformulated by Janet under the name of completion and in connection with the notion of multiplicative variables (Section 55.2) is behind a very effective approach for dealing with ideals that, on the basis of an historical misunderstanding, is labelled involutivity and produces involutive bases (Section 57.1).

2. • The non-dissimilar procedure proposed by Macaulay (Algorithm 30.1.2) but, mainly, his description of an ideal in terms of its dialytic equations presented through his matrix (Section 41.3) inspired Faugère to propose his F4 Algorithm; such a procedure completely changed its nature and was strongly improved when (F5) a signature was connected to each element of the considered ideal; such a signature allows us to trace the leading term of its representation in terms of the given Gaussian generating set of the ideal. This allows us to use a careful consideration of the trivial syzygies as a tool for detecting useless S-polynomials (Section 57.2).

After an historical perspective I introduce the original proposal of applying Janet's ideas as a solving tool (Section 57.1.1) and a related interesting description of the structure of the escalier of a zero dimensional ideal (Section 57.1.2); such a proposal was later strongly reformulated with a quite general notion of involutive monomial division (Section 57.1.3), leading to a theory and algorithm of involutive bases (Section 57.1.4).

After describing F4 (Section 57.2.1), I cover F5: I introduce the notation (Section 57.2.2), the original proposal by Faugère (Section 57.2.3), an improvement by Hashemi–Ars (Section 57.2.4) and the reformulation of Perry and his collaborators that grants termination (Section 57.2.5). I also discuss an alternative application of Macaulay's Matrix given by Bardet (Section 57.2.6) and take the occasion to correct a mistake in my presentation, in the second book (Section 25.4), of Gebauer– Möller's algorithm computing a Staggered Linear Bases (Section 57.2.7).

Finally I report the proposal by Mayr of a space-efficient approach for computing Gröbner bases (Section 57.3).

Gerdt

From Janet to Zharkov–Blinkov: Pommaret Bases

Historical Remark 57.1.1. In his treatise, Pommaret gives a short historical sketch.