Completion to Involution

In the previous chapter we only defined the notion of an involutive basis but we did not consider the question of the existence of such bases. Recall from Section 3.3 that (in the case of a coefficient field) the existence proof for Gröbner bases is straightforward. For involutive bases the situation is considerably more complicated. Indeed, we have already seen in Example 3.1.16 an (even monomial) ideal not possessing a finite Pommaret basis. Thus we surely cannot expect that an arbitrary polynomialideal has for every involutive division a finite involutive basis.

In Section 4.1 we introduce a special class of involutive divisions, the constructive divisions, which is naturally related to an algorithm for computing involutive bases (contained in Section 4.2). If such a division is in addition Noetherian, then the algorithm will always terminate with an involutive basis and thus provides us with a proof of the existence of such bases for many divisions (including in particular the Janet division). Unfortunately, both the definition of constructive divisions and the termination proof are highly technical and not very intuitive.

As a reward the underlying algorithm turns out to be surprisingly efficient despite its simplicity. However, in general, it does not produce a minimal basis and furthermore still contains some redundancies. In Section 4.4 we show how it can be modified such that the output is always minimal and simultaneously introduce a number of optimisations. This optimised algorithm underlies most implementations of involutive bases in computer algebra systems.