Involutive Bases in the Weyl Algebra

Coauthor(s): Marcus Hausdorf and Rainer Steinwandt

Reference: Journal of Symbolic Computation34 (2002) 181 -198

Description: In this article we extend the theory of involutive bases to (left) ideals in the Weyl algebra. The main point is that we do not only consider term orders but also the more general multiplicative monomial orders which makes things much more difficult, as normal form computations do not terminate in general. As for Gröbner bases, this is tackled by lifting to the homogenised Weyl algebra. However, the lift of the involutive division to the homogenisation is technically quite demanding and we could solve this problem completely only for globally defined divisions and the Janet division. It turns out that in general we cannot expect the existence of strong involutive bases (or reduced Gröbner bases). Only for the Janet division we designed a modified completion algorithm that always produces a strong basis.

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