Dynamical Aspects of Involutive Bases Computations

Abstract

The article is a contribution to a more efficient computation of involutive bases. We present an algorithm which computes a ‘sliced division’. A sliced division is an admissible partial division in the sense of Apel. Admissibility requires a certain order on the terms. Instead of ordering the terms in advance, our algorithm additionally returns such an order for which the computed sliced division is admissible. Our algorithm gives rise to a whole class of sliced divisions since there is some freedom to choose certain elements in the course of its run. We show that each sliced division refines the Thomas division and thus leads to terminating completion algorithms for the computation of involutive bases. A sliced division is such that its cones ‘cover’ a relatively ‘big’ part of the term monoid generated by the given terms. The number of prolongations that must be considered during the involutive basis algorithm is tightly connected to the dimensions and number of the cones. By some computer experiments, we show how this new division can be fruitful for the involutive basis algorithm.

We generalise the sliced division algorithm so that it can be seen as an algorithm which is parameterised by two choice functions and give particular choice functions for the computation of the classical divisions of Janet, Pommaret, and Thomas.