Improved Computation of Involutive Bases

Abstract

Abstract. In this paper, we describe improved algorithms to compute Janet and Pommaret bases. To this end, based on the method proposed by Möller et al. [21], we present a more efficient variant of Gerdt's algorithm (than the algorithm presented in [17]) to compute minimal involutive bases. Further, by using the involutive version of Hilbert driven technique, along with the new variant of Gerdt's algorithm, we modify the algorithm, given in [24], to compute a linear change of coordinates for a given homogeneous ideal so that the new ideal (after performing this change) possesses a finite Pommaret basis. All the proposed algorithms have been implemented in Maple and their efficiency is discussed via a set of benchmark polynomials.

Computation of Pommaret Bases Using Syzygies

“…We denote the set of all thus obtained syzygies by We now present a non-trivial variant of Gerdt's algorithm [10] computing simultaneously a minimal involutive basis for the input ideal and an involutive basis for the syzygy module of this basis. It uses an analogous idea as the algorithm given in [1]. However, since we aim at determining also a syzygy module, we must save the traces of all reductions and for this reason we cannot use the syzygies to remove useless reductions.…”

“…We have implemented the algorithm QuasiStable in Maple 17 4 and compared its performance with our implementation of the HDQuasiStable algorithm presented in [1] (it is a similar procedure applying a Hilbert driven technique). For this, we used some well-known examples from computer algebra literature.…”

“…The columns corresponding to C 1 and C 2 show, respectively, the number of polynomials removed by C 1 and C 2 criteria. The seventh column denotes the number of polynomials eliminated by the criterion related to signature applied in NextInvBasis algorithm (see [1] for more details). The eighth column shows the number of polynomials eliminated by the Hilbert driven technique which may be applied in NextInvBasis algorithm to remove useless reductions, (see [1] for more details).…”

“…The seventh column denotes the number of polynomials eliminated by the criterion related to signature applied in NextInvBasis algorithm (see [1] for more details). The eighth column shows the number of polynomials eliminated by the Hilbert driven technique which may be applied in NextInvBasis algorithm to remove useless reductions, (see [1] for more details). The ninth column shows the number of polynomials eliminated by the syzygy criterion described in NextIn-vBasis algorithm.…”

Deterministic genericity for polynomial ideals

“…Thus ideas like a Hilbert-driven Buchberger algorithm (Traverso, 1996) or exploiting syzygies for the detection of reductions to zero (Möller et al, 1992) (see more generally (Eder and Faugère, 2017) for a recent survey on signature based algorithms) can significantly increase the efficiency. Binaei et al (2016) report on some preliminary results in particular concerning the first point.…”