Computation of Pommaret Bases Using Syzygies

TL;DR

This paper introduces new algorithms leveraging syzygies for more efficient computation of Pommaret bases, reducing unnecessary reductions and enabling simultaneous Grobner basis computation, with implementation and benchmarking in Maple.

Contribution

It presents novel algorithms for computing Pommaret bases using syzygies, including an involutive variant of a signature-based Grobner basis algorithm, improving efficiency.

Findings

Algorithms successfully implemented in Maple.

Performance benchmarks show improved efficiency.

New methods enable simultaneous Grobner and syzygy basis computation.

Abstract

We investigate the application of syzygies for efficiently computing (finite) Pommaret bases. For this purpose, we first describe a non-trivial variant of Gerdt's algorithm to construct an involutive basis for the input ideal as well as an involutive basis for the syzygy module of the output basis. Then we apply this new algorithm in the context of Seiler's method to transform a given ideal into quasi stable position to ensure the existence of a finite Pommaret basis. This new approach allows us to avoid superfluous reductions in the iterative computation of Janet bases required by this method. We conclude the paper by proposing an involutive variant of the signature based algorithm of Gao et al. to compute simultaneously a Grobner basis for a given ideal and for the syzygy module of the input basis. All the presented algorithms have been implemented in Maple and their performance is evaluated via a set of benchmark ideals.