Involutive Bases Algorithm Incorporating F5 Criterion

TL;DR

This paper introduces a new involutive basis algorithm that integrates the F5 criterion, enhancing the efficiency of computing minimal involutive bases by combining signature-based and involutive methods.

Contribution

It presents a nonincremental involutive completion algorithm that incorporates the F5 criterion, improving upon existing involutive algorithms by reducing unnecessary reductions.

Findings

The new algorithm computes minimal involutive bases more efficiently.

Benchmark results show improved performance over the Gerdt-Blinkov involutive algorithm.

Implementation in Maple demonstrates practical applicability.

Abstract

Faugere's F5 algorithm is the fastest known algorithm to compute Groebner bases. It has a signature-based and an incremental structure that allow to apply the F5 criterion for deletion of unnecessary reductions. In this paper, we present an involutive completion algorithm which outputs a minimal involutive basis. Our completion algorithm has a nonincremental structure and in addition to the involutive form of Buchberger's criteria it applies the F5 criterion whenever this criterion is applicable in the course of completion to involution. In doing so, we use the G2V form of the F5 criterion developed by Gao, Guan and Volny IV. To compare the proposed algorithm, via a set of benchmarks, with the Gerdt-Blinkov involutive algorithm (which does not apply the F5 criterion) we use implementations of both algorithms done on the same platform in Maple.