# Improved Computation of Involutive Bases

## TL;DR

This paper introduces improved algorithms for computing Janet and Pommaret bases, enhancing efficiency and enabling finite Pommaret bases through coordinate changes, with implementations and benchmarks demonstrating performance gains.

## Contribution

It presents a more efficient variant of Gerdt's algorithm and a modified Seiler algorithm for computing involutive bases and finite Pommaret bases.

## Key findings

- Algorithms implemented in Maple show improved efficiency.
- New methods successfully compute minimal involutive bases.
- Coordinate change technique ensures finite Pommaret bases.

## Abstract

In this paper, we describe improved algorithms to compute Janet and Pommaret bases. To this end, based on the method proposed by Moller et al., we present a more efficient variant of Gerdt's algorithm (than the algorithm presented by Gerdt-Hashemi-M.Alizadeh) 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 by Seiler, 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.