Modular Algorithms For Computing Gr\"obner Bases in Free Algebras

TL;DR

This paper introduces a modular, signature-based algorithm for computing Gr"obner bases in free algebras, overcoming classical limitations and achieving significant speedups with practical implementation in SageMath.

Contribution

It extends modular techniques to free algebras and develops a new signature-based method for efficient Gr"obner basis computation.

Findings

Significant speedups over non-modular approaches.

The new algorithm is more general and efficient in verification.

First implementation demonstrates practical feasibility.

Abstract

In this work, we extend modular techniques for computing Gr\"obner bases involving rational coefficients to (two-sided) ideals in free algebras. We show that the infinite nature of Gr\"obner bases in this setting renders the classical approach infeasible. Therefore, we propose a new method that relies on signature-based algorithms. Using the data of signatures, we can overcome the limitations of the classical approach and obtain a practical modular algorithm. Moreover, the final verification test in this setting is both more general and more efficient than the classical one. We provide a first implementation of our modular algorithm in SageMath. Initial experiments show that the new algorithm can yield significant speedups over the non-modular approach.