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

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.