Modular Algorithms For Computing Gröbner Bases in Free Algebras
Clemens Hofstadler, Viktor Levandovskyy
TL;DR
This work tackles computing Gröbner bases for two-sided ideals in non-Noetherian free algebras, where bases can be infinite and classical modular reconstruction fails due to an abundance of unlucky primes. The authors develop a signature-based modular framework that works with a bimodule setting and computes strong Gröbner bases up to a signature $\sigma$, enabling finite modular computations and robust verification via the cover criterion. They introduce $\sigma$-lucky primes to preserve leading data across modular reductions and reconstruct the target basis from modular images using Chinese remaindering and rational reconstruction, with a majority-vote mechanism to handle unlucky primes. The approach is implemented in SageMath under the signature_gb package and demonstrates significant speedups over non-modular approaches, with practical verification that applies to both homogeneous and inhomogeneous inputs. This yields a scalable, verifiable method for noncommutative Gröbner bases in free algebras and broadens the toolkit for computational algebra in non-Noetherian settings.
Abstract
In this work, we extend modular techniques for computing Gröbner bases involving rational coefficients to (two-sided) ideals in free algebras. We show that the infinite nature of Gröbner 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.