Signature Gr\"obner bases, bases of syzygies and cofactor reconstruction in the free algebra

TL;DR

This paper extends signature-based Gr"obner basis algorithms from commutative to noncommutative free algebra, providing a new method with proven correctness, efficiency improvements, and reconstruction techniques.

Contribution

It introduces the first signature-based algorithm for noncommutative Gr"obner bases, adapting existing criteria and reconstruction methods to this setting.

Findings

Algorithm correctly enumerates signature Gr"obner bases.

Incorporates signature-based criteria like syzygy and F5.

Demonstrates efficiency improvements over classical methods.

Abstract

Signature-based algorithms have become a standard approach for computing Gr\"obner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gr\"obner bases in precisely this setting. The algorithm is an adaptation of Buchberger's algorithm including signatures. We prove that our algorithm correctly enumerates a signature Gr\"obner basis as well as a Gr\"obner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gr\"obner basis. Additionally, we adapt well-known signature-based criteria eliminating redundant reductions, such as the syzygy criterion, the F5 criterion and the singular criterion, to the case of noncommutative polynomials. We also generalize reconstruction methods from the commutative setting that allow to recover, from partial information about signatures, the coordinates of elements of a Gr\"obner basis in terms of the input polynomials, as well as a basis of the syzygy module of the generators. We have written a toy implementation of all the algorithms in the Mathematica package OperatorGB and we compare our signature-based algorithm to the classical Buchberger algorithm for noncommutative polynomials.