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Signature Gröbner bases, bases of syzygies and cofactor reconstruction in the free algebra

Signature-based algorithms have become a standard approach for computing Gröbner 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 Gröbner 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 Gröbner basis as well as a Gröbner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gröbner 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 Gröbner 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.