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 syzygy module, 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.

翻译：基于签名的算法已成为一种标准的方法,用于在交替多元圆环中计算 Gr\“obner 基础 ” 。 然而, 目前还不清楚如何将这一概念扩展至自由代数中的非对称性多元比值设置。 在本文中, 我们给出了一种基于签名的算法, 精确地计算 Gr\“ obner 基础 ” 。 该算法是对Buchberger 算法( 包括签名) 的调整。 我们证明, 我们的算法正确地罗列了一个签名 Gr\“ obner 基础 ” 以及一个“ gr\” 基数, 并且当理想的模块接受有限签名 Gr\“ obner 基础 ” 时, 这一概念就终止了。 此外, 我们调整了一个众所周知的基于签名的标准, 消除了多余的削减, 例如 共性标准、 F5 标准 和单项标准 。 该算法是非对 Buchberg 算法案例的调整。 我们还将基于通俗的重建方法的设置方法,, 从部分签名信息中, Grobnernernerner 矩阵元素运的元素运要素基础, 我们的计算算算算算算算算算法中, 的所有输入了我们的一个基础, 我们的计算算算算算算算算算算算算算算法, 。