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.

中文翻译：

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签名 Gröbner 基、syzygies 基和自由代数中的辅因子重建

基于签名的算法已经成为计算可交换多项式环中 Gr\"obner 基的标准方法。但是，到目前为止，还不清楚如何将这个概念扩展到自由代数中的非交换多项式的设置。在本文中，我们提出了一种基于签名的算法，用于在这种情况下计算 Gr\"obner 基。该算法是 Buchberger 算法的改编版，包括签名。我们证明了我们的算法正确地枚举了 syzygy 模块的签名 Gr\"obner 基和 Gr\"obner 基，并且只要理想允许有限签名 Gr\"obner 基，它就会终止。此外，我们适应得很好- 已知的基于签名的标准消除冗余减少，如syzygy标准、F5标准和奇异标准，非对易多项式的情况。我们还从交换设置中推广了重建方法，该方法允许从关于签名的部分信息中恢复基于输入多项式的 Gr\"obner 基元素的坐标，以及生成器的 syzygy 模块的基础. 我们已经编写了 Mathematica 包 OperatorGB 中所有算法的玩具实现，并将我们的基于签名的算法与用于非交换多项式的经典 Buchberger 算法进行了比较。