We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over $\mathbb{Z}$ as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when $R$ is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.

翻译：本文将签名Gröbner基（先前仅在域上自由代数或环上多项式环中研究）推广至混合代数$R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$中的理想，其中$R$为主理想整环。我们提出一种结合交换与非交换（签名）Gröbner基理论要素的计算算法，并证明其正确性。应用包括含交换变量的自由代数扩展（例如用于齐次化处理或交运算等理想论操作），以及关于$\mathbb{Z}$的计算（作为任意特征域上的泛化证明）。通过将签名覆盖准则推广至本文框架，该算法还解除了此前非交换签名基算法中的若干技术限制，例如现在允许消元序。我们针对$R$为域的情形给出了原型实现，并证明混合代数的算法相较于使用现有算法的经典方法具有更高效率。