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（此前已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。我们给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。应用包括带交换变量的自由代数扩张（例如为齐次化目的或执行交运算等理想论操作），以及在$\mathbb{Z}$上的计算（作为任意特征域上通用证明）。通过将签名覆盖准则推广至我们的设定，该算法还消除了先前非交换签名型算法中的某些技术限制，现允许使用消去序等序型。针对$R$为域的情形，我们提供了原型实现，并表明混合代数算法比使用现有算法的经典方法更高效。