Signature Gr\"obner bases in free algebras over rings

TL;DR

This paper extends signature Gr"obner bases to mixed algebras over rings, providing an algorithm with correctness proof, and demonstrates improved efficiency over classical methods in certain cases.

Contribution

It introduces a generalized algorithm for signature Gr"obner bases in mixed algebras over rings, with correctness proof and practical implementation.

Findings

Algorithm is correct and efficient for mixed algebras over rings.

Extends signature cover criterion to new algebraic settings.

Prototype implementation shows improved performance over classical methods.

Abstract

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.