Modular Techniques For Noncommutative Gröbner Bases

Abstract: In this note, we extend modular techniques for computing Gr\"obner bases from the commutative setting to the vast class of noncommutative $G$-algebras. As in the commutative case, an effective verification test is only known to us in the graded case. In the general case, our algorithm is probabilistic in the sense that the resulting Gr\"obner basis can only be expected to generate the given ideal, with high probability. We have implemented our algorithm in the computer algebra system {\sc{Singular}} and give timings to compare its performance with that of other instances of Buchberger's algorithm, testing examples from $D$-module theory as well as classical benchmark examples. A particular feature of the modular algorithm is that it allows parallel runs.