Computation of Dimension in Filtered Free Modules by Gröbner Reduction

We present an axiomatic approach to Gröbner basis techniques in free multi-filtered modules over a not necessarily commutative multi-filtered ring. It is shown that classical Gröbner basis concepts can be viewed as models of our axioms. Within this theory it is possible to prove a general theorem about the dimension of filter spaces in multi-filtered modules. We use these ideas for computing the Hilbert function of finitely generated multi-filtered modules over difference-differential rings. Thus the presented method allows to compute a multivariate generalization of the univariate and the bivariate dimension polynomial considered in the papers of Winkler and Zhou.