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The Concept of Gröbner Reduction for Dimension in filtered free modules

We present the concept of Gröbner reduction that is a Gröbner basis technique on filtered free modules. It allows to compute the dimension of a filtered free module viewn as a K-vector space. We apply the developed technique to the computation of a generalization of Hilbert-type dimension polynomials in K[X] as well as in finitely generated differencedifferential modules. The latter allows us to determine a multivariate dimension polynomial where we partition the set of derivations and the set of automorphism in a difference-differential ring in an arbitrary way.