Grobner bases in difference-differential modules and difference-differential dimension polynomials

In this paper we extend the theory of Grobner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for constructing these Grobner bases counterparts. To this aim we introduce the concept of "generalized term order" on N(m) x Z(n) and on difference-differential modules. Using Grobner bases on difference-differential modules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations.