Grobner bases in difference-differential modules and their applications

In this paper we will extend the theory of Gr\"obner bases to difference-differential modules which were introduced by Levin(2000) as a generalization of modules over rings of differential operators. The main goal of this paper is to present and verify algorithms for constructing these Gr\"obner basis counterparts. To this aim we define the concept of ''generalized term order'' on ${\Bbb N}^m \times {\Bbb Z}^n$ and on difference-differential modules. The relation between the Gr\"obner bases and some characteristic sets in the modules is also considered. As applications, we can compute the difference-differential dimension polynomial of a difference-differential module and of a system of linear partial difference-differential equations via the Gr\"obner bases.