Computing difference-differential Groebner Bases and difference-differential dimension polynomials

Dfference-differential Groebner bases and the algorithms were introduced by M.Zhou and F.Winkler (2006). In this paper we will make further investigations for the key concept of S-polynomials in the algorithm and we will improve tech- nically the algorithm. Then we apply the algorithm to compute the difference- differential dimension polynomial of a difference-differential module and of a system of linear partial difference-differential equations. Also, in cyclic module case, we present an algorithm to compute the difference-differential dimension polynomials in two variables with the Groebner basis.