Computing difference-differential Gröbner bases and difference-differential dimension polynomials

Difference-differential Gröbner 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 technically the algorithm. Then we apply the algorithm to compute the differencedifferential 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 Gröbner basis.