Multivariate Difference-Differential Dimension Polynomials

In this paper we generalize the Ritt-Kolchin method of characteristic sets and the classical Gr"obner basis technique to prove the existence and obtain methods of computation of multivariate difference-differential dimension polynomials associated with a finitely generated difference-differential field extension. We also give an interpretation of such polynomials in the spirit of the A. Einstein’s concept of strength of a system of PDEs and determine their invariants, that is, characteristics of a finitely generated difference-differential field extension carried by every its dimension polynomial.