Gröbner bases with respect to several orderings and multivariable dimension polynomials

Let D=K[X] be a ring of Ore polynomials over a field K and let a partition of the set of indeterminates into p disjoint subsets be fixed. Considering D as a filtered ring with the natural p-dimensional filtration, we introduce a special type of reduction in a free D-module and develop the corresponding Gröbner basis technique (in particular, we obtain a generalization of the Buchberger Algorithm). Using such a modification of the Gröbner basis method, we prove the existence of a Hilbert-type dimension polynomial in p variables associated with a finitely generated filtered D-module, give a method of computation and describe invariants of such a polynomial. The results obtained are applied in differential algebra where the classical theorems on differential dimension polynomials are generalized to the case of differential structures with several basic sets of derivation operators.