Computation of Hilbert Polynomials in Two Variables

Let R be a ring of polynomials in m + n indeterminatesx1 , . . . , xm, y1, . . . ,yn over a field K and let M be a finitely generated R -module. Furthermore, let (Rrs)r, s Nbe the natural double filtration of the ring R and let (Mrs)r,s Nbe the corresponding double filtration of the module M associated with the given system of generators. We introduce a special type of reduction in a free R -module and develop the appropriate technique of characteristic sets that allows us to prove the existence and find methods and algorithms of computation of a numerical polynomial in two variables φ(t1, t2) such thatφ (r, s) = KMrsfor all sufficienly large r, s N. The results obtained are applied in differential algebra where the classical theorems on differential dimension polynomials and methods of computation of such polynomials are generalized to the case of differential structures with two basic sets of derivation operators.