Computation of Hilbert polynomials in two variables (Q1974698)

Let \(R=K[x_1, \dots, x_m,y_1, \dots, y_n]\) be the ring of polynomials of indeterminates \(x_1,\dots, x_m\), \(y_1,\dots, y_n\) over a field \(K\), \(M\) be a finitely generated \(R\)-module with generators \(f_1,\dots, f_p\), \(M_{r,s}\) be the vector space over \(K\), generated by products \(\alpha\beta f_i\), where \(\alpha,\beta\) are monomials from \(K[x_1,\dots,x_m]\) and \(K[y_1, \dots, y_n]\) respectively such that \(\deg\alpha\leq r\), \(\deg\beta\leq s\) (instead of the usual notation deg, the author uses the notation Ord here). Existence of a polynomial \(\varphi (t_1,t_2)\) with rational coefficients such that \(\varphi (r,s)= \dim M_{r,s}\) for sufficiently large \((r\geq r_0,s\geq s_0)\) integers \(r,s\) is established. The form of this polynomial and an algorithm for its computation are indicated. As applied to extensions of a differential field with the system of derivation operators \(\Delta=\{ \delta_1, \delta_2,\dots\}\), a generalization of Kolchin's theorem on the differential dimension polynomial [\textit{E. R. Kolchin}, Bull. Am. Math. Soc. 70, 570--573 (1964; Zbl 0144.03702)] is obtained. The author presents an example where, using a partition \(\Delta=\Delta_1 \cup\Delta_2\) (with \(\Delta_1\cap \Delta_2= \emptyset)\) and the corresponding bifiltration, one can obtain more invariants of the extension than in the case of the Kolchin construction.