Bivariate Hilbert functions for the torsion functor

Let (R,P) be a commutative, local Noetherian ring, I, J ideals, M and N finitely generated R-modules. Suppose J+annR M+annR N is P-primary. The main result of this paper is Theorem 6, which gives necessary and sufficient conditions for the length of Tori(M/InM,N/JmN), to agree with a polynomial, for m,n 0. As a corollary, it is shown that the length of Tori(M/InM,N/InN) always agrees with a polynomial in n, for n 0, provided I+annR M+annR N is P-primary.