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 $J + ann_R M + ann_R N$ is $P$-primary. The main result of this paper is Theorem 6, which gives necessary and sufficient conditions for the length of $\t_i(M/I^nM,N/J^mN)$, to agree with a polynomial, for $m$, $n \gg 0$. As a corollary, it is shown that the length of $\t_i(M/I^nM,N/I^nN))$ always agrees with a polynomial in $n$, for $n \gg 0$, provided $I + ann_R M + ann_R N$ is $P$-primary.