Hilbert functions for two ideals

Let (R, m) denote a Noetherian, local ring R with maximal ideal m. Let I and J be ideals contained in m and assume I + J is m-primary. Then for all non-negative integers r and s, the R-module [R/(I′ + Js)] has finite length. We denote the length of this R-module by LR/(I′ + Js)] = λ(r, s). The function λ(r, s) is called the Hilbert function of I and J. Let denote the integers and for p ε , let [≥ p] = {n ε n ≥ p}. In this paper, we prove the following theorem: Suppose for some non-negative integer p, there exist ordered pairs (r1, s1), …, (rn, sn) ε [≥ p] × [≥ p] such that I′ ∩ Js [Σi=1n(Ir−ri Js−si)(Iri ∩ Jsusi)] + (Ir+1 + Js+1) for all r, s ≥ p. Then there exists a polynomial f(x, y) ε of [x, y] ( the rational numbers) such that λ(r, s) = f(r, s) for all r, s 0. Furthermore, ∂x(f) = dim(R/J), ∂y(f) = dim(R/I), and ∂(f) ≤ l(I) + l(J). Here ∂x(f)[∂y(f)] denotes the degree of f as a polynomial in x[y] and ∂(f) denotes the total degree of f. dim(S) is the Krull dimension of the ring S and l(U) is the analytical spread of the ideal U.