Hilbert functions for two ideals

Let (R, m) denote a Noetherian, local ring R with maximal ideal m. Let I an d J be ideals contained in m and assume I + J is m-primary. Then for all no n-negative integers r and s, the R-module [R/(I-r + J(s))] has finite lengt h. We denote the length of this R-module by l(R)[R/(I-r + J(S))] = lambda(r , s). The function MI,s) is called the Hilbert function of I and J. Let Z d enote the integers and for p is an element of Z, let Z[greater than or equa l to p] = {n is an element of Z\n greater than or equal to p}. In this pape r, we prove the following theorem: Suppose for some non-negative integer p, there exist ordered pairs (r(1), s(1)),...,(r(n), s(n)) is an element of Z [greater than or equal to p] X Z[greater than or equal to p] such that I-r boolean AND J(s) subset of or equal to [Sigma(i = 1)(n)(I(r - ri)J(s - si)) (I-ri boolean AND J(si))] + (Ir +1 + J(s + 1)) for all r, s greater than or equal to p. Then there exists a polynomial f(x, y) is an element of Q[x, y ] (Q the rational numbers) such that lambda(r,s) = f(r,s) for all r,s > 0. Furthermore, partial derivative(x)(f) = dim(R/J), partial derivative(y)(f) = dim(R/I), and partial derivative(f) less than or equal to l(I) + l(J). He re partial derivative(x)(f)[partial derivative(y)(f)] denotes the degree of f as a polynomial in x[y] and partial derivative(f) denotes the total degr ee of f. dim(S) is the Krull dimension of the ring S and l(II) is the analy tical spread of the ideal II. (C) 2000 Academic Press.