Hilbert functions for two ideals. II

Let (R, m, k) denote a Noetherian, local ring R with maximal ideal m and re sidue class field k. Let I and J be two ideals in R such that I + J is m-pr imary. Then for any integers r, s, the R-module R/I-r+J(i) has finite lengt h. The length of this module is denoted by H-I,H-J (r, s). The function H-I ,H-J : Z(2) --> Z is called the Hilbert function of I and J. Unlike the one variable case, there need be no polynomial f (x, y) is an element of Q [x, y] for which H-I,H-J (r, s) = f(r, s) for all r, s much greater than 0. In other words, a given Hilbert function may have no corresponding Hilbert po lynomial. However, there are many examples of I and J for which the domain of H-I,H-J breaks up into subregions Delta (1),..., Delta (n) such that H-I ,H-J (r, s) is given by a polynomial f(i) (r, s), (for r, s much greater th an 0) on each Delta (i). In this paper, we study those pairs I and J for wh ich H-I,H-J (r, s) is given by a polynomial in r, s, and min (r, s) for r,s much greater than 0. The domain of H-I,H-J breaks up into two regions Delt a (1) and Delta (2) and there are two polynomials f(1) (x, y) and f(2) (x, y) which describe H-I,H-J (r, s) for r, s much greater than 0. Examples wit h n > 3 are also discussed.