ON THE POLYNOMIAL PROPERTY OF A FUNCTION OF CERTAIN SYSTEMS OF PARAMETERS FOR ARTINIAN MODULES

Let (R, m) be a commutative Noetherian local ring with the maximal ideal m and A an Artinian R-module with N-dimA = d. Let n__- = (n_1, n_2,..., n_d) be a d-tuple of positive integers. For each system of parameters x__- = (x_1,..., x_2) of A, we consider the length ℓ_R(0 : _A (<x_1>^<n_1>,...,<x_d>^<n_d>)R) as a function d-variables on n_1,...,n_d. Then a necessary and sufficient condition for this length to be polynomial (in n_1,..., n_d) is given. Moreover, we shall introduce a system of parameters for which the length ℓ_R(0 : _A (<x_1>^<n_1>,...,<x_d>^<n_d>)R) can be computed by a nice formula