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 = (n1,n2,...,nd) be a d-tuple of positive integers. For each system of parameters x = (x1,...,xd) of A, we consider the length ℓ R(0 :A (x 1 n1 ,...,x d nd )R) as a function d-variables on n 1 ,...,n d . Then a necessary and sufficient condition for this length to be polynomial (in n1,...,nd) is given. Moreover, we shall introduce a system of parameters for which the length ℓ R(0 : A (x 1 n 1,...,x d nd )R) can be computed by a nice formula.