On the length of generalized fractions

Let M be a finitely generated module over a Noetherian local ring with dimM=d. Let (x1,…,xd) be a system of parameters of M and (n1,…,nd) a set of positive integers. Consider the length of generalized fraction 1/(x1n1,…,xdnd,1) as a function in n1,…,nd. Sharp and Hamieh [J. Pure Appl. Algebra 38 (1985) 323–336] asked whether this function is a polynomial for n1,…,nd large enough. In this paper, we will give counterexamples to this question. We also study conditions on the system of parameters in order to show that the length of the generalized fraction 1/(x1n1,…,xdnd,1) is not a polynomial for n1,…,nd large enough.