In a paper written in 1974, Amao proved that if I, J are ideals of a local ring such that \(J\subseteq I\) and \(\ell (J:I)<\infty\) then, for large n, \(\ell (I^ n/J^ n)\) is a polynomial \(\mu(n)\) in n. In the first section of this paper it is shown that the degree of this polynomial is at most the dimension d of the local ring.
In the remaining two sections the author gives interesting conditions over J to be a reduction of I in the case of a local quasi-unmixed ring. In particular J is a reduction of I if and only if \(\mu(n)\) has degree less than d. This generalizes an earlier result of the author and a further generalization is given, this time of a result of Böger which generalized the author’s earlier result.