Amao's Theorem and Reduction Criteria

In a paper written in 1974, Amao proved that if I , J are ideals of a local ring such that J ⊆ I and l ( J : I ) < ∞ then, for large n , l ( I n / J n ) is a polynomial μ( n ) in n . It is shown in the present paper that the degree of this polynomial is at most the dimension d of the local ring and, further, if the local ring is quasi‐unmixed, then J is a reduction of I if and only if μ( n ) has degree lessthan d . This generalises an earlier result of the author and a further generalisation is given, this time of a result of Böger which generalised the author's earlier result.