Degrees of multiplicity functions for equimultiple ideals

Let J be an equimultiple ideal of height a in a formally equidimensional local ring (R,m). If I is a proper ideal that contains J, we show that the degree of the multiplicity function fJ,I(n)=e(In/Jn) is at most a with equality if and only if J is not a reduction of I. As a consequence, we are able to define a unique filtration J⊆J[a]⊆…⊆J[1]⊆J‾ between the ideal J and its integral closure J‾ with J[k] being the largest ideal containing J such that deg⁡fJ,I(n)≤a−k−1. Further results consider the ideal J[1] and its relation to the S2-ification of the Rees algebra R[Jt].