Reduction numbers of equimultiple ideals

Let (A,m) be an unmixed local ring containing a field. If J is an m-primary ideal with Hilbert-Samuel multiplicity eA(J), a recent result of Hickel shows that every element in the integral closure J satisfies an equation of integral dependence over J of degree at most eA(J). We extend this result to equimultiple ideals J by showing that the degree of such an equation of integral dependence is at most cq(J), where cq(J) is one of the elements of the so-called multiplicity sequence introduced by Achilles and Manaresi. As a consequence, if the characteristic of the field contained in A is zero, it follows that the reduction number of an equimultiple ideal J with respect to any minimal reduction is at most cq(J)− 1.

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