© 1984 by London Mathematical Society
Generalizations of Reductions and Mixed Multiplicities
6 Hillcrest Park, Exeter EX4 4SH
Received 7 June 1983.
The first section of this paper is devoted to the definition, and proof of the existence, of what are called complete and joint reductions of a set of ideals of a d-dimensional local ring Q with maximal ideal m and residue field k. The former is defined for a set of ideals a1,..., as of Q, and consists of a set of sd elements xij (i = l,...,s;j = 1, ...,d) where xij 
ai and the elements yj = xljx2j... xsj(j = 1,..., d) form a reduction of a1 a2... as. The definition of a joint reduction applies to a set of, not necessarily distinct, ideals a1,..., ad, and d s a set of elements xi (i = 1,..., d) such that 
xia1...ai=1ai+1...ad is a reduction of a1a2...ad.
The second section commences with a treatment of Hilbert functions of several m-primary ideals based on deas of Buchsbaum and Auslander [1, corrections], and sketched in [8]. This is used to prove that the multiplicity of a joint reduction of d m-primary ideals a1,...,ad depends only on a1,...,ad and is a mixed multiplicity as defined by Teissier in [9]. The final part of Section 2 is devoted to a proof of the following result.
Suppose that Q is analytically unramified, and that L(Q / (an)'), where (an)' is the integral closure of an, with a m-primary, is equal to the polynomial(e(a)nd/d!–
(f(a)nd–1/(d–1)!)+... for large n. Then f(a) is a homogeneous polynomial over the set of m-primary ideals of Q in the sense that f(a1r1... asrs)can be expressed as a homogeneous polynomial of degree d – 1 for r1,..., rs 
0(and not merely for r1,..., rs all positive).
This extends a result proved in the case d = 2 in [7] to general d.