Generalizations of Reductions and Mixed Multiplicities

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 a 1 ,…, a s of Q , and consists of a set of sd elements x ij (i = l,…, s ;j = 1, …, d ) where x ij ⊂ a i and the elements y j = x l j x 2 j … x s j ( j = 1,…, d ) form a reduction of a 1 a 2 … a s . The definition of a joint reduction applies to a set of, not necessarily distinct, ideals a 1 ,…, a d , and d s a set of elements x i ( i = 1,…, d ) such that∑ d i = 1x i a 1 …a i=1 a i+1 …a d is a reduction of a 1 a 2 …a d . The second section commences with a treatment of Hilbert functions of several m‐primary ideals based on deas of B uchsbaum 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 a 1 ,…, a d depends only on a 1 ,…, a d 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 / ( a n )′), where ( a n )′ is the integral closure of a n , with a m ‐primary, is equal to the polynomial (e(a)n d /d!−½(f(a)n d−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 ( a 1 r 1 ⋯ a s r s ) can be expressed as a homogeneous polynomial of degree d − 1 for r 1 ,…, r s ⩾ 0(and not merely for r 1 ,…, r s all positive). This extends a result proved in the case d = 2 in [ 7 ] to general d .