Multidimensional residues and ideal membership

Abstract

Let $I(f)$ be a zero-dimensional ideal in $\bold C[z_1,\ldots,z_n]$ defined by a mapping $f$. We compute the logarithmic residue of a polynomial $g$ with respect to $f$. We adapt an idea introduced by Aizenberg to reduce the computation to a special case by means of a limiting process. We then consider the total sum of local residues of $g$ w.r.t. $f$. If the zeroes of $f$ are simple, this sum can be computed from a finite number of logarithmic residues. In the general case, you have to perturb the mapping $f$. Some applications are given. In particular, the global residue gives, for any polynomial, a canonical representative in the quotient space $\bold C[z]/I(f)$.