Let I(f) be a zero-dimensional ideal in C[z1, ..., zn] 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 C[z]/I(f).
@article{urn:eudml:doc:41331,
title = {Multidimensional residues and ideal membership.},
journal = {Publicacions Matem\`atiques},
volume = {42},
year = {1998},
pages = {143-152},
mrnumber = {MR1628158},
zbl = {0946.32002},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41331}
}
Perotti, Alessandro. Multidimensional residues and ideal membership.. Publicacions Matemàtiques, Tome 42 (1998) pp. 143-152. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41331/