Multidimensional residues and ideal membership.
Publicacions Matemàtiques, Tome 42 (1998) no. 1, p. 143-152.
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).
Source:
Zbl
Classification : 32C30, 32A25, 32A27 
@article{PMATES_1998__42_1_41331,
     author = {Alessandro Perotti},
     title = {Multidimensional residues and ideal membership.},
     journal = {Publicacions Matem\`atiques},
     pages = {143-152},
     volume = {42},
     number = {1},
     year = {1998},
     zbl = {an:0946.32002},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/PMATES_1998__42_1_41331/}
}
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Alessandro Perotti. Multidimensional residues and ideal membership.. Publicacions Matemàtiques, Tome 42 (1998) no. 1, p. 143-152. https://geodesic-test.mathdoc.fr/item/PMATES_1998__42_1_41331/