# Multidimensional residues and ideal membership.

• Volume: 42, Issue: 1, page 143-152
• ISSN: 0214-1493

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## Abstract

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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).

## How to cite

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Perotti, Alessandro. "Multidimensional residues and ideal membership.." Publicacions Matemàtiques 42.1 (1998): 143-152. <http://eudml.org/doc/41331>.

@article{Perotti1998,
abstract = {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).},
author = {Perotti, Alessandro},
journal = {Publicacions Matemàtiques},
keywords = {Funciones de variable compleja; Funciones de varias variables; Funciones holomorfas de varias variables; Residuos; integral representations; multidimensional residues; local residues},
language = {eng},
number = {1},
pages = {143-152},
title = {Multidimensional residues and ideal membership.},
url = {http://eudml.org/doc/41331},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Perotti, Alessandro
TI - Multidimensional residues and ideal membership.
JO - Publicacions Matemàtiques
PY - 1998
VL - 42
IS - 1
SP - 143
EP - 152
AB - 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).
LA - eng
KW - Funciones de variable compleja; Funciones de varias variables; Funciones holomorfas de varias variables; Residuos; integral representations; multidimensional residues; local residues
UR - http://eudml.org/doc/41331
ER -

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