Multidimensional residues and ideal membership

• Autores: Alessandro Perotti
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 42, Nº 1, 1998, págs. 143-152
• Idioma: inglés
• DOI: 10.5565/publmat_42198_07
• Títulos paralelos:
  • Residuos multidimensionales y miembros del ideal
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• Resumen

  • 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 $\bold C[z]/I(f)$.