Local residues in $\mathbf C^n$. Algebraic applications

A. K. Tsikh

Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 225-237

Résumé

Connected with a ingular point $a$ of an algebraic set $V=\{z\in\mathbf C^n:g(z)=0\}$ is the local residue \begin{equation} \operatorname{res}\limits_{\Gamma_a}(f/g)=\int_{\Gamma_a}\frac{f(z)}{g(z)}\,dz, \end{equation} of the rational function $f/g$, where $\Gamma_a$ is a cycle which has a representative in the $n$-dimensional homology group $H_n(\mathbf C^n\setminus V)$ in every neighborhood of the point $a$. The structure of the local residues of the form (1) is described in the case of an isolated singular point $a$: they are expressed in terms of finitely many derivatives of $f$ at $a$. As an application of local residues a theorem of Noether and Bertini is generalized to any number of variables. Bibliography: 17 titles.