On the multiplicity of a holomorphic mapping and on a formula for the logarithmic residue for overdetermined systems (Q2716173)

Let \(f=(f_1,\dots,f_n)\) be a holomorphic mapping from a domain \(G\subset\mathbb{C}^n\) into \(\mathbb{C}^n\) and assume that the analytic polyhedron \(\Pi=\{z\in G: |f_j(z)|<r_j\), \(j=1,\dots,n\}\) is compactly contained in \(G\). Then for any function \(\phi\) holomorphic on the closure of \(\Pi\), one has the logarithmic residue formula NEWLINE\[NEWLINE \int_\Gamma \phi \frac{df_1}{f_1} \wedge \dots \wedge \frac{df_n}{f_n} = (2\pi i)^n \sum_{a\in Z_f} \mu_a(f) \phi(a) NEWLINE\]NEWLINE where \(Z_f\) is the zero set of \(f\) in \(\Pi\), \(\Gamma=\{z\in G: |f_j(z)|=r_j\), \(j=1,\dots,n\}\) is the distinguished boundary of \(\Pi\), and \(\mu_a(f)\) is the multiplicity of \(f\) at \(a\). NEWLINENEWLINENEWLINEIn the paper under review, the author gives an extension of this result to overdetermined systems of mappings \(f:\mathbb{C}^n\supset G\to\mathbb{C}^m\), \(m>n\), and discusses some examples.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00006].