On proper holomorphic mappings between rigid polynomial domains in \(\mathbb C^{n+1}\)
A rigid polynomial domain in \(\mathbb{C}^{n+1}\) is one which is given as \[ \{(z_0, z)\in \mathbb{C}^{n+1}: r(z_0, z)= 2\text{\,Re}(z_0)+ P(z,\overline z)< 0\}, \] where \(P(z,\overline z)\) is a polynomial. In this paper the author proves the following theorem: Let \(D\) and \(D'\) be rigid polynomial nondegenerate domains in \(\mathbb{C}^{n+1}\). Then the branch locus of any proper holomorphic mapping \(f: D\to D'\) (denoted by \(V_f)\) satisfies \[ V_f\subset \bigcup_{1\leq k\leq N} \{(z_0, z)\in D: z\in\widehat B_k\}. \]
scientific article; zbMATH DE number 2141740
scientific article