Polynomial maps of the complex plane with the branched value sets isomorphic to the complex line (Q1873626)

Let \(f: \mathbb C^n\to\mathbb C^n\) be a polynomial dominating mapping and \(\text{deg}f \) its geometric degree, i.e. the number of elements in a generic fibre \(f^{-1}(a)\). Let \(E_f\subset \mathbb C^n\) be the branched value set of \(f\), i.e. \[ E_f=\{a\in\mathbb C^n:\#f^{-1}(a)\neq\text{deg}f \}. \] It is well-known that \(E_f=\emptyset\) or \(E_f\) is an algebraic hypersurface [\(E_f\) has been intensively investigated by \textit{Z. Jelonek}; see e.g. Math. Ann. 315, No.~1, 1--35 (1999; Zbl 0946.14039)]. The author completely classifies, up to polynomial automorphisms of \(\mathbb C^n\), polynomial mappings \(f: \mathbb C^2\to\mathbb C^2\), for which \(E_f\) is isomorphic to complex line \(\mathbb C\). In particular case when \(f\) has all fibers finite, then any such mapping is equivalent of \( (x,y)\to(x^{\text{deg}f },y)\).