Simple rational polynomials and the Jacobian conjecture
The paper under review studies the jacobian conjecture in dimension two, the question as to whether a polynomial function \(F=(f,g): \mathbb C^{2} \to \mathbb C^{2}\) with nonzero constant jacobian \(J(F)\) must be an isomorphism. The author focuses on one of the coordinate functions, say \(f: \mathbb C^{2} \to \mathbb C\). Homogenizing \(f\) yields a rational map \(\mathbb P^{2} \to \mathbb P^{1}\), and after a blow-up \(\pi: Z \to \mathbb P^{2}\) of points of indeterminacy one arrives at a morphism \(\varphi: Z \to \mathbb P^{1}\). The divisor \(\mathcal D = \pi^{-1}(\mathbb P^{2}- \mathbb C^{2})\) at infinity may have several components. A component \(V \subset \mathcal D\) is dicritical if the restriction \(\varphi|_{V}\) is non-constant and \(f\) is called \textit{simple} if the degree of each such restriction is one. The main theorem of the paper says that if \(f\) is simple, the generic fibre of \(f\) is rational, and \(f\) is not a locally trivial fibration, then \(f\) cannot belong to a Jacobian pair. In particular, the Jacobian conjecture holds for maps \(F=(f,g)\) in which \(f\) is simple with generically rational fibres.
scientific article; zbMATH DE number 5310673