The Jacobian conjecture for rational polynomials (Q2479938)

The paper is essentially a presentation of main results of the forthcoming paper of \textit{Lê Dũng Tráng} [Publ. Res. Inst. Math. Sci. 44, No. 2, 641--659 (2008; Zbl 1152.14050)]. Assume that \((f,g)\) is a Jacobian pair. The special case described in the paper is, when \(f\) is rational, i.e., its generic fiber is a punctured 2-sphere. The author, by studying mostly the divisor of \(f\) at infinity, conjectures that if \(f\) is not a locally trivial fibration over \(\mathbb{C}\) then \(f\) cannot belong to the Jacobian pair. The author restates this conjecture in terms of the combinatorial structure of the fiber at infinity and the dicritical components of \(f\).