A counterexample to Meisters' cubic-linear linearization conjecture
Let \(F:\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}\) be a polynomial mapping such that \(\text{Jac}(F)\in \mathbb{C}^{\ast }\), \(F(0)=0\), \(\text{J} F(0)=I_{n}\). \textit{B. Deng, G. H. Meisters} and \textit{G. Zampieri} [Z. Angew. Math. Phys. 46, 872-882 (1995; Zbl 0840.58013)] conjectured that there exists \(N\) such that for \(\lambda \in \mathbb{C}\) with \(\left|\lambda \right|>N\) the dilatation \(\lambda F\) is global analytic linearizable to its linear part, i.e. there exists an analytic automorphism \(\phi :\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}\) such that \[ \phi ^{-1}\circ (\lambda F)\circ \phi (X)=\lambda \text{J}F(0)X \] and proved that this implied the Jacobian conjecture. The author gives a counterexample to the above conjecture (for \(n\geq 5\)) of the form \(F=X+H\) where \(H\) is cubic and homogeneous (earlier such counterexamples were known with \(H\) of degree five). Moreover, a new counterexample to the Markus-Yamabe conjecture is also given.
scientific article; zbMATH DE number 1251195