Consider a mapping $f\colon\C^n\to\C^n$ of the form identity plus a term with polynomial components that are homogeneous of the third degree, and suppose that the Jacobian determinant of this mapping is constant throughout~$\C^n$ (polynomial mapping of Yagzhev type). As a stronger version of the classical Jacobian conjecture, the question has been posed whether for some values of~$\lambda\in\C\setminus\{0\}$ there exists a global change of variables (``conjugation'') on~$\C^n$ through which the mapping $\lambda f$ becomes its linear part at the origin. Van den~Essen has recently produced a simple Yagzhev mapping for which no such {\it polynomial} conjugation exists. We show here that van den~Essen's example still admits {\it global analytic} conjugations. The question on the existence of global conjugations for general Yagzhev maps is then still open.
On the existence of global analytic conjugations for polynomial mappings of Yagzhev type
ZAMPIERI, Gaetano
1996
Abstract
Consider a mapping $f\colon\C^n\to\C^n$ of the form identity plus a term with polynomial components that are homogeneous of the third degree, and suppose that the Jacobian determinant of this mapping is constant throughout~$\C^n$ (polynomial mapping of Yagzhev type). As a stronger version of the classical Jacobian conjecture, the question has been posed whether for some values of~$\lambda\in\C\setminus\{0\}$ there exists a global change of variables (``conjugation'') on~$\C^n$ through which the mapping $\lambda f$ becomes its linear part at the origin. Van den~Essen has recently produced a simple Yagzhev mapping for which no such {\it polynomial} conjugation exists. We show here that van den~Essen's example still admits {\it global analytic} conjugations. The question on the existence of global conjugations for general Yagzhev maps is then still open.
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