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

#### 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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Jacobian conjecture; cubic homogeneous maps; conjugations
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11562/393330
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