It is well known that the Jacobian conjecture follows if it is proved for the special polynomial mappings $f\colon\C^n\to\C^n$ of the Yagzhev type: $f(x)=x-G(x,x,x)$, where $G$ is a trilinear form and $\det f'(x)\equiv1$. Dru\D{z}kowski and Rusek~\cite{7} showed that if we take the local inverse of~$f$ at the origin and expand it into a Taylor series $\sum_{k\ge1}\Phi_k$ of homogeneous terms~$\Phi_k$ of degree~$k$, we find that $\Phi_{2m+1}$ is a linear combination of certain nested compositions'' of~$G$ with itself $m$ times. If~the Jacobian Conjecture were true, $f^{-1}$ should be a polynomial mapping of degree~$\le3^{n-1}$ and the terms $\Phi_k$ ought to vanish identically for $k>3^{n-1}$. We may wonder whether the reason why $\Phi_{2m+1}$ vanishes is that {\it each} of the nested compositions is somehow zero for large~$m$. In this note we show that this is not at all the case, using a polynomial mapping found by van den Essen for other purposes.

### Yagzhev polynomial mappings: on the structure of the Taylor expansion of their local inverse

#### Abstract

It is well known that the Jacobian conjecture follows if it is proved for the special polynomial mappings $f\colon\C^n\to\C^n$ of the Yagzhev type: $f(x)=x-G(x,x,x)$, where $G$ is a trilinear form and $\det f'(x)\equiv1$. Dru\D{z}kowski and Rusek~\cite{7} showed that if we take the local inverse of~$f$ at the origin and expand it into a Taylor series $\sum_{k\ge1}\Phi_k$ of homogeneous terms~$\Phi_k$ of degree~$k$, we find that $\Phi_{2m+1}$ is a linear combination of certain nested compositions'' of~$G$ with itself $m$ times. If~the Jacobian Conjecture were true, $f^{-1}$ should be a polynomial mapping of degree~$\le3^{n-1}$ and the terms $\Phi_k$ ought to vanish identically for $k>3^{n-1}$. We may wonder whether the reason why $\Phi_{2m+1}$ vanishes is that {\it each} of the nested compositions is somehow zero for large~$m$. In this note we show that this is not at all the case, using a polynomial mapping found by van den Essen for other purposes.
##### Scheda breve Scheda completa Scheda completa (DC)
1996
Jacobian conjecture; cubic homogeneous maps.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/393327
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