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arXiv · 1205.5853

A note on the Jacobian Conjecture

Dan Yan
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Abstract

In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX)^{*3}$, i.e. $F(X)=(x_1+(a_{11}x_1+...+a_{1n}x_n)^3,...,x_n+(a_{n1}x_1+...+a_{nn}x_n)^3)$, satisfies $TrJ((AX)^{*3})=0$, then $rank(A)\leq 1/2(n+\delta)$ where $\delta$ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension $\leq 9$ in the case $\prod_{i=1}^{n}a_{ii}\neq0$.

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Dan Yan. 2012-05-26. A note on the Jacobian Conjecture. https://doi.org/10.1016/j.laa.2011.01.016

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