A note on the Jacobian Conjecture

In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX{)}^{\ast 3}$, i.e. $F(X)=(x_1+(a_{11}{x}_1+\mathrm{.}\mathrm{.}\mathrm{.}+a_{1n}{x}_n{)}^3,\mathrm{.}\mathrm{.}\mathrm{.},x_n+(a_{n1}{x}_1+\mathrm{.}\mathrm{.}\mathrm{.}+a_{nn}{x}_n{)}^3)$, satisfies $TrJ((AX{)}^{\ast 3})=0$, then $rank(A)\le 1\mathrm{/}2(n+\delta )$ 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 $\le 9$ in the case ${\prod }_{i=1}^n{a}_{ii}\mathrm{\ne }0$.