On invertibility of some polynomial maps

Over an algebraically closed field $\mathbb{F}$ of zero characteristic polynomial map $\xi :{\mathbb{F}}^n\to {\mathbb{F}}^n$ of the form $\xi (x)=x-((x{A}_1{)}^3,(x{A}_2{)}^3,\mathrm{.}\mathrm{.}\mathrm{.},(x{A}_n{)}^3)$, where $x=(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ a row vector of variables, $A_i\in {\mathbb{F}}^n$ are column vectors, is considered. It is shown that if $\det (\mathrm{\partial }\xi (x))=1$, then $\xi$ is injective.