On invertibility of some polynomial maps

Abstract: Over an algebraically closed field $\mathbb{F}$ of zero characteristic polynomial map $$ξ: \mathbb{F}^n\rightarrow \mathbb{F}<sup>n$$ of the form $$ξ(x)=x-((xA_1)^{3}, (xA_2)^{3},..., (xA_n)<sup>{3})$$, where $x=(x_1,x_2,...,x_n)$ a row vector of variables, $A_i\in \mathbb{F}<sup>n$ are column vectors, is considered. It is shown that if $\det(\partialξ(x))=1$, then $ξ$ is injective.