On invertibility of some polynomial maps

TL;DR

This paper investigates the invertibility of specific polynomial maps over algebraically closed fields of zero characteristic, establishing conditions under which such maps are injective.

Contribution

It proves that polynomial maps of a particular form are injective if their Jacobian determinant equals one.

Findings

If the Jacobian determinant of the map is 1, then the map is injective.

The polynomial maps considered are of the form x minus a vector of cubed linear forms.

Injectivity is guaranteed under the condition of Jacobian determinant being 1.

Abstract

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