Some observations on the properness of Identity plus linear powers

For $2$ vectors $x,y\in {\mathbb{R}}^m$, we use the notation $x * y =(x_1y_1,\ldots ,x_my_m)$,andif$x=y$wealsousethenotation$x^2=x*x$ and define by induction $x^k=x\ast (x^{k-1})$. We use $<,>$ for the usual inner product on ${\mathbb{R}}^m$. For $A$ an $m\times m$ matrix with coefficients in $\mathbb{R}$, we can assign a map $F_A(x)=x+(Ax)^3:~$\mathbb{R}$^m\rightarrow $\mathbb{R}$^m$\mathrm{.}Amatrix$A$isDruzkowskiiff$det(JF_A(x))=1$forall$x\in $\mathbb{R}$^m$. Recently, Jiang Liu posted a preprint on arXiv asserting a proof of the Jacobian conjecture, by showing the properness of $F_A(x)$ when $A$ is Druzkowski, via some inequalities in the real numbers. In the proof, indeed Liu asserted the properness of $F_A(x)$ under more general conditions on $A$, see the main body of this paper for more detail. Inspired by this preprint, we research in this paper on the question of to what extend the above maps $F_A(x)$ (even for matrices $A$ which are not Druzkowski) can be proper. We obtain various necessary conditions and sufficient conditions for both properness and non-properness properties. A complete characterisation of the properness, in terms of the existence of non-zero solutions to a system of polynomial equations of degree at most $3$, in the case where $A$ has corank $1$, is obtained. Extending this, we propose a new conjecture, and discuss some applications to the (real) Jacobian conjecture. We also consider the properness of more general maps $x\pm (Ax{)}^k$ or $x\pm A(x^k)$. By a result of Druzkowski, our results can be applied to all polynomial self-mappings of ${\mathbb{C}}^m$ or ${\mathbb{R}}^m$.