Some observations on the properness of Identity plus linear powers

Abstract

For $2$ vectors $x,y\in \mathbb{R}^m$, we use the notation $x * y =(x_1y_1,\ldots ,x_my_m)$, and if $x=y$ we also use the notation $x^2=x*x$ and define by induction $x^k=x*(x^{k-1})$. 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 $F_A(x)=x+(Ax)^3:~\mathbb{R}^m\rightarrow \mathbb{R}^m$. A matrix $A$ is Druzkowski iff $det(JF_A(x))=1$ for all $x\in \mathbb{R}^m$. 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 $\mathbb{R}^m$.