This paper studies the properness of polynomial selfmaps \(F:\mathbb {R}^m \rightarrow \mathbb {R}^m\) or \(F:\mathbb {C}^m\rightarrow \mathbb {C}^m\) . By results of Druzkowski, this question is reduced to that of properness of maps of the following special form, which we call identity plus linear powers. For two 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 by induction \(x^k=x*(x^{k-1})\) . We use \(\langle ,\,\rangle \) 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 if and only if \(\det (JF_A(x))=1\) for all \(x\in \mathbb {R}^m\) . In this paper we research on the question of to what extent the above maps \(F_A(x)\) can be proper, and obtain various necessary conditions and sufficient conditions which suit very well the special form of the maps \(F_A\) . 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 \(x\pm A(x^k)\) . The advantage of our method, compared to existing methods such as Newton’s polyhedron, is that our criteria, which are tailored for the maps \(F_A\) , are easy to check and applicable to parametrised families of matrices. Moreover, we can work on \(\mathbb {R}\) , while other known results requiring working over algebraically closed fields such as \(\mathbb {C}\) .
