Let $P:\mathbb{R}^m\to \mathbb{R}^m$ be a regular map, i.e. a map whose components are polynomials. I was wondering whether we can say anything about the the component polynomials, their degrees or growth rates that'll guarantee that the inverse of the regular map $P$ exists (but doesn't have to be a regular map itself, like in the Jacobian conjecture).
As a quick example, if the regular map is "diagonal" in some sense, i.e.
$P(x_1,\dots x_m):=(P_1(x_1) \dots P_m(x_m)), P_i$'s are polynomials of one variable only, then invertibility of $P$ is equivalent to the invertibility of each $P_i, 1\le i \le m,$ which guarantees that $\deg(P_i)$ must be odd to guarantee the existence but not uniqueness of a real root of $P_i$. But is there a sufficient condition for invertibility of a general, or at least more general (than the "diagonal" one mentioned above) $P?$
