The $\mathbb{Z}_2$ topological degree of a (non-constant) polynomial in one variable, clearly, coincides with its degree as a polynomial, $\mod 2$.

Consider further a polynomial self-mapping $F$ on $\mathbb{R}^2$, and assume it is a proper map (in case, even more generally a map in higher dimension)

Is there a short way to decide what's the parity of the topological degree of $F$?

E.g. it's odd if $F$ is an odd map, or more generally, if $F$ can be transformed into an odd map by a proper homotopy. Actually: is there a short way to understand if a polynomial map is a proper map? What about the case of a gradient map (I mean, the gradient of a polynomial $f\in\mathbb{R}[x,y]$)?

I'm somehow confident that there may be a simple criterion known, at least in $\mathbb{R}^2.$ After all, what is required is just a one-bit information (well this argument doesn't convince me either).

lowerbound on the complexity of the question... – Thierry Zell Sep 14 '10 at 19:23