dg.differential geometry - A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n?

Question: Is there a polynomial map from ℝⁿ to ℝⁿ under which the image of the positive orthant (the set of points with all coordinates positive) is all of ℝⁿ?

Some observations:

My intuition is that the answer must be 'no'... but I confess my intuition for this sort of geometric problem is not very well-developed.

Of course it is relatively easy to show that the answer is 'no' when n=1. (In fact it seems like a nice homework problem for some calculus students.) But I can't seem to get any traction for n>1.

This feels like the sort of thing that should have an easy proof, but then I remember feeling that way the first time I saw the Jacobian conjecture... now I'm wary of statements about polynomial maps of ℝⁿ!