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?
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!