A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n?

2010-01-15T19:34:36Z

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!

Answer by Mariano Suárez-Alvarez for A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n?

2010-01-15T19:41:53Z

The map $z\in\mathbb C\mapsto z^4\in\mathbb C$, when written out in coordinates, is a polynomial map which sends the closed first quadrant to the whole of $\mathbb R^2$---and by considering cartesian products you get the same for $\mathbb R^{2n}=\mathbb C^n$.

Later: as observed in a comment by Charles, this can be turned into a solution for the open quadrant by composing with a translation, as in $z\in\mathbb C\mapsto (z-z_0)^4\in\mathbb C$ with $z_0$ in the open first quadrant.

A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n? - MathOverflow

A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n?

Answer by Mariano Suárez-Alvarez for A polynomial map from ℝ^n to ℝ^n mapping the positive orthant onto ℝ^n?