TITLE: Vector valued functions
QUESTION [5 upvotes]: Question : Does  there exist a surjective function $F$ that maps $\mathbb{R}^n_+$ to $\mathbb{R}^n$ (where $\mathbb{R}^n_+$ denotes the set of vectors of length $n$ with only positive entries). The answer is yes by considering the function $F$:
$$(x_1,\ldots,x_n)\to(\log x_1,\ldots,\log x_n)$$
It is easy to see that $F$ is surjective.
Now my real question is if there is such a function $F$ whose components are restricted to be polynomials of degree at most $n$ (and $F$ maps $\mathbb{R}^n_+$ to $\mathbb{R}^n$ and is surjective). For example:
$$(x_1,x_2,x_3,x_4)\to(1-x_1,x_1\cdot x_2+x_3,x_3^2+x_2-x_4,5x_4)$$
has polynomial components, but is not surjective as the restriction $x_i>0$ implies we cannot get a $1$ in the first coordinate.
Any help or advice is greatly appreciated. I think the answer is no but am not sure how to prove it.

REPLY [2 votes]: For n = 1, it is well-known and obvious that a polynomial $P(x)$ approaches either $+ \infty$ or else $-\infty$ as $x \to \infty$, so in particular it has all its values in either the positive half-axis or else in the negative half-axis for say $x > M$. But then it cannot cover the whole other half-axis on the compact interval $[0,M].