TITLE: When are polynomial maps open or closed?
QUESTION [5 upvotes]: 1.$f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=x^4+x^3$ is a closed map(or an open map)?
2.$f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=x^5+x^4$ is a closed map?
3.$f:\mathbb{R}^2 \rightarrow \mathbb{R}$, $f(x,y)=x^2+y^2$ is a closed map?
Generally, $f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=a_nx^n+...+a_1x+a_0$ is a closed map or an open map?
Generally, if $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a multivariable polynomials, $f$ is a closed map or an open map?
We have known that $x^2$ is a closed map but not an open map, $x^3+x$ is a homeomorphism, $x^3+2x$ is a closed map but not a homeomorphism.
Thank you very much.

REPLY [4 votes]: This is a small addition to Eric Wofsey's wonderful answer. Here is a useful and often easily checked criterion for properness of a polynomial map.
Proposition. Let $\Bbbk\in \left\{\mathbb R,\mathbb C \right\}$ and let  $\Bbbk^n\overset{f}{\to} \Bbbk^\ell$ be a polynomial mapping. Suppose the top degree homogeneous summand of each component has the property that its only root is the origin. Then $f$ is proper.
Here is a proof.