Version 1(solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$.
Version 2: If $f$ : $\mathbb R^n \rightarrow \mathbb R$ is a polynomial, $D^2f(x)$ is non-degenerate for every $x$, then $f(x)$ has at least one critical point.
The problem is how to prove Version 1 or prove it is not true. The result of Version 2 is what I need to prove another problem.

Version 1  (solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$.
Version 2: If $f$ : $\mathbb R^n \rightarrow \mathbb R$ is a polynomial, $D^2f(x)$ is non-degenerate for every $x$, then $f(x)$ has at least one critical point.
The problem is how to prove or disprove Version 1. The result of Version 2 is what I need to prove another problem.

Version 1: If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$.
Version 2: If $f$ : $\mathbb R^n \rightarrow \mathbb R$ is a polynomial, $D^2f(x)$ is non-degenerate for every $x$, then $f(x)$ has at least one critical point.
The problem is how to prove Version 1 or prove it is not true. The result of Version 2 is what I need to prove another problem.

Version 1(solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$.
Version 2: If $f$ : $\mathbb R^n \rightarrow \mathbb R$ is a polynomial, $D^2f(x)$ is non-degenerate for every $x$, then $f(x)$ has at least one critical point.
The problem is how to prove Version 1 or prove it is not true. The result of Version 2 is what I need to prove another problem.