What conditions would guarantee $\Psi: \mathbb{R}^n \rightarrow \mathbb{R}^2$ to be non-zero in a small neighbourhood? Suppose I have $\Psi: \mathbb{R}^n \rightarrow \mathbb{R}^2$ where 
$\Psi(\mathbf{0}) = \mathbf{0}$. I would like to show that there exists a small open set $U$ around $\mathbf{0}$ such that it is non-zero for all points in $U \backslash \{ \mathbf{0} \}$. 
I am wondering what kind of conditions on $\Psi$ would ensure this is satisfied? Any comments are appreciated. Thank you.  

If $n\leq2$ and ${\rm rank}\bigl(d\Psi({\bf 0})\bigr)=n$ then $\Psi$ is injective in a neighborhood of ${\bf 0}\in{\mathbb R}^n$, by the inverse function theorem.
If $n\geq3$ and ${\rm rank}\bigl(d\Psi({\bf 0})\bigr)=2$ then $\Psi^{-1}({\bf 0})$ is an $(n-2)$-dimensional submanifold of ${\mathbb R}^n$, by the implicit function theorem.
If $d\Psi({\bf 0})$ does not have maximal rank things are more complicated. I don't know of a simple answer.
