What conditions would guarantee $\Psi: \mathbb{R}^n \rightarrow \mathbb{R}^2$ to be non-zero in a small neighbourhood?

Question

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.

Answer 1

Let $h(x)=\|\Psi(x)\|^2_2$. Then $h(x)=0$ iff $\Psi(x)=0$ and $h:\mathbb{R}^n \to \mathbb{R}$.

If $\Psi \in C^2$, $\nabla h(0)=0$, and $\text{Hess}_0 h$ (the Hessian of $h$ at $x=0$) is positive definite. Then $0$ is a local minimum. Hence there is no other point such that $h(x) = 0 $ (i.e. $\Psi(x)=0$) for $x$ in a neighborhood of $0$.

Answer 2

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.

Answer 3

Here is a relevant fact:

Theorem Suppose that $\psi\colon \mathbb{R}^n \to \mathbb{R}$ is $C^3$, satisfies $\psi(\mathbf{0}) = 0$, and that the matrix of second partial derivatives $\left[\frac{\partial^2 \psi}{\partial x_i\,\partial x_j}(\mathbf{0})\right]$ is positive-definite. Then $\psi$ has a local minimum at $\mathbf{0}$, meaning there is an open neighborhood $U$ of $\mathbf{0}$ such that $\psi(\mathbf{x}) > 0$ for all $\mathbf{x} \in U$, $\mathbf{x} \neq \mathbf{0}$. Similarly, if the matrix of second partial derivatives is negative definite, then $\psi$ has a local maximum at $\mathbf{0}$.

The proof requires Taylor's theorem and some linear algebra.

Now suppose $\mathbf{\Psi}\colon \mathbb{R}^n \to \mathbb{R}^2$ is of class $C^3$, and $\mathbf{\Psi}(\mathbf{0}) = \mathbf{0}$. Writing $\mathbf{\Psi} = (\psi_1,\psi_2)$, suppose further that the matrix of second partial derivatives $\left[\frac{\partial^2 \psi_k}{\partial x_i\,\partial x_j}(\mathbf{0})\right]$ is positive-definite or negative-definite for each $k$. Then applying the theorem to each component gives neighborhoods $U_1$ and $U_2$ of $\mathbf{0}$ for which $\psi_k(\mathbf{x}) \neq 0$ for all $\mathbf{x} \in U_k \setminus\{\mathbf{0}\}$. Let $U = U_1 \cap U_2$.