Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n+m}$ be a real-analytic function (at least in a neighbourhood of a point). This question establishes that if the Jacobian matrix at a point is full rank, i.e. rank $n$, then the function is locally injective at that point.
I guess the converse (non full rank implies non locally injective) is false, since generalising $x \mapsto x^3$ to $x \mapsto (x^3, x^3)$ gives a counterexample, as the Jacobian at $x = 0$ has rank zero but the function is injective. My question is are there any sufficient conditions on the Jacobian at a point that imply the function is not locally injective at a point? (Or sufficient conditions in general, with or without using the Jacobian.) I'm also interested in the complex case but perhaps that's best reserved for another question.
EDIT: Thanks to the comment, I stumbled across this question on MO - where it's argued for $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ that if the Jacobian simultaneously has negative and positive determinant somewhere, then $f$ is not injective. It seems reasonable that this extends locally, i.e. $f$ is not locally injective at a point if in every neighbourhood of that point there are two points where the determinant is negative and positive. Does anyone know if it possible to extend this to the case $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n+m}$?
