This question is related to this one [A question about the inverse theorem function in $\mathbb{R}^n$](https://mathoverflow.net/questions/140207/a-question-about-the-inverse-theorem-function-in-mathbbrn). In the response I received, I was told that the issue was related to the singularities, as I do not have a strong background in this area, I'll ask the question again stating exactly the situation.


Let $A\in\mathbb{R}^{n\times n}$ and $\eta\in\mathbb{R}^n$ such that the familly $(A^{n-1}\eta,\ldots,\eta)$ is a basis of $\mathbb{R}^n$. We define the mapping
\begin{equation\*}
 \begin{matrix}
 G\_{\eta}
 \quad : \quad&\mathbb{R}^{n}&\rightarrow&\mathbb{R}^{n}\\
 & (t\_{1},\cdots,t\_{n}) &\mapsto &\prod\_{i=1}^{n}(A+t\_{i}I)\eta
 \end{matrix}.
\end{equation\*}


$G\_{\eta}$ is continuously differentiable and one can readily see that the jacobian determinant of $G\_{\eta}$ is given by
 $$\mid\frac{\partial G\_{\eta}(t\_{1},\ldots,t\_{n})}{\partial(t\_{1},\ldots,t\_{n})}|=
 \mathrm{det}(A^{n-1}\eta,\quad A^{n-2}\eta,\quad\ldots\quad ,\eta)V(t\_{1},\cdots,t\_{n}),$$
where $V(t\_{1},\cdots,t\_{n})=\displaystyle\prod\_{1\leq i<j\leq n}(t\_i-t\_j)$.


Then for $(t\_{1},\cdots,t\_{n})\in\mathbb{R}^{n}$ such that $V(t\_{1},\cdots,t\_{n})\neq0$, we have from the Inverse theorem function the existence of a neighborhood V of $(t\_{1},\cdots,t\_{n})$ in $\mathbb{R}^{n}$ such that $G\_{\eta}(V)$ is a neighborhood of $G\_{\eta}(t\_{1},\cdots,t\_{n})$ in $\mathbb{R}^{n}$. I wonder if this reamain true even for $(t\_{1},\cdots,t\_{n})\in\mathbb{R}^{n}$ such that $V(t\_{1},\cdots,t\_{n})=0$. 



Application of the inverse theorem function on singular points
Near any point where $V$ has a zero of order 1, i.e. a point where precisely two of the $t\_i$ are equal, the map looks like a fold singularity, so doesn't have any such neighborhood. It looks like $(x,y,\dots) \mapsto (x^2,y,\dots)$. I think you can find everything you need in [this nice introduction](http://www.sissa.it/fa/download/publications/remizov.pdf)


