Is possible transform an analytic map into a monomial?

Question

I'm quite new with the concept of a monomialization of a map, which seems to be quite useful. I'm hoping that, given a map with analytic coordinates $\boldsymbol{c}:\mathbb{R}^n \rightarrow \mathbb{R}^p$ and a point $\boldsymbol{x}\in\mathbb{R}^n$, a monomialization is able of provide a continuously-differentiable surjective map $\phi: U \rightarrow V$ from an open set $U\subset \mathbb{R}^s$ to an open set $V\subset\mathbb{R}^n$ with $\phi(\boldsymbol{0}) = \boldsymbol{x}$ such that each coordinate of $\boldsymbol{c} \circ \phi$ is a monomial of degree bigger than 2. Is it possible?

Answer 1

This is possible, or almost. Each coordinate having a degree 2 or bigger is not necessarily possible, at least there are no well known theorems about it, as far I have looked on the internet. The $\textbf{Theorem 2.1}$ and introduction states what is possible to accomplish in my context. I hope it helps people in future.