this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \mathbb R^m$ is open at zero iff it is surjective and $\sup_{\xi\in \mathbb S^{m-1}}\min\left\{|x|\mid x\in f^{-1}(\xi)\right\}<C$ for some constant $C>0$.
The surjectivity of $f$ ensures the existence of a right inverse, which might be, however, not continuous around $0$. This is the case, for example, of quadrics $Q:\mathbb R^n\to \mathbb R^n$. Indeed a map $f:\mathbb R^n\to \mathbb R^n$ such that $f(x_0)=y_0$ admits a continuous right inverse in a neighborhood of $y_0$ iff $f$ is injective in a neighborhood of $x_0$ (it is essentially a consequence of the invariance of the domain); since for quadrics $Q(x)=Q(-x)$ the right inverse can never be continuous.
But this particular class of examples works because $n=m$ .
The question I have is then the following: are there any known sufficient conditions one can impose for a homogeneous polynomial $f:\mathbb R^n\to \mathbb R^m$ open at zero to have a continuous right inverse in a neighborhood of $0\in \mathbb R^m$?
Thanks in advance.
Gil
