Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and let $$ F=(P_1,\ldots, P_m):\mathbb{K}^n\to \mathbb{K}^m $$ be a polynomial map. I would like to know under what conditions the preimages $F^{-1}(y)$ of $F$ depend continuously on $y\in Im(F)\subset\mathbb{K}^m$, with respect to the Hausdorff topology.
For example, this holds when $\mathbb{K}=\mathbb{R}$, and $P_1,\ldots P_k$ generate the algebra of polynomials invariant under the action of some compact group $G\subset O(n)$. However, I do not even know if the condition holds in the complex setting (i.e., $\mathbb{K}=\mathbb{C}$ and $G$ complex reductive group).
Although my interest is closely related to the group action case, I am primarily looking for conditions on the algebra generated by $P_1,\ldots, P_n$.
Any reference would be welcome!