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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!

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  • $\begingroup$ Could you please give a definition or reference for the "Haudorff topology"? Thanks a lot. $\endgroup$ Feb 1, 2016 at 9:03
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    $\begingroup$ Given subsets $X,Y$ of a metric space $Z$, the Hausdorff distance $d_H(X,Y)$ is the infimum number $r$ such that $X$ is contained in the $r$-ball around $Y$, and vice versa. For example, two concentric circles in the plane, of radii $a$ and $b$, have Hausdorff distance $|b-a|$. $\endgroup$ Feb 2, 2016 at 0:22

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