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Let $f$ be a polynomial on $\mathbb{C}^n$. Denote $$X_{R,p} = \{|x|<R\}\cap \{|f(x)|<p\}.$$

In "On the polynomials of I. N. Bernstein" Malgrange writes that H. Hamm proved that $f^{-1}(0)\cap\{|x|<R\}$ is a deformation retract of $X_{R,p}$, for sufficiently large $R$ and small $p$.

Has a proof of this result since appeared in the literature somewhere?

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  • $\begingroup$ Note that sets like $\{|f(x)| < p\}$ are equivalent to $\{|f(x)|^2 < p^2\}$ and the norm-squared function is much friendlier to work with (eg it is smooth). I don't know if your desired result has appeared anywhere, but I think a good starting point is Durfee's paper (jstor.org/stable/1999065). This does what you want globally, ie., without the $\{|x|<R\}$ part; to make the equivalence local, you have to show that the gradient vector field of $-|f|^2$ points inwards along the boundary $\{|x|=R\} - \{f=0\}$. $\endgroup$
    – Vidit Nanda
    Commented Jun 15, 2020 at 11:48
  • $\begingroup$ @ViditNanda Thanks for your comment. In the article you mention it seems though that the singular locus of $f^{-1}(0)$ has to be compact. Now I'm really looking for a general result, so I do not think that this article really contains what I'm looking for. I will take a look at citing literature though, thanks. $\endgroup$
    – user2520938
    Commented Jun 15, 2020 at 12:26

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