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Let $f\in\mathbb{C}[x_1,\dots,x_n]$, and let $V(f)$ denote the vanishing locus. Is it true that for large enough $N$, there is a homotopy equivalence $$\mathbb{C}^n\setminus V(f)\simeq B(0,N)\setminus V(f),$$ where $B(0,N)=\{|x|<N\}$.

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    $\begingroup$ This is more generally true for semialgebraic subsets of $\mathbf R^n$ and follows from the fact that they are conical at infinity (see Bochnak, Coste, Roy: Real algebraic geometry, Corollary 9.3.7, p. 225) $\endgroup$ Dec 8 '18 at 20:15
  • $\begingroup$ @JohannesHuisman Thank you for the answer, I will look up the statement in the book. $\endgroup$ Dec 9 '18 at 1:33
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    $\begingroup$ @JohannesHuisman This does indeed answer my question. If you want you write this as an answer and I will accept it. $\endgroup$ Dec 10 '18 at 13:10
  • $\begingroup$ OK, thank you. No need to write it as an answer, I think. $\endgroup$ Dec 11 '18 at 8:41
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    $\begingroup$ @JohannesHuisman It is recommended to convert your comment to an answer in this situation, so that the question does not remain on the "unanswered" list. $\endgroup$ Dec 11 '18 at 19:11
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This is more generally true for semialgebraic subsets of $\mathbf R^n$ and follows from the fact that they are conical at infinity (see Bochnak, Coste, Roy: Real algebraic geometry, Corollary 9.3.7, p. 225)

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