# Is $\mathbb{C}^n\setminus V(f)$ homotopy equivalent with a "large ball complement"?

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|.

• 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) Dec 8 '18 at 20:15
• @JohannesHuisman Thank you for the answer, I will look up the statement in the book. Dec 9 '18 at 1:33
• @JohannesHuisman This does indeed answer my question. If you want you write this as an answer and I will accept it. Dec 10 '18 at 13:10
• OK, thank you. No need to write it as an answer, I think. Dec 11 '18 at 8:41
• @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. Dec 11 '18 at 19:11

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)