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 $B(0,N)=\{|x|<N\}$.



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


