# Is \$ mathbb {C} ^ n setminus V (f) \$ equivalent homotopy with a "big ball add-on"?

Leave $$f in mathbb {C}[x_1,dots,x_n]$$, and let $$V (f)$$ denotes the escape locus. Is it true that for quite some time? $$N$$, there is a homotopy equivalence
$$mathbb {C} ^ n setminus V (f) simeq B (0, N) setminus V (f),$$
where $$B (0, N) = {| x | .