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 | <N } $.