TITLE: Are complements of non-isomorphic affine hypersurfaces necessarily non-isomorphic?
QUESTION [10 upvotes]: Fix $n$ and let $H_1$ and $H_2$ be two hypersurfaces in ${\bf A}^n$ (not necessarily smooth or irreducible, but we'll assume reduced). If the complements $U_i = {\bf A}^n \setminus H_i$ are isomorphic as schemes, does this imply that the $H_i$ are isomorphic?
From asking other people, the answer is yes via Euler characteristic arguments when the $H_i$ are smooth, but I have in mind singular examples. 
As for the ground field, the polynomials I have in mind are defined over ${\bf Z}$, so choose whatever field you like. Probably I'd prefer an answer for the complex numbers though.

REPLY [8 votes]: The paper of Jeremy Blanc, "The correspondence between a plane curve and its complement" (Crelle 2009) shows that two irreducible plane curves can have isomorphic complements in P^2 without being isomorphic.  I would look at his argument and see if you can arrange for the affine complements to be isomorphic as well (if indeed you really care about the distinction between A^n and P^n here.)