Let $\mathbb{A}_k^n$ be the affine $n$-space over $k$. For two hypersufaces $Z(f)$ and $Z(g)$ of $\mathbb{A}_k^n$, where $f,g\in k[x_1,x_2,\cdots,x_n]$, what can we say about the Zarisky closure of their differences $\overline{Z(f)\backslash Z(g)}$?
More specifically, is $\overline{Z(f)\backslash Z(g)}$ also a hypersurface $Z(h)$ for some $h\in k[x_1,x_2,\cdots,x_n]$? How to determine $h$ from $f$ and $g$?