  We provide counterexamples to the stable equivalence problem in every
dimension $d\geq2$. That means that we construct hypersurfaces $H_1,
H_2\subset\mathbb{C}^{d+1}$ whose cylinders $H_1\times\mathbb{C}$ and
$H_2\times\mathbb{C}$ are equivalent hypersurfaces in $\mathbb{C}^{d+2}$,
although $H_1$ and $H_2$ themselves are not equivalent by an automorphism of
$\mathbb{C}^{d+1}$. We also give, for every $d\geq2$, examples of two
non-isomorphic algebraic varieties of dimension $d$ which are biholomorphic.
