I presented this (pdf, in french) in a short talk for a differential topology course.
I also dabbled with (pdf, in french) the approximation of compact hypersurfaces. I wasn’t able to get a constructive result in time, so I left it as a very rough draft. [I posted a much improved follow up in April.] In the document, I sketch a proof of the following.
Theorem. Let be a compact hypersuface of . There exists a sequence of polynomials defined on a compact of such that for sufficiently large, is a hypersurface and