Constructive approximation on compact manifolds

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 M be a compact hypersuface of \mathbb{R}^n. There exists a sequence of polynomials \{P_n\} defined on a compact of \mathbb{R}^n such that for n sufficiently large, P_n^{-1}(0) is a hypersurface and

\text{dist}(P_n^{-1}(0), M) \rightarrow 0.

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