# 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

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