Mathematics

Nash-Tognoli Theorem states that every compact smooth manifold is diffeomorphic to a real algebraic one. A simple version of this theorem can be proved in the case of a hypersurface of $\mathbb{R}^n$, using a combination of Thom's Isotopy Lemma and Weirestrass' approximation theorem. In this talk I will address this question from the quantitative point of view: given the "data" defining the hypersurface, what can be said about the smallest degree of the approximating polynomial? It turns out that it is possible to measure the "non-singularity" a smooth hypersurface $\left\lbrace f=0\right\rbrace$ is by looking at the distance $a(f)$, in the space of all smooth functions, of the defining function f from the "discriminant" (the set of all singular hypersurfaces, this will be made precise in the talk). As soon as the quantity $a(f)$ is positive, it is possible to produce a quantitative estimate on the degree of a defining polynomial. Estimates of this type imply a compactness result in the $C^2$ topology for the set $a(f)>c>0$, which has many interesting consequences: for instance, there are only finitely many diffeomorphism classes of smooth hypersurfaces with $a(f)>c$, and their Betti numbers can be estimated using $a(f)$ only. 

This is based on joint works with Daouda Niang Diatta and Michele Stecconi.