### Smoothing of real algebraic hypersurfaces by rigid isotopies

Define for a smooth compact hypersurface ${M}^{n}$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left({M}^{n}\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}({M}^{n})/r({M}^{n})$, where $r\left({M}^{n}\right)$ is the distance from ${M}^{n}$ to its central set. (In other words, $r\left({M}^{n}\right)$ is the maximal radius of an open non-selfintersecting tube around ${M}^{n}$ in ${\mathbf{R}}^{n+1}.)$
We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left){d}^{\alpha \left(n\right){d}^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and