The set of zeroes of an “almost polynomial” function

Abstract:

Let $f$ be a smooth function on the unit $n$-dimensional ball, with the ${C^0}$-norm, equal to one. We prove that if for some $k \geqslant 2$, the norm of the $k$ th derivative of $f$ is bounded by ${2^{ - k - 1}}$, then the set of zeroes $Y$ of $f$ is similar to that of a polynomial of degree $k - 1$. In particular, $Y$ is contained in a countable union of smooth hypersurfaces; "many" straight lines cross $Y$ in not more than $k - 1$ points, and the $n - 1$-volume of $Y$ is bounded by a constant, depending only on $n$ and $k$.