Higher derivatives of functions with zeros on algebraic curves

Abstract

Let $f:B^n\to \mathbb{R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with ${\max }_{z\in B^n}\Vert f(z)\Vert =1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty interior, then for each $k=1,\dots ,d+1$ the norm of the $k$-th derivative $\Vert f^{(k)}\Vert$ is at least $M=M(n,k)>0$. We show that this fact remains valid for all ``sufficiently dense'' sets $Z$ (including finite ones). The density of $Z$ is measured via the behavior of the covering numbers of $Z$. In particular, the bound $\Vert f^{(k)}\Vert \ge \tilde{M}=\tilde{M}(n,k)>0$ holds for each $Z$ with the box (or Minkowski, or entropy) dimension ${\dim }_e(Z)$ greater than $n-\frac{1}{k}$.