Higher derivatives of functions with zeros on algebraic curves

Abstract

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty interior, then for each $k=1,\ldots,d+1$ the norm of the $k$-th derivative $\|f^{(k)}\|$ 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 $\|f^{(k)}\|\ge \tilde M=\tilde M(n,k)>0$ holds for each $Z$ with the box (or Minkowski, or entropy) dimension $\dim_e(Z)$ greater than $n-\frac{1}{k}$.