Higher derivatives of functions with zeros on algebraic curves

TL;DR

This paper extends a known result about the lower bounds of derivatives of functions that vanish on sets with interior to sets that are sufficiently dense, characterized by their covering numbers and Minkowski dimension.

Contribution

It demonstrates that the lower bounds on derivatives hold for sets with high enough Minkowski dimension, generalizing previous results from sets with interior to more sparse sets.

Findings

Lower bounds on derivatives extend to dense sets with high Minkowski dimension.

The bounds depend on the set's covering numbers and dimension.

Results apply to finite and infinite dense sets.

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 $n-\frac{1}{k}$.