  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 $M=M(n,k)>0$. A natural question to
ask is: What happens for other sets $Z$? In particular, for finite, but
sufficiently dense sets?}
  This question was partially answered in ([16],[20-22]). This study can be
naturally related to a certain special settings of the classical Whitney's
smooth extension problem.
  Our goal in the present paper is threefold: first, to provide an overview of
the relevant questions and existing results in the general Whitney's problem.
Second, we provide an overview of our specific setting and some available
results. Third, we provide some new results in our direction. These new results
extend the recent result of [21], where an answer to the above question is
given via the topological information on $Z$.
