Lower Bounds for High Derivatives of Smooth Functions With Given Zeros

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 \(\Vert f^{(k)}\Vert \) 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] and [20–22]. This study is naturally related to a certain special settings of the Whitney’s smooth extension problem. Our goal in this 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, which extend the recent result of [21], where an answer to the above question is given via the topological information on Z.