Higher derivatives of functions with given critical points and values

Let \(f: B^n \rightarrow {{\mathbb {R}}}\) be a \(d+1\) times continuously differentiable function on the unit ball \(B^n\), with \(\mathrm{max\,}_{z\in B^n} \Vert f(z) \Vert =1\). A well-known fact is that if f vanishes on a set \(Z\subset B^n\) with a non-empty interior, then for each \(k=1,\ldots ,d+1\) 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?”. This question was partially answered in Goldman and Yomdin (Lower bounds for high derivatives of smooth functions with given zeros. arXiv:2402.01388), Yomdin (Anal Math Phys 11:89, 2021), Yomdin (J Singul 25:443–455, 2022) and Yomdin (Higher derivatives of functions vanishing on a given set. arXiv:2108.02459v1). In the present paper, we ask a similar (and closely related) question: what happens with the high-order derivatives of f, if its gradient vanishes on a given set \(\Sigma \)? And what conclusions for the high-order derivatives of f can be obtained from the analysis of the metric geometry of the “critical values set” \(f(\Sigma )\)? In the present paper, we provide some initial answers to these questions.