higher derivatives of functions with given critical points and values

TL;DR

This paper investigates how the high-order derivatives of a smooth function are constrained when its gradient vanishes on a set, linking these constraints to the geometric properties of the critical value set.

Contribution

It introduces new insights into the relationship between the vanishing gradient set and the behavior of high-order derivatives, emphasizing the role of the geometry of the critical value set.

Findings

High-order derivatives are constrained by the geometry of the critical value set.

The analysis connects the metric geometry of the critical values to derivative bounds.

Initial results suggest geometric properties influence derivative behavior.

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 $M=M(n,k)>0$. A natural question to ask is ``what happens for other sets $Z$?''. This question was partially answered in [16]-[18]. In the present paper we ask for 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.