G. Goldman, Y. Yomdin, Derivatives of all orders of smooth functions with given zeros, Vol. 2026 (2026), No. 31, pp. 1-10

Full Text: PDF
DOI: 10.23952/cot.2026.31

Received February 5, 2025; Accepted May 20, 2025; Published online March 23, 2026

Abstract. Let $f: B^n \rightarrow {\mathbb R}$ be an infinitely 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 $d=1,2,\ldots,$ the norm of the d-th derivative $||f^{(d)}||$ is at least $\frac{(d+1)!}{2^{d+1}}$ . A natural question to ask is: What happens for other sets Z? For finite, but sufficiently dense sets Z? This question was partially answered by the second author of this paper. This study can be naturally related to a certain special settings of the classical Whitney smooth extension problem. However, in our results, as well as in most of publications on the Whitney problem, the differentiability degree d was assumed to be fixed. Our main goal in the present paper is to allow d to run to $\infty$ in our results above, and to present the corresponding lower bounds for the asymptotic behavior of the high-order derivatives of f in terms of simple geometric characteristics of the zero set Z of f.

How to Cite this Article:
G. Goldman, Y. Yomdin, Derivatives of all orders of smooth functions with given zeros, Commun. Optim. Theory 2026 (2026) 31.