Smooth rigidity and Remez inequalities via Topology of level sets

Y. Yomdin

Journal of Singularities
volume 25 (2022), 443-455

Received: 31 January 2021. Received in revised form: 14 March 2022.

DOI: 10.5427/jsing.2022.25v

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Abstract:

A smooth rigidity inequality provides an explicit lower bound for the $(d+1)$-st derivatives of a smooth function $f$, which holds, if $f$ exhibits certain patterns, forbidden for polynomials of degree $d$. The main goal of the present paper is twofold: first, we provide an overview of some recent results and questions related to smooth rigidity, which recently were obtained in Singularity Theory, in Approximation Theory, and in Whitney smooth extensions. Second, we prove some new results, specifically, a new Remez-type inequality, and on this base we obtain a new rigidity inequality. In both parts of the paper we stress the topology of the level sets, as the input information.

Here is a very informal statement of the main new result of this paper: the classical Remez-type inequality compares the maxima of a polynomial on the unit ball B and on its sub-domain Z. The answer is given in terms of the volume of Z. We replace Z by its boundary (whose volume is zero), but require Z to have sufficiently many connected components. Then the answer is given in terms of the minimal volume of the connected components of Z.

Author(s) information:

Y. Yomdin
The Weizmann Institute of Science
Rehovot 76100, Israel
yosef.yomdin@weizmann.ac.il