Smooth rigidity and Remez inequalities via Topology of level sets

TL;DR

This paper explores the topology of level sets to establish new Remez-type inequalities and smooth rigidity bounds, connecting approximation theory, singularity theory, and Whitney extensions with explicit quantitative results.

Contribution

It introduces a novel Remez-type inequality and a corresponding smooth rigidity inequality based on the topology of level sets, advancing the understanding of polynomial and smooth function behavior.

Findings

New Remez-type inequality for polynomials with level set topology

Explicit lower bounds for derivatives of smooth functions vanishing on level sets

Topological approach to smooth rigidity and polynomial approximation

Abstract

A smooth rigidity inequalitiy 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 are the main new results of the paper: \smallskip Let $B^n$ be the unit $n$-dimensional ball. For a given integer $d$ let $Z\subset B^n$ be a smooth compact hypersurface with $N=(d-1)^n+1$ connected components $Z_j$. Let $\mu_j$ be the $n$-volume of the interior of $Z_j$, and put $\mu=\min \mu_j, \ j=1,\ldots, N$. Then for each polynomial $P$ of degree $d$ on ${\mathbb R}^n$ we have $$ \frac{\max_{B^n}|P|}{\max_{Z}|P|}\le (\frac{4n}{\mu})^d. $$ As a consequence, we provide an explicit lower bound for the $(d+1)$-st derivatives of any smooth function $f$, which vanishes on $Z$, while being of order $1$ on $B^n$ (smooth rigidity)}: $$ ||f^{(d+1)}||\ge \frac{1}{(d+1)!}(\frac{4n}{\mu})^d. $$ We also provide an interpretation, in terms of smooth rigidity, of one of the simplest versions of the results in \cite{Ler.Ste}.