Geodesic
Smooth rigidity and Remez inequalities via Topology of level sets
Journal of Singularities, Tome 25 (2022), pp. 443-455

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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. 
@article{10_5427_jsing_2022_25v,
     author = {Y. Yomdin},
     title = {Smooth rigidity and {Remez} inequalities via {Topology} of level sets},
     journal = {Journal of Singularities},
     pages = {443--455},
     publisher = {mathdoc},
     volume = {25},
     year = {2022},
     doi = {10.5427/jsing.2022.25v},
     url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2022.25v/}
}
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Y. Yomdin. Smooth rigidity and Remez inequalities via Topology of level sets. Journal of Singularities, Tome 25 (2022), pp. 443-455. doi: 10.5427/jsing.2022.25v

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