“Smooth rigidity” and Remez-type inequalities

Abstract

If a \((d+1)\)-smooth function f(x) on \([-1,1]\), with \(\mathrm{max\,}_{[-1,1]}|f(x)|\ge 1,\) has \(d+1\) or more distinct zeroes on \([-1,1]\), then \(\mathrm{max\,}_{[-1,1]}|f^{(d+1)}(x)|\ge 2^{-d-1}(d+1)!\). This follows from the polynomial interpolation of f at its zeroes, with Lagrange’s remainder formula. This is one of the simplest examples of what we call “smooth rigidity”: certain geometric properties of zero sets of smooth functions f imply explicit lower bounds on the high-order derivatives of f. In dimensions greater than one, the powerful one-dimension tools such as Lagrange’s remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroes of smooth functions of several variables (Yomdin in Proc AMS 90(4):538–542, 1984). In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of Yomdin (1984), including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a pure multi-dimensional approach to smooth rigidity, based on polynomial Remez-type inequalities (which compare the maxima of a polynomial on the unit ball, and on its subset). Very informally, one of our main results is that the smooth rigidity of a zeroes set Z is approximately the reciprocal Remez constant of Z.