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.

如果\（[-1,1] \）上的\（（d + 1）\）-平滑函数f（x）具有\（\ mathrm {max \，} _ {[-1,1]} || f（x）| \ ge 1，\）在\（[-1,1] \）上具有\（d + 1 \）或更多不同的零，然后是\（\ mathrm {max \，} _ {[-1 ，1]} | f ^ {（d + 1）}（x）| \ ge 2 ^ {-d-1}（d + 1）！\）。这是根据f在零处的多项式插值以及拉格朗日的余数公式得出的。这是我们所说的“平滑刚性”最简单的例子之一：零套光滑函数某些几何性质˚F暗示对的高阶导数显式下界˚F。如果尺寸大于1，则不能直接使用功能强大的一维工具（例如Lagrange的余数公式）和划分的有限差分。尽管如此，上面的结果仍暗示着通过线段对多个变量的平滑函数的零值有很强的限制（Yomdin，Proc AMS 90（4）：538-542，1984）。在本文中，我们研究了平滑函数零集的几何形状，并显着扩展了Yomdin（1984）的结果，尤其是考虑了有限零集（对于这些部分，线段通常不起作用）。我们的主要目标是基于多项式Remez型不等式（将单位球及其子集上的多项式的最大值进行比较），开发出一种用于平滑刚度的纯多维方法。非常非正式地我们的主要结果之一是， 置 零点Z 的平滑刚度近似为 Z的倒数Remez常数。