论文标题
光滑的刚性和雷姆型不平等现象
Smooth rigidity and Remez-type inequalities
论文作者
论文摘要
如果一个变量的平滑函数在单位间隔上具有最大一个，并且在那里具有$ d $零，则其$（d+1）$ - st derivative必须是“大”。这是我们所谓的“平滑刚度”的最简单示例之一：零平滑功能集的某些几何特性$ f $意味着在$ f $的高阶导数上显式下限。在大于一个的维度中，强大的一维工具，例如拉格朗日的剩余公式和有限的差异，并不直接适用。尽管如此，上面的结果意味着通过线路部分，对几个变量的平滑函数的零限制\ cite {yom1}）。 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\ cite {yom1}的结果，包括考虑到有限的零集（为此，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（将单位球上多项式的最大值和子集比较）。非常非正式的情况，我们的主要结果之一是，零Z $的“平滑刚度”大约是$ z $的“ Inverse Remez常数”。
If a smooth function of one variable has maximum one on the unit interval, and has there $d$ zeroes, then its $(d+1)$-st derivative must be "big". 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, like 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 \cite{Yom1}). In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of \cite{Yom1}, including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a truly 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 a "smooth rigidity" of a zeroes set $Z$ is approximately the "inverse Remez constant" of $Z$.
