ASZ-SIMONOVITS LEMMA, DIMENSION-FREE ESTIMATES FOR VOLUMES OF SUBLEVEL SETS OF POLYNOMIALS, AND DISTRIBUTION OF ZEROES OF RANDOM ANALYTIC FUNCTIONS.
ASZ-SIMONOVITS LEMMA, DIMENSION-FREE ESTIMATES FOR VOLUMES OF SUBLEVEL SETS OF POLYNOMIALS, AND DISTRIBUTION OF ZEROES OF RANDOM ANALYTIC FUNCTIONS.
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发表时间：
2001-08
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通讯作者：
F. Nazarov;M. Sodin;A. Volberg
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作者：
F. Nazarov;M. Sodin;A. Volberg
本文的目的是吸引读者的注意力，一个无量纲的几何不等式，可以证明使用经典的针分解。这个不等式使我们能够得到清晰的无量纲估计的分布多项式的值在n维凸体。这样的估计，反过来，导致一个令人惊讶的结果分布的零的随机解析函数;非正式地说，我们表明，简单的家庭的解析函数，存在一个“典型的”分布的零，这样的家庭的一部分所占据的职能，其分布的零偏离，典型的一个由一些固定量，是约常数exp{-大小的偏差}。该文件基本上是独立的。在选择风格时，我们试图使它成为一个愉快的阅读都为高年级本科生和专家。至于“论文有什么新内容？”“，我们相信它的答案是两个变量的函数，第一个是“写了什么”，第二个是“谁在阅读”。由于我们不知道第二个变量的值，我们只能在第一个变量固定的情况下给出答案的范围。对于目标受众，它将是标准范围[无，一切]（包括两个端点）。
The goal of this paper is to attract attention of the reader to a dimension-free geometric inequality that can be proved using the classical needle decomposition. This inequality allows us to derive sharp dimension-free estimates for the distribution of values of polynomials in n-dimensional convex bodies. Such estimates, in their turn, lead to a surprising result about the distribution of zeroes of random analytic functions; informally speaking, we show that for simple families of analytic functions, there exists a "typical" distribution of zeroes such that the portion of the family occupied by the functions whose distribution of zeroes deviates from that typical one by some fixed amount, is about Const exp{-size of the deviation}. The paper is essentially self-contained. When choosing the style, we tried to make it an enjoyable reading for both a senior undergraduate student and an expert. As to the question "What is new in the paper?", we beleive that the answer to it is a function of two variables, the first being "what is written" and the second being "who is reading". Since we have no knowledge of the value of the second variable, we can only give the range of answers with the first variable fixed. For the targeted audience it will be the standard range [Nothing, Everything] (with both endpoints included).