几何Kannan-Lovasz-Simonovits引理，多项式子级集体积的无量纲估计，随机解析函数零点的分布
The geometric Kannan-Lovasz-Simonovits lemma, dimension-free estimates for volumes of sublevel sets of polynomials, and distribution of zeroes of random analytic functions
摘要：
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n维凸体中多项式值分布的精确无量纲估计。这样的估计，反过来，导致一个令人惊讶的结果，关于零点的分布随机解析函数;非正式地说，我们表明，对于简单的家庭的解析函数，存在一个“典型”的零点分布，这样的家庭所占的部分的功能，其零点分布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高年级本科生和专家的愉快阅读。至于“报纸上有什么新内容？我们相信，这个问题的答案是两个变量的函数，第一个变量是“写了什么”，第二个变量是“谁在读”。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）。
英文摘要：
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
