论文标题
分化下多项式根的密度的保护法
Conservation Laws for the Density of Roots of Polynomials under Differentiation
论文作者
论文摘要
令$ p_n（x）$为$ n $具有$ n $ dintist的$ n $的多项式，根据$ \ mathbb {r} $上的nice概率分布$ u（0，x）dx $分发的真实根。一个自然的问题是了解$（t \ cdot n）的根源的密度$ u（t，x）$，$ p_n $的$ th衍生品$ 0 <t <1 $ as $ n \ rightarrow \ rightarrow \ infty $。我们得出了$ u（t，x）$的演变的对话定律的\ textit {infinite}。前三个是\ begin {align*} \ int _ {\ MathBb {r}} {u（t，x）〜dx} = 1-t，\ qquad \ qquad \ qquad \ qquad \ int _ {\ m athbb {r}} {u（t，t，t，t，x〜dx} x〜dx} x〜dx} = \ left（1-t \ right（1-t \ right） x〜dx}，\ qquad \ int _ {\ Mathbb {r}} \ int _ {\ Mathbb {r}} u（t，x）（x-y）^2 u（t，y） u（0，y）〜dx dy。 \ end {align*}作者建议$ u（t，x）$可能会根据涉及希尔伯特变换的非本地进化方程而发展；这已通过两种特殊的封闭形式解决方案进行了验证 - 这些保护定律因此指出了希尔伯特转型的有趣身份。我们讨论许多开放问题。
Let $p_n(x)$ be a polynomial of degree $n$ having $n$ distinct, real roots distributed according to a nice probability distribution $u(0,x)dx$ on $\mathbb{R}$. One natural problem is to understand the density $u(t,x)$ of the roots of the $(t\cdot n)-$th derivative of $p_n$ where $0 < t < 1$ as $n \rightarrow \infty$. We derive an \textit{infinite} number of conversation laws for the evolution of $u(t,x)$. The first three are \begin{align*} \int_{\mathbb{R}}{ u(t,x) ~ dx} = 1-t, \qquad \qquad \int_{\mathbb{R}}{ u(t,x) x ~ dx} = \left(1-t\right)\int_{\mathbb{R}}{ u(0,x) x~ dx}, \qquad \int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy = (1-t)^3 \int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy. \end{align*} The author suggested that $u(t,x)$ might evolve according to a nonlocal evolution equation involving the Hilbert transform; this has been verified for two special closed form solutions -- these conservation laws thus point to interesting identities for the Hilbert transform. We discuss many open problems.
