论文标题
在分化下对真实随机多项式的复杂根运动建模
Modeling complex root motion of real random polynomials under differentiation
论文作者
论文摘要
在本文中，我们将非局部，非线性偏微分方程视为模拟在分化下随机多项式的复杂根系组的各向异性动力学。这些方程式旨在概括Stefan Steinerberger（2019）在实际情况下获得的最新PDE，以及Sean O'Rourke和Stefan Steinerberger（2020）获得的PDE在径向案例中，相当于1d。这些PDE近似于足够高度n的随机多项式的复合根动力学。时间t的单位对应于n个区分，增量$δ$ t对应于1 n。尚未解决2D的一般情况，特别是对于实际多项式的复杂根部，尚未解决。本文的目的是朝着这个方向提出第一次尝试。我们假设根部是根据具有局部均匀性属性（在文本中定义）的常规分布分布的，并且该属性是在差异下维持的。这使我们能够得出两个耦合方程式的系统来对运动进行建模。对于其他应用程序，我们的系统可能很有趣。用枫树系统计算的示例说明了本文。
In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment $Δ$t corresponds to 1 n. The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.
