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 $\Delta$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.
翻译:在本文中,我们考虑的是非局部、非线性部分差异方程式,以模拟在差异下随机多元体的复杂根组的异常现象变化。这些方程式旨在概括Stefan Steinerberger(2019年)在真实案件中最近获得的PDE(2019年),Sean O'Rourke和Stefan Steinerberger(202020年)在辐射案件中获得的PDE(202020年),这相当于1D。这些PDEs为足够高水平的随机多元体的复杂根组的复杂根组的动态。时间单位与n差异相对应,加值$\Delta$t对应于1n。2D的一般情况,特别是真实多元体的复杂根组,尚未得到解决。本文的目的是首次朝这个方向提出尝试。我们假设根的分布与本地同质属性(在文本中定义)的定期分配,而该属性在差异下维持。这使我们得以产生两个相配对的公式系统来模拟该运动。我们的系统与其它应用的图图图能被展示。