论文标题
重复分化下的零动力学
Dynamics of zeroes under repeated differentiation
论文作者
论文摘要
考虑一个随机的多项式$ p_n $ $ n $，其根部是根据某些概率分布$μ_0$在复杂平面上$ \ mathbb c $采样的独立随机变量。自然而然地认为，对于[0,1）$中的固定$ t \而言，$ n \ to \ infty $，$ [tn] $ - $ p_n $的零元素的零是根据$ \ mathbb c $的$ \ \ mathbb c $的$μ_t$进行分配的。假设$μ_0$集中在真实线上，或者是旋转不变的，Steinerberger [Proc。 AMS，2019年]以及O'Rourke和Steinerberger [Arxiv：1910.12161]衍生出根密度的非局部传输方程。我们介绍了一种治疗此类问题的不同方法。在旋转不变的情况下，我们获得了$ψ（x，t）$的封闭公式，这是$ [tn] $ -p_n $的$ [tn] $ -th衍生物的径向部分的渐近密度。尽管它的推导是不合格的，但我们为其正确性提供了数值证据，并证明它解决了O'Rourke和Steinerberger的PDE。此外，我们提出了几个示例，其中解决方案是完全显式的（包括特殊情况，其中初始条件$ψ（x，0）$是Delta函数的任意凸组合），并分析了解决方案的某些特性，例如void annuli和零圆圈的行为。作为对方法正确性的额外支持，我们表明，当$μ_0$集中在实际线上时，应用于适用的情况，给出了正确的结果，该结果已知，该结果可以从自由概率方面具有解释。
Consider a random polynomial $P_n$ of degree $n$ whose roots are independent random variables sampled according to some probability distribution $μ_0$ on the complex plane $\mathbb C$. It is natural to conjecture that, for a fixed $t\in [0,1)$ and as $n\to\infty$, the zeroes of the $[tn]$-th derivative of $P_n$ are distributed according to some measure $μ_t$ on $\mathbb C$. Assuming either that $μ_0$ is concentrated on the real line or that it is rotationally invariant, Steinerberger [Proc. AMS, 2019] and O'Rourke and Steinerberger [arXiv:1910.12161] derived nonlocal transport equations for the density of roots. We introduce a different method to treat such problems. In the rotationally invariant case, we obtain a closed formula for $ψ(x,t)$, the asymptotic density of the radial parts of the roots of the $[tn]$-th derivative of $P_n$. Although its derivation is non-rigorous, we provide numerical evidence for its correctness and prove that it solves the PDE of O'Rourke and Steinerberger. Moreover, we present several examples in which the solution is fully explicit (including the special case in which the initial condition $ψ(x,0)$ is an arbitrary convex combination of delta functions) and analyze some properties of the solutions such as the behavior of void annuli and circles of zeroes. As an additional support for the correctness of the method, we show that a similar method, applied to the case when $μ_0$ is concentrated on the real line, gives a correct result which is known to have an interpretation in terms of free probability.
