A nonlocal transport equation modeling complex roots of polynomials under differentiation,Proceedings of the American Mathematical Society

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A nonlocal transport equation modeling complex roots of polynomials under differentiation
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-02-10 , DOI: 10.1090/proc/15314
Sean O’Rourke , Stefan Steinerberger

Abstract:Let $p_n:\mathbb{C} \rightarrow \mathbb{C}$ be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution $2\pi r u(r) dr$ in the complex plane. A natural question is how the distribution of roots evolves under repeated (say

times) differentiation of the polynomial. We conjecture a mean-field expansion for the evolution of $\psi (s) = u(s) s$ :

$\displaystyle \frac {\partial \psi }{\partial t} = \frac {\partial }{\partial x... ...\left ( \frac {1}{x} \int _{0}^{x} \psi (s) ds \right )^{-1} \psi (x) \right ).$

The evolution of $\psi (s) \equiv 1$ corresponds to the evolution of random Taylor polynomials

$\displaystyle p_n(z) = \sum _{k=0}^{n}{ \gamma _k \frac {z^k}{k!}}$ $\displaystyle \quad \text {where} \quad \gamma _k \sim \mathcal {N}_{\mathbb{C}}(0,1).$

We discuss some numerical examples suggesting that this particular solution may be stable. We prove that the solution is linearly stable. The linear stability analysis reduces to the classical Hardy integral inequality. Many open problems are discussed.

中文翻译：

微分下的多项式复数根的非局部输运方程

摘要：让一个随机复数多项式，其根是从复数平面中的径向分布中采样的。一个自然的问题是，根的分布在多项式的重复（说时间）微分下如何演化。我们推测平均场扩展为的演化： $p_n：\ mathbb {C} \ rightarrow \ mathbb {C}$ $2 \ pi ru（r）dr$

$\ psi（s）= u（s）s$

$\ displaystyle \ frac {\ partial \ psi} {\ partial t} = \ frac {\ partial} {\ partial x ... ... \ left（\ frac {1} {x} \ int _ {0} ^ {x} \ psi（s）ds \ right）^ {-1} \ psi（x）\ right）。$

的演化对应于随机泰勒多项式的演化 $\ psi（s）\ equiv 1$

$\ displaystyle p_n（z）= \ sum _ {k = 0} ^ {n} {\ gamma _k \ frac {z ^ k} {k！}}$ $\ displaystyle \ quad \ text {where} \ quad \ gamma _k \ sim \ mathcal {N} _ {\ mathbb {C}}（0,1）。$

我们讨论一些数值示例，表明此特定解决方案可能是稳定的。我们证明该解是线性稳定的。线性稳定性分析简化为经典的Hardy积分不等式。讨论了许多未解决的问题。

更新日期：2021-04-12

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