Dynamics of Zeroes Under Repeated Differentiation

ABSTRACT

Consider a random polynomial P_n of degree n whose roots are independent random variables sampled according to some probability distribution μ₀ 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 μ₀ is concentrated on the real line or that it is rotationally invariant, Steinerberger [Proc. AMS, 2019] and O’Rourke and Steinerberger [Proc. AMS, 2021] 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 $\psi (x,t)$, 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 $\psi (x,0)$ 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 μ₀ is concentrated on the real line, gives a correct result which is known to have an interpretation in terms of free probability.

KEYWORDS:

• Random polynomials
• zeroes
• critical points
• repeated differentiation
• PDE’s
• Cauchy–Stieltjes transform
• logarithmic potentials
• Legendre–Fenchel transform
• free probability
• free binomial distribution

2010 MATHEMATICS SUBJECT CLASSIFICATION::

• Primary: 30C15
• Secondary: 35A25
• 60B10
• 60B20
• 60F10
• 82C70
• 35Q70
• 46L54
• 44A15
• 31A99

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Acknowledgements

We are grateful to the unknown referees for enlightening comments, in particular for suggesting an interpretation of the real case in terms of finite free probability. ZK has been supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics—Geometry—Structure.

Additional information

Funding

We are grateful to the unknown referees for enlightening comments, in particular for suggesting an interpretation of the real case in terms of finite free probability. ZK has been supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics—Geometry—Structure.