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 µ 0 on the complex plane C. It is natural to conjecture that, for a fixed t ∈ [0, 1) and as n → ∞, the zeroes of the [tn]-th derivative of P n are distributed according to some measure µ t on 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. A similar method is applied to the case when µ 0 is concentrated on the real line. In this setting, Steinerberger [arXiv:2009.03869] obtained a representation of µ t as a free convolution power of µ 0 , for which we provide a rigorous proof. As an illustration, we compute the asymptotic density of zeroes of the [tn]-th derivative of the polynomial (x + 1) [m1n] (x − 1) [m2n] with fixed m 1 , m 2 > 0. The result is essentially the free binomial distribution.

Rodrigues' descendants of a polynomial and Boutroux curves

“…This text has been mainly written, but for various reasons not completely finished already in Spring 2018; its content has been presented during a workshop "Hausdorff geometry of polynomials and polynomial sequences" at the Mittag-Leffler institute in Stockholm. Since then several relevant papers discussing similar questions about the behavior of roots of polynomials under consecutive differentiations appeared, see e.g., [St1,St2,HoKa,KiTa]. In particular, paper [St2] contains a heuristic deduction of an intriguing partial differential equation satisfied (under several additional assumptions) by the density of roots under differentiation.…”

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

“…Secondly, we derive the interesting relation between derivatives of a polynomial and free additive convolution powers, as observed by Steinerberger [Ste20] and proved by Hoskins and Kobluchko [HK20]. The main observation here, which we prove in Section 3.3, is that when…”

The Flow of Polynomial Roots Under Differentiation

“…See also [38] on a direct link between the spectrum of sub-matrices and roots of the derivatives of characteristic polynomials. In addition to these observations, [23] establishes a direct link between evolution of roots of a polynomial under differentiation and free fractional convolution. The result of [23] applies for each fixed time in a limit of n → ∞, and does not directly involve the PDE (1.1).…”

“…In addition to these observations, [23] establishes a direct link between evolution of roots of a polynomial under differentiation and free fractional convolution. The result of [23] applies for each fixed time in a limit of n → ∞, and does not directly involve the PDE (1.1). We point out several more recent papers that are also related to this circle of ideas, either linking evolution of roots under differentiation and minor process [24], establishing connections between limiting distributions of Bessel and Dunkl processes modeling particle systems and free convolutions [70] or proving a version of "crystallization" for a class of random matrix ensembles [18].…”