Zeros of repeated derivatives of random polynomials

Feng, Renjie; Yao, Dong

doi:10.2140/apde.2019.12.1489

Cited by 11 publications

(10 citation statements)

References 14 publications

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“…There seem to be no results on this general PDE, but in the case when the initial distribution µ is rotationally invariant, the PDE simplifies considerably. The PDE corresponding to this isotropic special case has been first non-rigorously derived by O'Rourke and Steinerberger [42] and can be solved explicitly, as has been shown in [29]; see also [20] for a particular case when µ is the uniform distribution on the unit circle.…”

Section: Resultsmentioning

confidence: 99%

Repeated differentiation and free unitary Poisson process

Kabluchko¹

2021

Preprint

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We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degree d trigonometric polynomial are distributed according to some probability measure ν in the large d limit, then the zeroes of its [2td]-th derivative, where t > 0 is fixed, are distributed according to the free multiplicative convolution of ν and the free unitary Poisson distribution with parameter t. In the simplest special case, our result states that the zeroes of the [2td]-th derivative of the trigonometric polynomial (sin θ2 ) 2d (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter t, in the large d limit. The latter distribution is defined in terms of the function ζ = ζt(θ) which solves the implicit equation ζ − t tan ζ = θ and satisfies ζt(θ) = θ + t tan(θ + t tan(θ + t tan(θ + . . .))), Im θ > 0, t > 0.

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Section: Resultsmentioning

confidence: 99%

Repeated differentiation and free unitary Poisson process

Kabluchko¹

2021

Preprint

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“…We provide the details of the argument since it will be needed in the following. We take some t ∈ [0, 1) and look at the [tn]-th derivative of G n as defined in (6):…”

Section: 2mentioning

confidence: 99%

“…Here, t ∈ [0, 1) stays fixed, and [x] denotes the integer part of x. This question has been raised and studied in the papers of Steinerberger [28], O'Rourke and Steinerberger [21] and Feng and Yao [6]; see also [11,30,31]. Assigning a weight 1/n to each zero of the [tn]-th derivative, one can construct a subprobability measure on C denoted by µ (n) t :=…”

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confidence: 99%

Dynamics of zeroes under repeated differentiation

Hoskins¹,

Kabluchko²

2020

Preprint

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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.

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“…In particular, do they obey a regular spacing at a local scale? We refer to [24,25,26,49] and references therein.…”

Section: Open Problemsmentioning

confidence: 99%

“…A result commonly attributed to Riesz [56] implies that the minimum gap between consecutive roots of p n is bigger than that of p n : zeroes even out and become more regular. We refer to results of Farmer & Rhoades [24], Farmer & Yerrington [25], Feng & Yao [26] and Pemantle & Subramnian [49]. Our result is inspired by a one-dimensional investigation due to the second author [55]:…”

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confidence: 92%

A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

O’Rourke¹,

Steinerberger²

2019

Preprint

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Let pn : C → C be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution u(r)rdr in the complex plane. A natural question is how the distribution of roots evolves under repeated (say n/2−times) differentiation of the polynomial. We derive a mean-field expansion for the evolution of ψ(s) = u(s)sWe 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.

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Zeros of repeated derivatives of random polynomials

Cited by 11 publications

References 14 publications

Repeated differentiation and free unitary Poisson process

Repeated differentiation and free unitary Poisson process

Dynamics of zeroes under repeated differentiation

A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

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