A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

O’Rourke, Sean; Steinerberger, Stefan

doi:10.48550/arxiv.1910.12161

Cited by 6 publications

(12 citation statements)

References 36 publications

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“…Additionally, the analogous 'infinite degree' setting, in which polynomials are replaced by analytic functions, has also been considered -see Polya [34], Farmer & Rhoades [13] and Pemantle & Subramanian [33]. Another natural extension is to consider the dynamics of the roots of successive derivatives for complex-valued polynomials p n : C → C. If the roots are given by a smooth probability distribution u(0, z) : C → R ≥0 and the limiting measure u(0, z) is radial, O'Rourke & Steinerberger [31] suggest that the following nonlocal transport equation…”

Section: Related Resultsmentioning

confidence: 99%

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A Semicircle Law for Derivatives of Random Polynomials

Hoskins¹,

Steinerberger²

2020

Preprint

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Let x 1 , . . . , xn be n independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial pn having roots at x 1 , . . . , xn. We prove that for ∈ N fixed as n → ∞, the (n − )−th derivative of p n behaves like a Hermite polynomial: for x in a compact interval,where He is the −th probabilists' Hermite polynomial and γn is a random variable converging to the standard N (0, 1) Gaussian as n → ∞. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.

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Section: Related Resultsmentioning

confidence: 99%

“…The derivation was, however, based on heuristic arguments, a rigorous proof is still outstanding. It was recently shown [44] that there are an infinite number of conservation laws that are satisfied by both explicit closed form solutions, indicating that the PDE might have an abundance of interesting structure (see also [16,31]). (3) The case t = 1.…”

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confidence: 99%

A Semicircle Law for Derivatives of Random Polynomials

Hoskins¹,

Steinerberger²

2020

Preprint

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“…The same question is meaningful for general polynomials p n : C → C having a distribution of roots being given by a smooth probability distribution u(0, z) : C → R ≥0 in the complex plane. In case the limiting measure u(0, z) is radial, the problem was studied by O'Rourke and the author [23] who (non-rigorously) derived a nonlocal transport equation…”

Section: A Partial Differential Equationmentioning

confidence: 99%

A nonlocal transport equation describing roots of polynomials under differentiation

Steinerberger

2019

Proc. Amer. Math. Soc.

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Let pn be a polynomial of degree n having n distinct, real roots distributed according to a nice probability distribution u(0, x)dx on R. One natural problem is to understand the density u(t, x) of the roots of the (t·n)−th derivative of pn where 0 < t < 1 as n → ∞. The author suggested that these densities might satisfy the partial differential equation of transport typeIn particular, we present two closed-form solutions for which all infinitely many conservation laws are valid. This raises a number of interesting questions.2010 Mathematics Subject Classification. 35Q80, 70H33 (primary), 26C10, 42B20 (secondary).

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“…Polynomials. Roots of polynomials are a classical subject and there are many results we do not describe here, see [6,7,8,12,13,16,17,18,19,20,24,26,27,28,29,30,31,32,33,34,37,38,39,40]. Our problem will be as follows: let µ be a compactly supported probability measure on the real line and suppose x 1 , .…”

mentioning

confidence: 99%

“…Numerical simulations in [15] also suggested that the solution tends to become smoother. O'Rourke and the author [29] derived an analogous transport equation for polynomials with roots following a radial distribution in the complex plane.…”

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confidence: 99%

Free Convolution Powers via Roots of Polynomials

Steinerberger¹

2020

Preprint

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Let µ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power µ k for any real k ≥ 1. The purpose of this short note is to give an elementary description of µ k in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.Given G µ , we define the R−transform R µ (s) for sufficiently small complex s by demanding that 1for all sufficiently large z. The free convolution µ ν is then the unique compactly supported measure for whichfor all sufficiently small s. A fundamental result due to Voiculescu is the free central limit theorem: if µ is a compactly supported probability measure with mean 0 and variance 1, then suitably rescaled copies of µ k converge to the semicircular distribution. This notion can be extended to the reals.Theorem (Fractional Free Convolution Powers exist, [5,24]). Let µ be a compactly supported probability measure on R and assume k ≥ 1 is real. Then there exists a unique compactly supported probability measure µ k such thatfor all s sufficiently small.

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A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

Cited by 6 publications

References 36 publications

A Semicircle Law for Derivatives of Random Polynomials

A Semicircle Law for Derivatives of Random Polynomials

A nonlocal transport equation describing roots of polynomials under differentiation

Free Convolution Powers via Roots of Polynomials

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