Critical points of random polynomials with independent identically distributed roots

Kabluchko, Zakhar

doi:10.1090/s0002-9939-2014-12258-1

Cited by 47 publications

(58 citation statements)

References 9 publications

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“…The assumption on µ being compactly supported is clearly not sharp, our proof immediately transfers to probability measures having a certain rate of decay at infinity. The result is similar in spirit to a recent result of Kabluchko [9] (proving a conjecture of Pemantle & Rivin [11]) who showed that the distribution of critical points {z ∈ C : p n (z) = 0} reproduces µ for general probability measures µ. If µ is not radial, the situation is not quite as simple.…”

Section: Introduction and Main Resultssupporting

confidence: 86%

On Zeroes of Random Polynomials and an Application to Unwinding

Steinerberger

2019

International Mathematics Research Notices

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Let µ be a probability measure in C with a continuous and compactly supported density function, let z 1 , . . . , zn be independent random variables, z i ∼ µ, and consider the random polynomialWe determine the asymptotic distribution of {z ∈ C : pn(z) = pn(0)}. In particular, if µ is radial around the origin, then those solutions are also distributed according to µ as n → ∞. Generally, the distribution of the solutions will reproduce parts of µ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.

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Section: Introduction and Main Resultssupporting

confidence: 86%

On Zeroes of Random Polynomials and an Application to Unwinding

Steinerberger

2019

International Mathematics Research Notices

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“…To conclude, let us mention the works of Kabluchko [10], Pemantle-Rivlin [13], and Subramanian [16], which study the empirical measure of critical points for ensembles of random polynomials similar to the ones we consider here. We also point the reader to the work of Nazarov-Sodin-Volberg [12] and the recent article of Feng [6], which both concern the critical points of random holomorphic functions with respect to a smooth connection.…”

Section: Sendov's Conjecturementioning

confidence: 99%

Pairing of zeros and critical points for random polynomials

Hanin¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Let p N be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure µ on the Riemann sphere S 2 . This article proves that if we condition p N to have a zero at some fixed point ξ ∈ S 2 , then, with high probability, there will be a critical point w ξ a distance N −1 away from ξ. This N −1 distance is much smaller than the N −1/2 typical spacing between nearest neighbors for N i.i.d. points on S 2 . Moreover, with the same high probability, the argument of w ξ relative to ξ is a deterministic function of µ plus fluctuations on the order of N −1 .

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“…The proofs here are adapted from the proof of Kabluchko's theorem as presented in [5]. The proofs involve in analyzing the function L n (z) := P n (z) Pn(z) = n k=1 1 z−ξ k .…”

Section: Outline Of Proofsmentioning

confidence: 99%

Limiting empirical distribution of zeros and critical points of random polynomials agree in general

Reddy¹

2017

Electron. J. Probab.

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In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences {an} n≥1 and {bn} n≥1 of complex numbers whose limiting empirical measures are same. By choosing ξn = an or bn with equal probability, define the sequence of polynomials by Pn(z) = (z − ξ1) . . . (z − ξn). We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function) is also proved. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.

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Critical points of random polynomials with independent identically distributed roots

Cited by 47 publications

References 9 publications

On Zeroes of Random Polynomials and an Application to Unwinding

On Zeroes of Random Polynomials and an Application to Unwinding

Pairing of zeros and critical points for random polynomials

Limiting empirical distribution of zeros and critical points of random polynomials agree in general

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