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Critical points of random polynomials with independent identically distributed roots

Author: Zakhar Kabluchko
Journal: Proc. Amer. Math. Soc. 143 (2015), 695-702
MSC (2010): Primary 30C15; Secondary 60G57, 60B10
Published electronically: September 19, 2014
MathSciNet review: 3283656
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X_1,X_2,\ldots $ be independent identically distributed random
variables with values in $ \mathbb{C}$. Denote by $ \mu $ the probability distribution of $ X_1$. Consider a random polynomial $ P_n(z)=(z-X_1)\ldots (z-X_n)$. We prove a conjecture of Pemantle and Rivin [in: I. Kotsireas and E. V. Zima, eds., Advances in Combinatorics, Waterloo Workshop in Computer Algebra, $ 2011$] that the empirical measure

$\displaystyle \mu _n:=\frac 1{n-1}\sum _{P_n'(z)=0} \delta _z$

counting the complex zeros of the derivative $ P_n'$ converges in probability to $ \mu $, as $ n\to \infty $.

References [Enhancements On Off] (What's this?)

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Additional Information

Zakhar Kabluchko
Affiliation: Institute of Stochastics, Ulm University, Helmholtzstr. 18, 89069 Ulm, Germany

Keywords: Random polynomials, empirical distribution, critical points, zeros of the derivative, logarithmic potential
Received by editor(s): July 4, 2012
Received by editor(s) in revised form: February 12, 2013, and May 3, 2013
Published electronically: September 19, 2014
Communicated by: David Levin
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.