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

DOI:
https://doi.org/10.1090/S0002-9939-2014-12258-1

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 \[ \mu _n:=\frac 1{n-1}\sum _{P_n’(z)=0} \delta _z\] counting the complex zeros of the derivative $P_n’$ converges in probability to $\mu$, as $n\to \infty$.

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Additional Information

**Zakhar Kabluchko**

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

MR Author ID:
696619

ORCID:
0000-0001-8483-3373

Email:
zakhar.kabluchko@uni-ulm.de

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.