# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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## Critical points of random polynomials with independent identically distributed rootsHTML articles powered by AMS MathViewer

by Zakhar Kabluchko
Proc. Amer. Math. Soc. 143 (2015), 695-702 Request permission

## 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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• 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
• 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