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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Pairing between zeros and critical points of random polynomials with independent roots


Authors: Sean O’Rourke and Noah Williams
Journal: Trans. Amer. Math. Soc. 371 (2019), 2343-2381
MSC (2010): Primary 30C15; Secondary 60G57, 60B10
DOI: https://doi.org/10.1090/tran/7496
Published electronically: October 23, 2018
MathSciNet review: 3896083
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Abstract: Let $ p_n$ be a random, degree $ n$ polynomial whose roots are chosen independently according to the probability measure $ \mu $ on the complex plane. For a deterministic point $ \xi $ lying outside the support of $ \mu $, we show that almost surely the polynomial $ q_n(z):=p_n(z)(z - \xi )$ has a critical point at distance $ O(1/n)$ from $ \xi $. In other words, conditioning the random polynomials $ p_n$ to have a root at $ \xi $ almost surely forces a critical point near $ \xi $. More generally, we prove an analogous result for the critical points of $ q_n(z):=p_n(z)(z - \xi _1)\linebreak\cdots (z - \xi _k)$, where $ \xi _1, \ldots , \xi _k$ are deterministic. In addition, when $ k=o(n)$, we show that the empirical distribution constructed from the critical points of $ q_n$ converges to $ \mu $ in probability as the degree tends to infinity, extending a recent result of Kabluchko [Proc. Amer. Math. Soc. 143 (2015), no. 2, 695-702].


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Additional Information

Sean O’Rourke
Affiliation: Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, Colorado 80309
Email: sean.d.orourke@colorado.edu

Noah Williams
Affiliation: Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, Colorado 80309
Email: noah.williams@colorado.edu

DOI: https://doi.org/10.1090/tran/7496
Keywords: Random polynomials, critical points, zeros of the derivative, outlier critical points, empirical distribution
Received by editor(s): July 28, 2017
Published electronically: October 23, 2018
Additional Notes: The first author was supported in part by NSF grant ECCS-1610003.
Article copyright: © Copyright 2018 American Mathematical Society