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Expected discrepancy for zeros of random algebraic polynomials


Authors: Igor E. Pritsker and Alan A. Sola
Journal: Proc. Amer. Math. Soc. 142 (2014), 4251-4263
MSC (2010): Primary 30C15; Secondary 30B20, 60B10
DOI: https://doi.org/10.1090/S0002-9939-2014-12147-2
Published electronically: August 7, 2014
MathSciNet review: 3266993
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Abstract: We study asymptotic clustering of zeros of random polynomials and show that the expected discrepancy of roots of a polynomial of degree $ n$, with not necessarily independent coefficients, decays like $ \sqrt {\log n/n}$. Our proofs rely on discrepancy results for deterministic polynomials and on order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of the plane, such as circles centered on the unit circumference, and polygons inscribed in the unit circumference.


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Additional Information

Igor E. Pritsker
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: igor@math.okstate.edu

Alan A. Sola
Affiliation: Statistical Laboratory, University of Cambridge, Cambridge CB3 0WB, United Kingdom
Email: a.sola@statslab.cam.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2014-12147-2
Keywords: Random polynomials, expected discrepancy of roots, expected number of roots in special sets.
Received by editor(s): August 19, 2012
Received by editor(s) in revised form: January 13, 2013
Published electronically: August 7, 2014
Additional Notes: The first author acknowledges support from the NSA under grant H98230-12-1-0227
The second author acknowledges support from the EPSRC under grant EP/103372X/1
Dedicated: Dedicated to Vladimir Andrievskii on his 60th birthday
Communicated by: Walter Van Assche
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.