Expected discrepancy for zeros of random algebraic polynomials
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- by Igor E. Pritsker and Alan A. Sola PDF
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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.References
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Additional Information
- Igor E. Pritsker
- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
- MR Author ID: 319712
- Email: igor@math.okstate.edu
- Alan A. Sola
- Affiliation: Statistical Laboratory, University of Cambridge, Cambridge CB3 0WB, United Kingdom
- MR Author ID: 804661
- Email: a.sola@statslab.cam.ac.uk
- 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 - Communicated by: Walter Van Assche
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3266993
Dedicated: Dedicated to Vladimir Andrievskii on his 60th birthday