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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lower bounds for the absolute value
of random polynomials on a neighborhood
of the unit circle


Authors: S. V. Konyagin and W. Schlag
Journal: Trans. Amer. Math. Soc. 351 (1999), 4963-4980
MSC (1991): Primary 42A05, 42A61; Secondary 30C15, 60F05
Published electronically: August 27, 1999
MathSciNet review: 1487621
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $T(x)=\sum _{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon>0$ and most choices of sign, $\min _{x\in[0,2\pi)}|T(x)|<n^{-1/2+\epsilon}$, provided $n$ is large. In this paper we show that the power $n^{-1/2}$ is optimal. More precisely, for sufficiently small $\epsilon>0$ and large $n$ most choices of sign satisfy $\min _{x\in[0,2\pi)}|T(x)|> \epsilon n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials.


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Additional Information

S. V. Konyagin
Affiliation: Institute for Advanced Study, School of Mathematics, Olden Lane, Princeton, New Jersey 08540
Address at time of publication: Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia
Email: kon@nw.math.msu.su

W. Schlag
Affiliation: Institute for Advanced Study, School of Mathematics, Olden Lane, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
Email: schlag@math.princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02241-2
PII: S 0002-9947(99)02241-2
Received by editor(s): February 5, 1997
Received by editor(s) in revised form: September 24, 1997
Published electronically: August 27, 1999
Additional Notes: The authors were supported by the National Science Foundation, grant DMS 9304580. This research was carried out while the authors were members of the Institute for Advanced Study, Princeton. It is a pleasure to thank the Institute for its hospitality and generosity. The authors would like to thank A.\ G.\ Karapetian for comments on a preliminary version of this paper.
Article copyright: © Copyright 1999 American Mathematical Society