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

DOI:
https://doi.org/10.1090/S0002-9947-99-02241-2

Published electronically:
August 27, 1999

MathSciNet review:
1487621

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Abstract | References | Similar Articles | Additional Information

Abstract: Let where stands for a random choice of sign with equal probability. The first author recently showed that for any and most choices of sign, , provided is large. In this paper we show that the power is optimal. More precisely, for sufficiently small and large most choices of sign satisfy . Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials.

**1.**R. N. Bhattacharya, R. Rao.*Normal approximation and asymptotic expansions.*John Wiley & Sons, New York, 1976.MR**55:9219****2.**G. H. Hardy, E. M. Wright.*An introduction to the theory of numbers.*Fifth edition, Oxford University Press, Oxford, 1979.MR**81i:10002****3.**J-P. Kahane.*Some random series of functions.*Second Edition, Cambridge University Press, Cambridge, 1985.MR**87m:60119****4.**A. G. Karapetian.*On the minimum of the absolute value of trigonometric polynomials with random coefficients.*(Russian) Mat. Zametki 61 (1997), no. 3, 451-455. CMP**98:12****5.**B. S. Kashin.*The properties of random polynomials with coefficients .*Vestn. Mosk. Univ. Ser. Mat.-Mekh., No. 5 (1987), 40-46.MR**89a:60135****6.**S. V. Konyagin.*On the minimum modulus of random trigonometric polynomials with coefficients*. (Russian, Translation in Math. Notes) Mat. Zametki 56 (1994), no. 3, 80-101. MR**95k:42015****7.**J. E. Littlewood.*On polynomials*. J. London Math. Soc. 41 (1966), 367-376.MR**33:4237****8.**L. A. Shepp, R. J. Vanderbei.*The complex zeros of random polynomials.*Trans. Amer. Math. Soc. 347 (1995), no. 11, 4365-4384.MR**96a:30006****9.**A. Zygmund.*Trigonometric Series.*Second Edition, Cambridge University Press, Cambridge, 1959.MR**21:6498**

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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:
https://doi.org/10.1090/S0002-9947-99-02241-2

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