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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Random polynomials having few or no real zeros


Authors: Amir Dembo, Bjorn Poonen, Qi-Man Shao and Ofer Zeitouni
Journal: J. Amer. Math. Soc. 15 (2002), 857-892
MSC (2000): Primary 60G99; Secondary 12D10, 26C10
Published electronically: May 16, 2002
MathSciNet review: 1915821
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Abstract: Consider a polynomial of large degree $n$whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly $k$ real zeros with probability $n^{-b+o(1)}$ as $n \rightarrow \infty$through integers of the same parity as the fixed integer $k \ge 0$. In particular, the probability that a random polynomial of large even degree $n$ has no real zeros is $n^{-b+o(1)}$. The finite, positive constant $b$is characterized via the centered, stationary Gaussian process of correlation function ${\mathrm{sech}} (t/2)$. The value of $b$ depends neither on $k$ nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability $n^{-b+o(1)}$ one may specify also the approximate locations of the $k$ zeros on the real line. The constant $b$ is replaced by $b/2$in case the i.i.d. coefficients have a nonzero mean.


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Additional Information

Amir Dembo
Affiliation: Department of Mathematics & Statistics, Stanford University, Stanford, California 94305
Email: amir@math.stanford.edu

Bjorn Poonen
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: poonen@math.berkeley.edu

Qi-Man Shao
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Address at time of publication: Department of Mathematics, National University of Singapore, Singapore, 117543
Email: shao@math.uoregon.edu

Ofer Zeitouni
Affiliation: Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email: zeitouni@ee.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0894-0347-02-00386-7
PII: S 0894-0347(02)00386-7
Keywords: Random polynomials, Gaussian processes
Received by editor(s): May 30, 2000
Received by editor(s) in revised form: October 30, 2001
Published electronically: May 16, 2002
Additional Notes: The first author’s research was partially supported by NSF grant DMS-9704552
The second author was supported by NSF grant DMS-9801104, a Sloan Fellowship, and a Packard Fellowship.
The third author’s research was partially supported by NSF grant DMS-9802451
The fourth author’s research was partially supported by a grant from the Israel Science Foundation and by the fund for promotion of research at the Technion
Article copyright: © Copyright 2002 American Mathematical Society