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 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 real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by 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

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