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

DOI:
https://doi.org/10.1090/S0894-0347-02-00386-7

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

MR Author ID:
250625

ORCID:
0000-0002-8593-2792

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

MR Author ID:
186850

ORCID:
0000-0002-2520-1525

Email:
zeitouni@ee.technion.ac.il

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