Random polynomials having few or no real zeros
Authors:
Amir Dembo, Bjorn Poonen, QiMan Shao and Ofer Zeitouni
Journal:
J. Amer. Math. Soc. 15 (2002), 857892
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 947203840
Email:
poonen@math.berkeley.edu
QiMan 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, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
zeitouni@ee.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0894034702003867
PII:
S 08940347(02)003867
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 DMS9704552
The second author was supported by NSF grant DMS9801104, a Sloan Fellowship, and a Packard Fellowship.
The third author’s research was partially supported by NSF grant DMS9802451
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
