## Random polynomials having few or no real zeros

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- by Amir Dembo, Bjorn Poonen, Qi-Man Shao and Ofer Zeitouni
- J. Amer. Math. Soc.
**15**(2002), 857-892 - DOI: https://doi.org/10.1090/S0894-0347-02-00386-7
- Published electronically: May 16, 2002
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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.## References

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## Bibliographic 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
- 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 - © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1915821