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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Random polynomials having few or no real zeros
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by Amir Dembo, Bjorn Poonen, Qi-Man Shao and Ofer Zeitouni PDF
J. Amer. Math. Soc. 15 (2002), 857-892 Request permission

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
  • 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