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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[Ad]**Robert J. Adler,*An introduction to continuity, extrema, and related topics for general Gaussian processes*, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. MR**1088478****[BR]**A. T. Bharucha-Reid and M. Sambandham,*Random polynomials*, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR**856019****[BP]**Bloch, A.; Pólya, G.*On the roots of certain algebraic equations*. Proc. London Math. Soc.**33**(1932), 102-114.**[CS]**M. Csörgő and P. Révész,*Strong approximations in probability and statistics*, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**666546****[DZ]**Amir Dembo and Ofer Zeitouni,*Large deviations techniques and applications*, 2nd ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. MR**1619036****[EK]**Alan Edelman and Eric Kostlan,*How many zeros of a random polynomial are real?*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 1, 1–37. MR**1290398**, https://doi.org/10.1090/S0273-0979-1995-00571-9**[EO]**Erdos, P.; Offord, A. C.*On the number of real roots of a random algebraic equation*. Proc. London Math. Soc.**6**(1956), 139-160. MR**17:500f****[Fa]**Kambiz Farahmand,*Topics in random polynomials*, Pitman Research Notes in Mathematics Series, vol. 393, Longman, Harlow, 1998. MR**1679392****[GR]**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR**582453****[IM1]**I. A. Ibragimov and N. B. Maslova,*The average number of zeros of random polynomials*, Vestnik Leningrad. Univ.**23**(1968), no. 19, 171–172 (Russian, with English summary). MR**0238376****[IM2]**I. A. Ibragimov and N. B. Maslova,*The mean number of real zeros of random polynomials. I. Coefficients with zero mean*, Teor. Verojatnost. i Primenen.**16**(1971), 229–248 (Russian, with English summary). MR**0286157****[IM3]**I. A. Ibragimov and N. B. Maslova,*The mean number of real zeros of random polynomials. II. Coefficients with a nonzero mean*, Teor. Verojatnost. i Primenen.**16**(1971), 495–503 (Russian, with English summary). MR**0288824****[IM4]**I. A. Ibragimov and N. B. Maslova,*The average number of real roots of random polynomials*, Dokl. Akad. Nauk SSSR**199**(1971), 13–16 (Russian). MR**0292134****[IZ]**Ildar Ibragimov and Ofer Zeitouni,*On roots of random polynomials*, Trans. Amer. Math. Soc.**349**(1997), no. 6, 2427–2441. MR**1390040**, https://doi.org/10.1090/S0002-9947-97-01766-2**[Ja]**B. R. Jamrom,*The average number of real zeros of random polynomials*, Dokl. Akad. Nauk SSSR**206**(1972), 1059–1060 (Russian). MR**0314114****[Ka1]**Kac, M.*On the average number of real roots of a random algebraic equation*. Bull. Amer. Math. Soc.**49**(1943), 314-320. Erratum: Bull. Amer. Math. Soc.**49**(1943), 938. MR**4:196d**; MR**5:179g****[Ka2]**Kac, M.*On the average number of real roots of a random algebraic equation. II.*Proc. London Math. Soc.**50**(1949), 390-408. MR**11:40e****[KMT]**J. Komlós, P. Major, and G. Tusnády,*An approximation of partial sums of independent RV’s, and the sample DF. II*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**34**(1976), no. 1, 33–58. MR**402883**, https://doi.org/10.1007/BF00532688**[LiS]**Li, W.V; Shao, Q. M.*A normal comparison inequality and its applications.*Probab. Theo. Rel. Fields, to appear.**[LO1]**Littlewood, J. E.; Offord, A. C.*On the number of real roots of a random algebraic equation. I.*J. London Math. Soc.**13**(1938), 288-295.**[LO2]**Littlewood, J. E.; Offord, A. C.*On the number of real roots of a random algebraic equation. II.*Proc. Cambridge Philos. Soc.**35**(1939), 133-148.**[LO3]**Littlewood, J. E.; Offord, A. C.*On the number of real roots of a random algebraic equation. III.*Rec. Math. [Mat. Sbornik] N.S.**54**(1943), 277-286. MR**5:179h****[LS1]**B. F. Logan and L. A. Shepp,*Real zeros of random polynomials*, Proc. London Math. Soc. (3)**18**(1968), 29–35. MR**0234512**, https://doi.org/10.1112/plms/s3-18.1.29**[LS2]**B. F. Logan and L. A. Shepp,*Real zeros of random polynomials. II*, Proc. London Math. Soc. (3)**18**(1968), 308–314. MR**0234513**, https://doi.org/10.1112/plms/s3-18.2.308**[Ma1]**N. B. Maslova,*The variance of the number of real roots of random polynomials*, Teor. Verojatnost. i Primenen.**19**(1974), 36–51 (Russian, with English summary). MR**0334327****[Ma2]**N. B. Maslova,*The distribution of the number of real roots of random polynomials*, Teor. Verojatnost. i Primenen.**19**(1974), 488–500 (Russian, with English summary). MR**0368136****[PS]**Bjorn Poonen and Michael Stoll,*The Cassels-Tate pairing on polarized abelian varieties*, Ann. of Math. (2)**150**(1999), no. 3, 1109–1149. MR**1740984**, https://doi.org/10.2307/121064**[Sh]**Shao, Q. M.*A Gaussian correlation inequality and its applications to the existence of small ball constant.*Preprint, (1999).**[St]**D. C. Stevens,*The average number of real zeros of a random polynomial*, Comm. Pure Appl. Math.**22**(1969), 457–477. MR**251003**, https://doi.org/10.1002/cpa.3160220403**[Sto]**William F. Stout,*Almost sure convergence*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 24. MR**0455094****[Str]**V. Strassen,*An invariance principle for the law of the iterated logarithm*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**3**(1964), 211–226 (1964). MR**175194**, https://doi.org/10.1007/BF00534910**[To]**Todhunter, I.*A history of the mathematical theory of probability.*Stechert, New York, 1931.**[Wa]**You Jing Wang,*Bounds on the average number of real roots of a random algebraic equation*, Chinese Ann. Math. Ser. A**4**(1983), no. 5, 601–605 (Chinese). An English summary appears in Chinese Ann. Math. Ser. B 4 (1983), no. 4, 527. MR**742181****[Wi]**J. Ernest Wilkins Jr.,*An asymptotic expansion for the expected number of real zeros of a random polynomial*, Proc. Amer. Math. Soc.**103**(1988), no. 4, 1249–1258. MR**955018**, https://doi.org/10.1090/S0002-9939-1988-0955018-1

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
60G99,
12D10,
26C10

Retrieve articles in all journals with MSC (2000): 60G99, 12D10, 26C10

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:
https://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