Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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

 
 

 

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

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Ad] Adler, R. J. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics Lecture Notes--Monograph Series, 12. Institute of Mathematical Statistics, Hayward, CA, 1990. MR 92g:60053
  • [BR] Bharucha-Reid, A. T.; Sambandham, M. Random polynomials. Probability and Mathematical Statistics. Academic Press, Inc., Orlando, Fla., 1986. MR 87m:60118
  • [BP] Bloch, A.; Pólya, G. On the roots of certain algebraic equations. Proc. London Math. Soc. 33 (1932), 102-114.
  • [CS] Csörgo, M.; Révész, P. Strong approximations in probability and statistics. Academic Press, New York, 1981. MR 84d:60050
  • [DZ] Dembo, A; Zeitouni, O. Large Deviations Techniques and Applications, second ed. Springer-Verlag, New York, 1998. MR 99d:60030
  • [EK] Edelman, A.; Kostlan, E. How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. (N.S.) 32 (1995), 1-37. Erratum: Bull. Amer. Math. Soc. (N.S.) 33 (1996), 325. MR 95m:60082; CMP 96:11
  • [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] Farahmand, K. Topics in random polynomials. Pitman research notes in mathematics series 393. Longman, Harlow, 1998. MR 2000d:60092
  • [GR] Gradshteyn, I. S.; Ryzhuk, I. M. Tables of integrals, series, and products, 4-th ed. Academic Press, San Diego, 1980. MR 81g:33001
  • [IM1] Ibragimov, I. A.; Maslova, N. B. The average number of zeros of random polynomials. Vestnik Leningrad. Univ. 23 (1968), 171-172. MR 38:6652
  • [IM2] Ibragimov, I. A.; Maslova, N. B. The mean number of real zeros of random polynomials. I. Coefficients with zero mean. Theor. Probability Appl. 16 (1971), 228-248. MR 44:3371
  • [IM3] Ibragimov, I. A.; Maslova, N. B. The mean number of real zeros of random polynomials. II. Coefficients with a nonzero mean. Theor. Probability Appl. 16 (1971), 485-493. MR 44:6019
  • [IM4] Ibragimov, I. A.; Maslova, N. B. The average number of real roots of random polynomials. Soviet Math. Dokl. 12 (1971), 1004-1008. MR 45:1221
  • [IZ] Ibragimov, I. A.; Zeitouni, O. On roots of random polynomials. Trans. American Math. Soc. 349 (1997), 2427-2441. MR 97h:60050
  • [Ja] Jamrom, B. R. The average number of real zeros of random polynomials. Soviet Math. Dokl. 13 (1972), 1381-1383. MR 47:2666
  • [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] Komlós, J.; Major, P.; Tusnády, G. An approximation of partial sums of independent R.V.'s and the sample D.F. II. Z. Wahr. verw. Gebiete 34 (1976), 35-58. MR 53:6697
  • [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] Logan, B. F.; Shepp, L. A. Real zeros of random polynomials. Proc. London Math. Soc. 18 (1968), 29-35. MR 38:2829
  • [LS2] Logan, B. F.; Shepp, L. A. Real zeros of random polynomials. II. Proc. London Math. Soc. 18 (1968), 308-314. MR 38:2830
  • [Ma1] Maslova, N. B. The variance of the number of real roots of random polynomials. Teor. Verojatnost. i Primenen. 19 (1974), 36-51. MR 48:12646
  • [Ma2] Maslova, N. B. The distribution of the number of real roots of random polynomials. Theor. Probability Appl. 19 (1974), 461-473 (1975). MR 51:4378
  • [PS] Poonen, B.; Stoll, M. The Cassels-Tate pairing on polarized abelian varieties. Annals of Math. 150 (1999), 1109-1149. MR 2000m:11048
  • [Sh] Shao, Q. M. A Gaussian correlation inequality and its applications to the existence of small ball constant. Preprint, (1999).
  • [St] Stevens, D. C. The average number of real zeros of a random polynomial. Comm. Pure Appl. Math. 22 (1969), 457-477. MR 40:4234
  • [Sto] Stout, W. F. Almost sure convergence. Academic Press, New York, 1974. MR 56:13334
  • [Str] Strassen, V. An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Gebiete 3 (1964), 211-226. MR 30:5379
  • [To] Todhunter, I. A history of the mathematical theory of probability. Stechert, New York, 1931.
  • [Wa] Wang, Y. J. Bounds on the average number of real roots of a random algebraic equation. Chinese Ann. Math. Ser. A 4 (1983), 601-605. An English summary appears in Chinese Ann. Math. Ser. B 4 (1983), 527. MR 85c:60081
  • [Wi] Wilkins, J. Ernest, Jr. An asymptotic expansion for the expected number of real zeros of a random polynomial. Proc. Amer. Math. Soc. 103 (1988), 1249-1258. MR 90f:60105

Similar Articles

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

American Mathematical Society