Real zeros of random algebraic polynomials

Author:
K. Farahmand

Journal:
Proc. Amer. Math. Soc. **113** (1991), 1077-1084

MSC:
Primary 60G99

DOI:
https://doi.org/10.1090/S0002-9939-1991-1077787-5

MathSciNet review:
1077787

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: There are many known asymptotic estimates of the expected number of real zeros of algebraic polynomials with independent random coefficients of equal means. The present paper considers the case when the means of the coefficients are not all necessarily equal. The expected number of crossings of two algebraic polynomials with unequal degree flows from the results.

**[1]**Harald Cramér and M. R. Leadbetter,*Stationary and related stochastic processes. Sample function properties and their applications*, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0217860****[2]**Kambiz Farahmand,*On the average number of real roots of a random algebraic equation*, Ann. Probab.**14**(1986), no. 2, 702–709. MR**832032****[3]**K. Farahmand,*The average number of level crossings of a random algebraic polynomial*, Stochastic Anal. Appl.**6**(1988), no. 3, 247–272. MR**949678**, https://doi.org/10.1080/07362998808809147**[4]**I. A. Ibragimov and N. B. Maslova,*On the expected number of real zeros of random polynomials*. I.*Coefficients with zero means*, Theory Probab. Appl.**16**(1971), 228-248.**[5]**-,*On the expected number of real zeros of random polynomials*. II.*Coefficients with non-zero means*, Theory Probab. Appl.**16**(1971), 485-493.**[6]**M. Kac,*On the average number of real roots of a random algebraic equation*, Bull. Amer. Math. Soc.**49**(1943), 314–320. MR**0007812**, https://doi.org/10.1090/S0002-9904-1943-07912-8**[7]**M. Kac,*On the average number of real roots of a random algebraic equation. II*, Proc. London Math. Soc. (2)**50**(1949), 390–408. MR**0030713**, https://doi.org/10.1112/plms/s2-50.5.390**[8]**J. E. Littlewood and A. C. Offord,*On the number of real roots of a random algebraic equation*, Proc. Cambridge Philos. Soc.**35**(1939), 133-148.**[9]**S. O. Rice,*Mathematical analysis of random noise*, Bell System Tech. J.**24**(1945), 46–156. MR**0011918**, https://doi.org/10.1002/j.1538-7305.1945.tb00453.x

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60G99

Retrieve articles in all journals with MSC: 60G99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1077787-5

Keywords:
Number of real roots,
number of crossings,
Kac-Rice formula,
random algebraic polynomial

Article copyright:
© Copyright 1991
American Mathematical Society