Algebraic polynomials with non-identical random coefficients

Authors:
K. Farahmand and A. Nezakati

Journal:
Proc. Amer. Math. Soc. **133** (2005), 275-283

MSC (2000):
Primary 60G99; Secondary 60H99

DOI:
https://doi.org/10.1090/S0002-9939-04-07501-X

Published electronically:
July 26, 2004

MathSciNet review:
2086220

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial The coefficients are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all sufficiently large, the above expected value is shown to be . Also, it is known that if the have non-identical variance , then the expected number of real zeros increases to . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain . In fact, so far the case of is the only case that can significantly increase the expected number of real zeros.

**1.**A. T. Bharucha-Reid and M. Sambandham,*Random polynomials*, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR**856019****2.**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**3.**Kambiz Farahmand,*Topics in random polynomials*, Pitman Research Notes in Mathematics Series, vol. 393, Longman, Harlow, 1998. MR**1679392****4.**K. Farahmand,*On random algebraic polynomials*, Proc. Amer. Math. Soc.**127**(1999), no. 11, 3339–3344. MR**1610956**, https://doi.org/10.1090/S0002-9939-99-04912-6**5.**K. Farahmand and P. Hannigan,*The expected number of local maxima of a random algebraic polynomial*, J. Theoret. Probab.**10**(1997), no. 4, 991–1002. MR**1481657**, https://doi.org/10.1023/A:1022618801587**6.**K. Farahmand and P. Hannigan,*Large level crossings of random polynomials*, Stochastic Anal. Appl.**20**(2002), no. 2, 299–309. MR**1900362**, https://doi.org/10.1081/SAP-120003436**7.**K. Farahmand and M. Sambandham.

Real zeros of classes of random algebraic polynomials.*J. Appl. Math. and Stochastic Analysis*, 16:249-255, 2003.**8.**M. Kac.

On the average number of real roots of a random algebraic equation.*Bull. Amer. Math. Soc.*, 49:314-320, 1943. MR**4:196d****9.**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**10.**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**11.**S.O. Rice.

Mathematical theory of random noise.*Bell System Tech. J.*, 25:46-156, 1945.

Reprinted in: Selected Papers on Noise and Stochastic Processes (ed. N. Wax), Dover, New York, 1954, 133-294.MR**6:233i****12.**M. Sambandham,*On a random algebraic equation*, J. Indian Math. Soc. (N.S.)**41**(1977), no. 1-2, 83–97. MR**0651565****13.**M. Sambandham,*On the average number of real zeros of a class of random algebraic curves*, Pacific J. Math.**81**(1979), no. 1, 207–215. MR**543744****14.**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 *Proceedings of the American Mathematical Society*
with MSC (2000):
60G99,
60H99

Retrieve articles in all journals with MSC (2000): 60G99, 60H99

Additional Information

**K. Farahmand**

Affiliation:
Department of Mathematics, University of Ulster at Jordanstown, Co. Antrim, BT37 0QB, United Kingdom

Email:
K.Farahmand@ulst.ac.uk

**A. Nezakati**

Affiliation:
Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhan, Iran

Email:
Nezakati@math.um.ac.ir

DOI:
https://doi.org/10.1090/S0002-9939-04-07501-X

Keywords:
Number of real zeros,
real roots,
random algebraic polynomials,
Kac-Rice formula,
non-identical random variables

Received by editor(s):
February 18, 2003

Received by editor(s) in revised form:
September 8, 2003

Published electronically:
July 26, 2004

Additional Notes:
This work was completed while the second author was visiting the Department of Mathematics at the University of Ulster. The hospitality of the Department of Mathematics at the University of Ulster is appreciated. The financial support for this visit was supported by the Ministry of Science, Research and Technology of I. R. Iran

Communicated by:
Richard C. Bradley

Article copyright:
© Copyright 2004
American Mathematical Society