Algebraic polynomials with nonidentical random coefficients
Authors:
K. Farahmand and A. Nezakati
Journal:
Proc. Amer. Math. Soc. 133 (2005), 275283
MSC (2000):
Primary 60G99; Secondary 60H99
Published electronically:
July 26, 2004
MathSciNet review:
2086220
Fulltext PDF Free Access
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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 nonidentical 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. BharuchaReid and M. Sambandham.
Random Polynomials. Academic Press, New York, 1986. MR 87m:60118
 2.
A. Edelman and E. Kostlan.
How many zeros of a random polynomial are real? Bull. Amer. Math. Soc., 32:137, 1995. MR 95m:60082
 3.
K. Farahmand.
Topics in Random Polynomials. Addison Wesley Longman, London, 1998. MR 2000d:60092
 4.
K. Farahmand.
On random algebraic polynomials. Proc. Amer. Math. Soc., 127:33393344, 1999. MR 2000b:60130
 5.
K. Farahmand and P. Hannigan.
The expected number of local maxima of a random algebraic polynomial. J. Theoretical Probability, 10:9911002, 1997. MR 99a:60053
 6.
K. Farahmand and P. Hannigan.
Large level crossings of random polynomials. Stochastic Analysis and Applications, 20:299309, 2002. MR 2003d:60107
 7.
K. Farahmand and M. Sambandham.
Real zeros of classes of random algebraic polynomials. J. Appl. Math. and Stochastic Analysis, 16:249255, 2003.
 8.
M. Kac.
On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc., 49:314320, 1943. MR 4:196d
 9.
B.F. Logan and L.A. Shepp.
Real zeros of random polynomials. Proc. London Math. Soc., 18:2935, 1968. MR 38:2829
 10.
B.F. Logan and L.A. Shepp.
Real zeros of random polynomials. II. Proc. London Math. Soc., 18:308314, 1968. MR 38:2830
 11.
S.O. Rice.
Mathematical theory of random noise. Bell System Tech. J., 25:46156, 1945. Reprinted in: Selected Papers on Noise and Stochastic Processes (ed. N. Wax), Dover, New York, 1954, 133294.MR 6:233i
 12.
M. Sambandham.
On a random algebraic polynomial. J. Indian Math. Soc., 41:8397, 1977. MR 58:31391
 13.
M. Sambandham.
On the average number of real zeros of a class of random algebraic curves. Pacific J. Math., 81:207215, 1979.MR 80h:60055
 14.
J.E. Wilkins.
An asymptotic expansion for the expected number of real zeros of a random polynomial. Proc. Amer. Math. Soc., 103:12491258, 1988. MR 90f:60105
 1.
 A.T. BharuchaReid and M. Sambandham.
Random Polynomials. Academic Press, New York, 1986. MR 87m:60118
 2.
 A. Edelman and E. Kostlan.
How many zeros of a random polynomial are real? Bull. Amer. Math. Soc., 32:137, 1995. MR 95m:60082
 3.
 K. Farahmand.
Topics in Random Polynomials. Addison Wesley Longman, London, 1998. MR 2000d:60092
 4.
 K. Farahmand.
On random algebraic polynomials. Proc. Amer. Math. Soc., 127:33393344, 1999. MR 2000b:60130
 5.
 K. Farahmand and P. Hannigan.
The expected number of local maxima of a random algebraic polynomial. J. Theoretical Probability, 10:9911002, 1997. MR 99a:60053
 6.
 K. Farahmand and P. Hannigan.
Large level crossings of random polynomials. Stochastic Analysis and Applications, 20:299309, 2002. MR 2003d:60107
 7.
 K. Farahmand and M. Sambandham.
Real zeros of classes of random algebraic polynomials. J. Appl. Math. and Stochastic Analysis, 16:249255, 2003.
 8.
 M. Kac.
On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc., 49:314320, 1943. MR 4:196d
 9.
 B.F. Logan and L.A. Shepp.
Real zeros of random polynomials. Proc. London Math. Soc., 18:2935, 1968. MR 38:2829
 10.
 B.F. Logan and L.A. Shepp.
Real zeros of random polynomials. II. Proc. London Math. Soc., 18:308314, 1968. MR 38:2830
 11.
 S.O. Rice.
Mathematical theory of random noise. Bell System Tech. J., 25:46156, 1945. Reprinted in: Selected Papers on Noise and Stochastic Processes (ed. N. Wax), Dover, New York, 1954, 133294.MR 6:233i
 12.
 M. Sambandham.
On a random algebraic polynomial. J. Indian Math. Soc., 41:8397, 1977. MR 58:31391
 13.
 M. Sambandham.
On the average number of real zeros of a class of random algebraic curves. Pacific J. Math., 81:207215, 1979.MR 80h:60055
 14.
 J.E. Wilkins.
An asymptotic expansion for the expected number of real zeros of a random polynomial. Proc. Amer. Math. Soc., 103:12491258, 1988. MR 90f:60105
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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 115991775, Mashhan, Iran
Email:
Nezakati@math.um.ac.ir
DOI:
http://dx.doi.org/10.1090/S000299390407501X
PII:
S 00029939(04)07501X
Keywords:
Number of real zeros,
real roots,
random algebraic polynomials,
KacRice formula,
nonidentical 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
