Real zeros of random algebraic polynomials
Author:
K. Farahmand
Journal:
Proc. Amer. Math. Soc. 113 (1991), 10771084
MSC:
Primary 60G99
MathSciNet review:
1077787
Fulltext PDF Free Access
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 YorkLondonSydney, 1967. MR 0217860
(36 #949)
 [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
(87k:60140)
 [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
(89m:60122), http://dx.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), 228248.
 [5]
, On the expected number of real zeros of random polynomials. II. Coefficients with nonzero means, Theory Probab. Appl. 16 (1971), 485493.
 [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
(4,196d), http://dx.doi.org/10.1090/S000299041943079128
 [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
(11,40e)
 [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), 133148.
 [9]
S.
O. Rice, Mathematical analysis of random noise, Bell System
Tech. J. 24 (1945), 46–156. MR 0011918
(6,233i)
 [1]
 H. Cramer and M. R. Leadbetter, Stationary and related stochastic process, Wiley, New York, 1967. MR 0217860 (36:949)
 [2]
 K. Farahmand, On the average number of real roots of a random equation, Ann. Probab. 14 (1986), 702709. MR 832032 (87k:60140)
 [3]
 , The average number of level crossings of a random algebraic polynomial, Stochastic Anal. Appl. 6 (1988), 247272. MR 949678 (89m:60122)
 [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), 228248.
 [5]
 , On the expected number of real zeros of random polynomials. II. Coefficients with nonzero means, Theory Probab. Appl. 16 (1971), 485493.
 [6]
 M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314320. MR 0007812 (4:196d)
 [7]
 , On the average number of real roots of a random algebraic equation, Proc. London Math. Soc. 50 (1949), 390408. MR 0030713 (11:40e)
 [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), 133148.
 [9]
 S. O. Rice, Mathematical theory of random noise, Bull. System Tech. J. 25 (1945), 46156. MR 0011918 (6:233i)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
60G99
Retrieve articles in all journals
with MSC:
60G99
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110777875
PII:
S 00029939(1991)10777875
Keywords:
Number of real roots,
number of crossings,
KacRice formula,
random algebraic polynomial
Article copyright:
© Copyright 1991
American Mathematical Society
