On random algebraic polynomials
Author:
K. Farahmand
Journal:
Proc. Amer. Math. Soc. 127 (1999), 33393344
MSC (1991):
Primary 60H99; Secondary 42Bxx
Published electronically:
May 6, 1999
MathSciNet review:
1610956
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper provides asymptotic estimates for the expected number of real zeros and level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .
 1.
E.
Bogomolny, O.
Bohigas, and P.
Lebœuf, Distribution of roots of random polynomials,
Phys. Rev. Lett. 68 (1992), no. 18, 2726–2729.
MR
1160289 (92m:81054), http://dx.doi.org/10.1103/PhysRevLett.68.2726
 2.
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)
 3.
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 (95m:60082), http://dx.doi.org/10.1090/S027309791995005719
 4.
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)
 5.
Kambiz
Farahmand, On the average number of level crossings of a random
trigonometric polynomial, Ann. Probab. 18 (1990),
no. 3, 1403–1409. MR 1062074
(91i:60140)
 6.
Kambiz
Farahmand, Level crossings of a random
trigonometric polynomial, Proc. Amer. Math.
Soc. 111 (1991), no. 2, 551–557. MR 1015677
(91f:60092), http://dx.doi.org/10.1090/S00029939199110156774
 7.
I.
S. Gradshteyn and I.
M. Ryzhik, Table of integrals, series, and products, Academic
Press [Harcourt Brace Jovanovich, Publishers], New YorkLondonToronto,
Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey;
Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V.
Geronimus]\ and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from
the Russian. MR
582453 (81g:33001)
 8.
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
 9.
J.E. Littlewood and A.C. Offord. On the number of real roots of a random algebraic equation. J. London Math. Soc., 13:288295, 1938.
 10.
J.E. Littlewood and A.C. Offord. On the number of real roots of a random algebraic equation II. Proc. Camb. Phil. Soc., 35:133148, 1939.
 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.
 12.
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
(90f:60105), http://dx.doi.org/10.1090/S00029939198809550181
 1.
 E. Bogomolny, O. Bohigas, and P. Leboeuf. Distribution of roots of random polynomials. Phys. Rev. Lett., 68:27262729, 1992. MR 92m:81054
 2.
 H. Cramér and M.R. Leadbetter. Stationary and Related Stochastic Processes. Wiley, N.Y., 1967. MR 36:949
 3.
 A. Edelman and E. Kostlan. How many zeros of a random polynomial are real? Bull. Amer. Math. Soc., 32:137, 1995. MR 95m:60082
 4.
 K. Farahmand. On the average number of real roots of a random algebraic equation. Ann. Probab., 14:702709, 1986. MR 87k:60140
 5.
 K. Farahmand. On the average number of level crossings of a random trigonometric polynomial. Ann. Probab., 18:14031409, 1990. MR 91i:60140
 6.
 K. Farahmand. Level crossings of a random trigonometric polynomial. Proc. Amer. Math. Soc., 111:551557, 1991. MR 91f:60092
 7.
 I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series and Products. Academic Press, London, 1980, xv+1160 pp. MR 81g:33001
 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.
 J.E. Littlewood and A.C. Offord. On the number of real roots of a random algebraic equation. J. London Math. Soc., 13:288295, 1938.
 10.
 J.E. Littlewood and A.C. Offord. On the number of real roots of a random algebraic equation II. Proc. Camb. Phil. Soc., 35:133148, 1939.
 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.
 12.
 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
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
60H99,
42Bxx
Retrieve articles in all journals
with MSC (1991):
60H99,
42Bxx
Additional Information
K. Farahmand
Affiliation:
Department of Mathematics, University of Ulster, Jordanstown, Co. Antrim BT37 0QB, United Kingdom
Email:
k.farahmand@ulst.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002993999049126
PII:
S 00029939(99)049126
Keywords:
Number of real roots,
real zeros,
random algebraic polynomials,
random trigonometric polynomials,
KacRice formula
Received by editor(s):
July 9, 1997
Received by editor(s) in revised form:
December 17, 1997, and February 5, 1998
Published electronically:
May 6, 1999
Communicated by:
Stanley Sawyer
Article copyright:
© Copyright 1999
American Mathematical Society
