On algebraic polynomials with random coefficients

Author:
K. Farahmand

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2763-2769

MSC (2000):
Primary 60H99; Secondary 60G15

DOI:
https://doi.org/10.1090/S0002-9939-01-05836-1

Published electronically:
March 15, 2001

MathSciNet review:
1838800

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

**1.**A. Edelman and E. Kostlan,*How many zeros of a random polynomial are real?*, Bull. Amer. Math. Soc.,**32**(1995), 1-37. MR**95m:60082****2.**K. Farahmand and P. Hannigan,*The expected number of local maxima of a random algebraic polynomial*, J. Theoretical Probability,**10**(1997), 991-1002. MR**99a:60053****3.**K. Farahmand,*Local maxima of a random trigonometric polynomial*, J. Theor. Prob.,**7**(1994), 175-185. MR**95b:60059****4.**K. Farahmand,*Sharp crossings of a non-stationary stochastic process and its application to random polynomials*, Stoch. Anal. Appl.,**14**(1996), 89-100. MR**97d:60071****5.**K. Farahmand,*Topics in Random Polynomials*, Addison Wesley Longman, 1998, London. MR**2000d:60092****6.**K. Farahmand,*On random algebraic polynomials*, Proc. Amer. Math. Soc.,**127**(1999), 3339-3344. MR**2000b:60130****7.**J. E. Littlewood and A. C. Offord,*On the number of real roots of a random algebraic equation*, J. London Math. Soc.,**13**(1938), 288-295.**8.**J. E. Littlewood and A. C. Offord,*On the number of real roots of a random algebraic equation*II, Proc. Camb. Phil. Soc.,**35**(1939), 133-148.**9.**J. E. Wilkins.,*An asymptotic expansion for the expected number of real zeros of a random polynomial*, Proc. Amer. Math. Soc.,**103**(1988), 1249-1258. MR**90f:60105**

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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:
https://doi.org/10.1090/S0002-9939-01-05836-1

Keywords:
Number of real roots,
real zeros,
number of maxima,
random algebraic polynomials,
Kac-Rice formula

Received by editor(s):
September 1, 1999

Received by editor(s) in revised form:
January 26, 2000

Published electronically:
March 15, 2001

Communicated by:
Claudia M. Neuhauser

Article copyright:
© Copyright 2001
American Mathematical Society