Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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

Abstract | References | Similar Articles | Additional Information

Abstract:

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

$a_0\binom{n-1}{0}^{1/2}+a_1\binom{n-1}{1}^{1/2}x +a_2\binom{n-1}{2}^{1/2}x^2+\cdots+a_{n-1}\binom{n-1}{n-1}^{1/2}x^{n-1}$where $a_{j}, j= 0, 1, 2, \ldots, n-1$, 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 $x$ axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form $a_{0}+a_{1}x +a_{2}x^{2}+\cdots +a_{n- 1}x^{n-1}$which was previously the most studied.


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60H99, 60G15

Retrieve articles in all journals with MSC (2000): 60H99, 60G15


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

American Mathematical Society