An asymptotic expansion for the expected number of real zeros of a random polynomial

Author:
J. Ernest Wilkins

Journal:
Proc. Amer. Math. Soc. **103** (1988), 1249-1258

MSC:
Primary 60G99

DOI:
https://doi.org/10.1090/S0002-9939-1988-0955018-1

MathSciNet review:
955018

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Abstract: Let be the expected number of real zeros of a polynomial of degree whose coefficients are independent random variables, normally distributed with mean 0 and variance 1. We find an asymptotic expansion for of the form

**[1]**M. Kac,*On the average number of real roots of a random algebraic equation*, Bull. Amer. Math. Soc.**49**(1943), 314–320. MR**0007812**, https://doi.org/10.1090/S0002-9904-1943-07912-8**[2]**B. R. Jamrom,*The average number of real zeros of random polynomials*, Dokl. Akad. Nauk SSSR**206**(1972), 1059–1060 (Russian). MR**0314114****[3]**B. R. Jamrom,*The average number of real roots of a random algebraic polynomial*, Vestnik Leningrad. Univ.**19**(1971), 152–156 (Russian, with English summary). MR**0298742****[4]**You Jing Wang,*Bounds on the average number of real roots of a random algebraic equation*, Chinese Ann. Math. Ser. A**4**(1983), no. 5, 601–605 (Chinese). An English summary appears in Chinese Ann. Math. Ser. B 4 (1983), no. 4, 527. MR**742181****[5]**D. C. Stevens,*The average and variance of the number of real zeros of random functions*, Ph.D. dissertation, New York Univ., 1965.**[6]**J. Ernest Wilkins Jr.,*An upper bound for the expected number of real zeros of a random polynomial*, J. Math. Anal. Appl.**42**(1973), 569–577. MR**0326842**, https://doi.org/10.1016/0022-247X(73)90164-9**[7]**Zhen Hua Luo,*The average number of real roots of a random algebraic equation*, Chinese Ann. Math.**1**(1980), no. 3-4, 541–544 (Chinese, with English summary). MR**619600****[8]**M. J. Christensen and M. Sambandham,*An improved lower bound for the expected number of real zeros of a random algebraic polynomials*, Stochastic Anal. Appl.**2**(1984), no. 4, 431–436. MR**769280**, https://doi.org/10.1080/07362998408809046**[9]**Zhong Ming Yu,*Bounds on the average number of real roots for a class of random algebraic equations*, J. Math. Res. Exposition**2**(1982), no. 2, 81–85 (Chinese, with English summary). MR**669828****[10]**D. K. Kahaner,*Some computations of expected number of real zeros of random polynomials*, J. Math. Anal. Appl.**48**(1974), 780–784. MR**0418402**, https://doi.org/10.1016/0022-247X(74)90151-6

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0955018-1

Article copyright:
© Copyright 1988
American Mathematical Society