# Abstract of Talk by Dr. J. Ernest Wilkins

## Real Zeros of Random Polynomials

Dr. J. Ernest Wilkins
Department of Mathematics
Clark-Atlanta University
Box J
Atlanta, GA 30314-4839

## Office Telephone Number:

(404)-880-8834
## Fax Number:

(404)-880-8222
## Electronic Mail Address:

We present a survey of results that estimate the mean value of the number
of real roots of an algebraic polynomial of degree n when the coefficients
of that polynomial are random variables and n is a large integer. Of
particular interest is the special case in which the coefficients are
independent, normally distributed random variables, each with mean 0
and variance 1, although we will discuss some results for several other
coefficient distributions. In all the cases considered the mean number
of real roots is O(log n), so that there are relatively few real roots.
We also mention the small number of results known for estimating the
variance of the number of real roots. We then consider analogous issues
for trigonometric cosine polynomials of the form,
$a_1 \cos x + a_2 \cos 2x + \cdots + a_n \cos nx$, and estimate the
mean number of
zeros of this polynomial that lie in the interval (0, 2p). In contrast
to the algebraic case, this mean number is An + o(n) for some constant
A that depends on the coefficient distribution.

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