Zeros of an algebraic polynomial with nonequal means random coefficients

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a 0 + a 1 x + a 2 x 2 + ⋯ + a n − 1 x n − 1 . The coefficients a j ( j = 0 , 1 , 2 , … , n − 1 ) are assumed to be independent normal random variables with nonidentical means. Previous results are mainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are divided into many groups of equal sizes. We show that the behaviour of the random polynomial is dictated by the mean of the first group of the coefficients in the interval ( − 1 , 1 ) and the mean of the last group in ( − ∞ , − 1 ) ∪ ( 1 , ∞ ) .