Zeros of a random algebraic polynomial with coefficient means in geometric progression

Abstract

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients { a j } n−1 j=0 of the polynomial T( x)= a 0+ a 1 x+ a 2 x 2+⋯+ a n−1 x n−1 are normally distributed, with mean E( a j )= μ j+1 , where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of | μ|<1 and | μ|>1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0.