Algebraic polynomials with non-identical random coefficients

Abstract:

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial $a_0 +a_1 x+ a_2 x^2+ \cdots +a_{n-1}x^{n-1}.$ The coefficients $a_j$ $(j=0, 1, 2, \dotsc , n-1)$ are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all $n$ sufficiently large, the above expected value is shown to be $O(\log n)$. Also, it is known that if the $a_j$ have non-identical variance $\binom {n-1}{j}$, then the expected number of real zeros increases to $O(\sqrt {n})$. It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than $O(\log n)$. In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain $O(\log n)$. In fact, so far the case of $\mbox {var}(a_j)={\binom {n-1}{j}}$ is the only case that can significantly increase the expected number of real zeros.