Random algebraic polynomials with increasing variance of the coefficients (Q2869516)

The average number of real zeros of a random algebraic polynomial of the form \(A_0 + A_1 x + A_2 x^2 + \cdots + A_n x^n\) is well known. This average number is proved, for \(n\) large, to be asymptotic to \(EN(-\infty, \infty) \sim(2/\pi) \log n\), when \(\{A_i\}_{i=0}^n\) is a sequence of independent identical normal standard random variables. Recently, there have been many interesting developments in this topic which introduced a new class of random algebraic polynomials by imposing a new assumption on the distribution of the above coefficients. However, for most cases, the above asymptotic value remains persistent. Only when the variance of coefficients changes, for example to \(\mathrm {var } (A_j)= \;{n\choose j}\), the order of the above asymptotic value increases significantly to \(\sqrt n\).NEWLINENEWLINEHowever, in this paper the authors consider a polynomial in an interesting form of \(Q_n(x) = \sum _{k=0}^ n \exp\{ (k(2n-k)/2n)\sqrt{n}\}A_k x^k\), where, as previously, \(A_k\) (\(k=0, 1, 2 \dots n\)) are independent identically distributed random variables with standard normal distribution. It is shown that for this case the average number of real zeros is less than \(\sqrt{n}\) and higher than \((2/\pi) \log n\). In fact, this expected number is \(EN(-\infty, \infty) \sim (2\sqrt{\pi -2})n^{1/4}/ \sqrt{2 \pi}\).NEWLINENEWLINEEarlier results on this subject are discussed by \textit{A. T. Bharucha-Reid} and \textit{M. Sambandham} [Random polynomials. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers) (1986; Zbl 0615.60058)], which has a comprehensive list of references and by the reviewer [Topics in random polynomials. Harlow: Addison Wesley Longman (1998; Zbl 0949.60010)].