On different classes of algebraic polynomials with random coefficients (Q936990)

Summary: The expected number of real zeros of the polynomial of the form \(a_0+ a_1x+ a_2x^2+\cdots+ a_nx^n\), where \(a_0,a_1,a_2,\dots,a_n\) is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in \((-\infty,\infty)\) is asymptotic to \((2/\pi)\log n\). In this paper, we show that this asymptotic value increases significantly to \(\sqrt{n+1}\) when we consider a polynomial in the form \[ a_0 \binom n0^{1/2} x/\sqrt 1+ a_1\binom n1^{1/2}x^2/\sqrt 2+ a_2 \binom n2^{1/2} x^3/\sqrt 3+\cdots+ a_n\binom nn^{1/2}x^{n+1}/ \sqrt{n+1} \] instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.