On Different Classes of Algebraic Polynomials with Random Coefficients

The expected number of real zeros of the polynomial of the form 𝑎0+𝑎1𝑥+𝑎2𝑥2+⋯+𝑎𝑛𝑥𝑛, where 𝑎0,𝑎1,𝑎2,…,𝑎𝑛 is a sequence of standard Gaussian random variables, is known. For 𝑛 large it is shown that this expected number in (−∞,∞) is asymptotic to (2/𝜋)log𝑛. In this paper, we show that this asymptotic value increases significantly to √𝑛+1 when we consider a polynomial in the form 𝑎0𝑛01/2√𝑥/1+𝑎1𝑛11/2𝑥2/√2+𝑎2𝑛21/2𝑥3/√3+⋯+𝑎𝑛𝑛𝑛1/2𝑥𝑛+1/√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.