On Different Classes of Algebraic Polynomials with Random Coefficients

[摘要] The expected number of real zeros of the polynomial of the forma0+a1x+a2x2+⋯+anxn, wherea0,a1,a2,…,anis a sequence of standardGaussian random variables, is known. Fornlarge it is shown that this expectednumber in(−∞,∞)is asymptotic to(2/π)log n. In this paper, we show thatthis asymptotic value increases significantly ton+1when we consider apolynomial in the forma0(n0)1/2x/1+a1(n1)1/2x2/2+a2(n2)1/2x3/3+⋯+an(nn)1/2xn+1/n+1instead. We give the motivation for our choice ofpolynomial and also obtain some other characteristics for the polynomial, suchas the expected number of level crossings or maxima. We note, and present,a small modification to the definition of our polynomial which improves ourresult from the above asymptotic relation to the equality.

[学科分类] 应用数学