Real zeros of random algebraic polynomials with binomial elements
Summary: This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial \(a_0+a_1x+a_2 x^2+\cdots+a_{n-1}x^{n-1}\). The coefficients \(a_j\) \((j=0,1,2,\dots, n-1)\) are assumed to be independent normal random variables with mean zero. For integers \(m\) and \(k=O(\log n)^2\) the variances of the coefficients are assumed to have nonidentical value \(\text{var} (a_j)={k-1\choose j-ik}\), where \(n=k\cdot m\) and \(i=0,1,2,\dots,m-1\). Previous results are mainly for identically distributed coefficients or when \(\text{var} (a_j)= {n\choose j}\). We show that the latter is a special case of our general theorem.
scientific article; zbMATH DE number 5134599
scientific article