Complex roots of a class of random algebraic polynomials (Q1268749)

Let \(P_n(z)=\sum_{j=0}^{n-1}\eta_jg_jz^j\) be a complex polynomial with real coefficients \(g_j\) and \(\eta_j=a_j+ib_j\), where \(a_j\), \(b_j\) are independent standard normal random variables. For \(K=K_1+iK_2\) and a set \(\Phi\) of complex numbers, let \(\nu_K^n(\Phi)\) be the number of roots in \(\Phi\) of \(P(z)=K\). Then the expectation \(E\nu_K^n(\Phi)=\smallint_{\Phi}h_K^n(z) dz\). A formula of values of the density \(h_K^n\) is given. Also, the limiting behaviour of this density function as \(n\) tends to infinity for several special selected \(g_j\)'s is obtained.