Complex Roots of a Random Algebraic Polynomial

This paper, for any constant K, provides an exact formula for the average density of the distribution of the complex roots of equation h qh zqh z 2 0 1 2 q???qh z ny1sK where h sa qib and a 4ny1 and b 4ny1 are sequences ny1 j j j jjs0 j js0 of independent identically and normally distributed random variables and K is a complex number with K as its real and imaginary parts. The case of real roots of the above equation with real coefficients and K, zgR is well known. Further we obtain the limiting behaviour of this distribution function as n tends to infinity.