Complex Roots of a Random Algebraic Polynomial

This paper, for any constantK, provides an exact formula for the average density of the distribution of the complex roots of equation η₀ + η₁z + η₂z² + ··· + η_{n − 1}z^{n − 1} = Kwhere η_j = a_j + ib_jand {a_j}^{n − 1}_{j = 0}and {b_j}^{n − 1}_{j = 0}are sequences of independent identically and normally distributed random variables andKis a complex number withKas its real and imaginary parts. The case of real roots of the above equation with real coefficients andK,z Ris well known. Further we obtain the limiting behaviour of this distribution function asntends to infinity.