Complex Roots of a Class of Random Algebraic Polynomials

The average density of the distribution of complex roots of equation h0g0q h1g1zqh2g2z2q???qhny1gny1zny1sK with random coefficients hj and for zgC, gjʹ1, js0, 1, . . . , ny1, and K a complex constant with equal real and imaginary parts is known. The present paper provides a formula for such an average density for any constant real gj and hjsajqbj, where the ajs and bj s, js0, 1, . . . , ny1, are real standard normal independent random variables, and KsK1qiK2, where K1and K2are constants independent of z. The limiting behaviour of this density function as n tends to infinity for several special selected gjs is obtained.