Algebraic polynomials with random non-symmetric coefficients

This paper provides an asymptotic formula for the expected number of zeros of a polynomial of the form a0(ω)+a1(ω)n11/2x+a2(ω)n21/2x2+⋯+an(ω)nn1/2xn for large n. The coefficients {aj(ω)}j=0n are assumed to be a sequence of independent normally distributed random variables with fixed mean μ and variance one. It is shown that for μ non-zero this expected number is half of that for μ=0. This behavior is similar to that of classical random algebraic polynomials but differs from that of random trigonometric polynomials.