Consider a random polynomial Gn(z) = ξnzn + · · · + ξ1 z + ξ0 with independent identically distributed complex-valued coefficients. Suppose that the distribution of log(1 + log(1 + |ξ0|)) has a slowly varying tail. Then the distribution of the complex roots of Gn concentrates in probability, as n → ∞, to two centered circles and is uniform in the argument as n → ∞. The radii of the circles are |ξ0/ξτ |1/τ and |ξτ /ξn |1/(n-τ), where ξτ denotes the coefficient with the maximum modulus. © 2012 Society for Industrial and Applied Mathematics.