This paper addresses the problem of establishing whether a complex matrix depending polynomially on a scalar parameter and its conjugate constrained over the complex unit circumference is robustly asymptotically stable in either the continuous-time case or the discrete-time case. A necessary and sufficient condition is proposed in terms of a linear matrix inequality (LMI) feasibility test based on complex Lyapunov functions depending polynomially on the uncertainty. Specifically, the condition is sufficient for any arbitrarily chosen degree of the Lyapunov function. Moreover, the condition is also necessary for a sufficiently large degree of the Lyapunov function, and an upper bound on the minimum degree required for achieving necessity is also provided. Some numerical examples illustrate the proposed results.