  The extragradient method has gained popularity due to its robust convergence
properties for differentiable games. Unlike single-objective optimization, game
dynamics involve complex interactions reflected by the eigenvalues of the game
vector field's Jacobian scattered across the complex plane. This complexity can
cause the simple gradient method to diverge, even for bilinear games, while the
extragradient method achieves convergence. Building on the recently proven
accelerated convergence of the momentum extragradient method for bilinear games
\citep{azizian2020accelerating}, we use a polynomial-based analysis to identify
three distinct scenarios where this method exhibits further accelerated
convergence. These scenarios encompass situations where the eigenvalues reside
on the (positive) real line, lie on the real line alongside complex conjugates,
or exist solely as complex conjugates. Furthermore, we derive the
hyperparameters for each scenario that achieve the fastest convergence rate.
