  We calculate the influence of dynamical exchange effects on the response
properties and the static properties of a two-dimensional many-polaron gas.
These effects are not manifested in the random-phase approximation which is
widely used in the analysis of the many-polaron system. Here they are taken
into account by using a dielectric function derived in the time-dependent
Hartree-Fock formalism. At weak electron-phonon coupling, we find that
dynamical exchange effects lead to substantial corrections to the random-phase
approximation results for the ground state energy, the effective mass, and the
optical conductivity of the polaron system. Furthermore, we show that the
reduction of the spectral weight of the optical absorption spectrum at
frequencies above the longitudinal optical phonon frequency, due to many-body
effects, is overestimated by the random-phase approximation.
