  A new window is opening in high-energy astronomy: X-ray polarimetry. With
many missions currently under development and scheduled to launch as early as
2021, observations of the X-ray polarisation of accreting X-ray pulsars will
soon be available. As polarisation is particularly sensitive to the geometry of
the emission region, the upcoming polarimeters will shed new light on the
emission mechanism of these objects, provided that we have sound theoretical
models that agree with current spectroscopic and timing observation and that
can make predictions of the polarisation parameters of the emission. We here
present a new model for the polarised emission of accreting X-ray pulsars in
the accretion column scenario that for the first time takes into account the
macroscopic structure and dynamics of the accretion region and the propagation
of the radiation toward the observer, including relativistic beaming,
gravitational lensing and quantum electrodynamics. In this paper we present all
the details of the model, while in a companion paper, we apply our model to
predict the polarisation parameters of the bright X-ray pulsar Hercules X-1.
