We investigate propagation and amplification of the auroral kilometric radiation over the geomagnetic poles using a particular model for the physical parameters in the auroral zone, which takes into account density gradients perpendicular to the geomagnetic field. The propagation of the waves is investigated with the canonical set of equations of the geometrical optics, taking into account thermal effects and considering both the energetic and the background electron populations, with the components of the dielectric tensor calculated in the locally homogeneous plasma approximation. It is shown that the spatial scale of inhomogeneity perpendicular to the magnetic field is an important factor in the amplification, although less favorable than indicated by previous studies using the method of Poeverlein to obtain the trajectory of the radiation.