  We prove analogues of several well-known results concerning rational
morphisms between quadrics for the class of so-called quasilinear
$p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field,
so many of the geometric methods which have been successfully applied to the
study of projective homogeneous varieties over fields cannot be used. We are
therefore forced to take an alternative approach, which is partly facilitated
by the appearance of several non-traditional features in the study of these
objects from an algebraic perspective. Our main results were previously known
for the class of quasilinear quadrics. We provide new proofs here, because the
original proofs do not immediately generalise for quasilinear hypersurfaces of
higher degree.
