  In this paper we explore and expand the connection between two modern
descriptions of scattering amplitudes, the CHY formalism and the framework of
positive geometries, facilitated by the scattering equations. For theories in
the CHY family whose $S$-matrix is captured by some positive geometry in the
kinematic space, the corresponding canonical form can be obtained as the
pushforward via the scattering equations of the canonical form of a positive
geometry defined in the CHY moduli space. In order to compute these canonical
forms in kinematic spaces, we study the general problem of pushing forward
arbitrary rational differential forms via the scattering equations. We develop
three methods which achieve this without ever needing to explicitly solve any
scattering equations. Our results use techniques from computational algebraic
geometry, including companion matrices and the global duality of residues, and
they extend the application of similar results for rational functions to
rational differential forms.
