  A recent experiment reported for the first time the preparation of a Fermi
degenerate gas of polar molecules and observed a suppression of their chemical
reaction rate compared to the one expected from a purely classical treatment.
While it was hypothesized that the suppression in the ultracold regime had its
roots in the Fermi statistics of the molecules, this argument is inconsistent
with the fact that the Fermi pressure should set a lower bound for the chemical
reaction rate. Therefore it can not be explained from standard two-body
$p$-wave inelastic collisions. Here we develop a simple model of chemical
reactions that occur via the formation and decay of molecular complexes. We
indeed find that pure two-body molecule losses are unable to explain the
observed suppression. Instead we extend our description beyond two-body physics
by including effective complex-molecule interactions possible emerging from
many-body and effective medium effects at finite densities and in the presence
of trapping light. %Under this framework we observe that additional
complex-molecule collisions, which manifest as a net three-body molecular
interaction could give rise to the additional suppression. Although our
effective model is able to quantitatively reproduce recent experimental
observations, a detailed understanding of the actual physical mechanism
responsible for these higher-order interaction processes is still pending.
