  The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has been
extended and generalised in many ways. In this paper, we describe properties of
the characteristic polynomial of a weighted lattice and show that it has a
recursive description, which we use to obtain results on the critical exponent
of $q$-polymatroids. We prove a Critical Theorem for representable
$q$-polymatroids and we provide a lower bound on the critical exponent. We show
that $q$-polymatroids arising from certain families of rank-metric codes attain
this lower bound.
