  We identify two recursively defined polynomial conditions for FI-modules in
the literature. We characterize these conditions using homological invariants
of FI-modules (namely the local degree and regularity, together with the stable
degree) and clarify their relationship. For one of these conditions, we give
improved twisted homological stability ranges for the symmetric groups. As
another application, we improve the representation stability ranges for
congruence subgroups with respect to the action of an appropriate linear group
by a factor of 2 in its slope.
