  We use the stabilization functors to study the combinatorial aspects of the
$F$-polynomial of a representation of any finite-dimensional basic algebra. We
characterize the vertices of their Newton polytopes. We give an explicit
formula for the $F$-polynomial restricting to any face of its Newton polytope.
For acyclic quivers, we give a complete description of all facets of the Newton
polytope when the representation is general. We also prove that the support of
the $F$-polynomial is saturated for any rigid representation. We provide many
examples and counterexamples, and pose several conjectures.
