  Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a
characterization for the existence of a factorization $f= h \odot g$ and the
construction of $h$. As a ramification of this result we obtain a parallel
result for the Minkowski factorization of polytopes. Using our construction we
show that for any given polytopal fan there is a polytope factorization basis,
i.e. a finite set of polytopes with respect to which any polytope whose normal
fan is refined by the original fan can be uniquely written as a signed
Minkowski sum. We explicitly study the factorization of polymatroids and their
generalizations, Coxeter matroid polytopes, and give a hyperplane description
of the cone of deformations for this class of polytopes.
