Combinatorics of F-Polynomials.

The article focuses on the combinatorial study of F-polynomials associated with representations of finite-dimensional basic algebras, particularly their Newton polytopes. It characterizes the vertices of these Newton polytopes as dimension vectors of unique subrepresentations and provides explicit formulas for restricting F-polynomials to faces of their Newton polytopes using stabilization functors and δ-stable reductions. For acyclic quivers, the paper gives a complete description of all facets of the Newton polytope for general representations, showing that the outer normal vectors correspond to certain extremal rays related to homological conditions, and proves that the support of the F-polynomial is saturated for any rigid representation. Several conjectures are posed, including that for cluster monomials, vertices of the Newton polytope correspond exactly to monomials with coefficient one, and that the support of their F-polynomials is saturated in general cluster algebras. The work connects representation theory, cluster algebras, and tropical geometry through functorial and homological methods.