Tropical F$F$‐polynomials and general presentations

We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. study its interplay with general presentation for any finite-dimensional basic algebra. give an interpretation evaluating at weight vector. As consequence, we Newton polytope ${\sf N}(M)$ dual fan and 1-skeleton N}(M)$. propose algorithm to determine generic polytopes, show it works path algebras. application, representation-theoretic Fock-Goncharov's duality pairing. explicit construction clusters, which consists real Schur representations. specialize above results cluster-finite algebras preprojective Dynkin type.