Tropical polytopes and cellular resolutions

Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This view-point gives rise to a family of cellular resolutions of monomial ideals that generalize the hull complex of Bayer and Sturmfels [Bayer and Sturmfels 98], instances of which improve upon the hull resolution in the sense of being smaller. We also suggest a new definition of a face of a tropical polytope, which has nicer properties than previous definitions; we give examples and provide many conjectures and directions for further research in this area.