Poincar\'e duality for singular tropical hypersurfaces

We establish a partial extension of the Poincar\'e duality theorem of Jell-Rau-Shaw to tropical hypersurfaces arising from non-primitive triangulations. We introduce a notion of level of primitivity for triangulations of lattice polytopes and show that tropical hypersurfaces satisfy a partial form of Poincar\'e duality determined by this level. This notion of primitivity is defined modulo a fixed integral domain and is weaker than the classical notion of primitivity. Moreover, we obtain a generalization of complete Poincar\'e duality over this integral domain for tropical hypersurfaces whose underlying triangulations are primitive modulo the integral domain. As a corollary, we show that any tropical hypersurface obtained by patchworking from a triangualtion of a simple lattice polytope satisfies complete Poincar\'e duality over the field of rational numbers, providing a converse to a theorem of Aksnes. Throughout, we allow triangulations that are not necessarily convex.