Root polytopes, tropical types, and toric edge ideals

We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its ‘type’ data, analogous to the covectors of an oriented matroid. By work of Develin–Sturmfels and Fink–Rincón, these ‘tropical complexes’ are dual to regular subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig–Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among ‘type ideals’ which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge rings of bipartite graphs, which have been extensively studied in the commutative algebra community.