Enumerating polytropes

Polytropes are both ordinary and tropical polytopes. We show that tropical types of polytropes in TPn−1 are in bijection with cones of a certain Gröbner fan GFn in Rn2−n restricted to a small cone called the polytrope region. These in turn are indexed by compatible sets of bipartite and triangle binomials. Geometrically, on the polytrope region, GFn is the refinement of two fans: the fan of linearity of the polytrope map appeared in [24], and the bipartite binomial fan. This gives two algorithms for enumerating tropical types of polytropes: one via general Gröbner fan software such as gfan, and another via checking compatibility of systems of bipartite and triangle binomials. We use these algorithms to compute types of full-dimensional polytropes for n=4, and maximal polytropes for n=5.