Enumerating Polytropes

Abstract

Polytropes are both ordinary and tropical polytopes. We show that tropical types of polytropes in ${\mathbb{TP}}^{n-1}$ are in bijection with cones of a certain Gröbner fan ${\mathcal{GF}}_n$ in ${\mathbb{R}}^{n^2-n}$ 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, ${\mathcal{GF}}_n$ is the refinement of two fans: the fan of linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite binomial fan. This gives two algorithms for enumerating tropical types of polytropes: one via a general Gröbner fan software such as \textsf{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 $n=5$.

Improved exposition, fixed error in reporting the number maximal polytropes for $n=6$, fixed error in definition of bipartite binomials

References in corpus (1)

• Tropical matrix duality and Green's D relation

Cited by in corpus (3)

• Weighted digraphs and tropical cones
• A tropical isoperimetric inequality
• Quasi--Euclidean classification of Alcoved Convex Polyhedra