Enumerating Polytropes

Algebra Seminar
Monday, October 6, 2014 - 15:05
1 hour (actually 50 minutes)
Skiles 005
UT Austin
Polytropes are both ordinary and tropical polytopes. Tropical types of polytropes in \R^n are in bijection with certain cones of a specific Gr\"obner fan in \R^{n^2-n}. Unfortunately, even for n = 5 the entire fan is too large to be computed by existing software. We show that the polytrope cones can be decomposed as the cones from the refinement of two fans, intersecting with a specific cone. This allows us to enumerate types of full-dimensional polytropes for $n = 4$, and maximal polytropes for $n = 5$ and $n = 6$. In this talk, I will prove the above result and describe the key difficulty in higher dimensions.