Tropical compactification via Ganter's algorithm

TL;DR

This paper introduces a canonical method for compactifying polyhedral complexes in Euclidean space, utilizing Ganter's algorithm to analyze their combinatorial structure, especially when related to tropical toric varieties.

Contribution

It presents a novel compactification approach for polyhedral complexes and demonstrates how Ganter's algorithm can compute their Hasse diagrams within polymake.

Findings

The compactification aligns with tropical toric varieties when recession cones form a fan.

Ganter's algorithm effectively computes the Hasse diagram of the compactified complex.

Implementation in polymake facilitates practical computation of these structures.

Abstract

We describe a canonical compactification of a polyhedral complex in Euclidean space. When the recession cones of the polyhedral complex form a fan, the compactified polyhedral complex is a subspace of a tropical toric variety. In this case, the procedure is analogous to the tropical compactifications of subvarieties of tori. We give an analysis of the combinatorial structure of the compactification and show that its Hasse diagram can be computed via Ganter's algorithm. Our algorithm is implemented in and shipped with polymake.