Computing Tropical Linear Spaces

TL;DR

This paper introduces the cyclic Bergman fan of a matroid, a computationally friendly structure on tropical linear spaces, along with a fast algorithm and software implementation for practical calculations in tropical geometry.

Contribution

It defines the cyclic Bergman fan, provides a fast algorithm for its computation, and implements it in software, enabling new computational applications in tropical geometry.

Findings

The cyclic Bergman fan refines the nested set structure on tropical linear spaces.

A fast algorithm for computing the cyclic Bergman fan is developed.

The implementation, TropLi, is available online for practical use.

Abstract

We define and study the cyclic Bergman fan of a matroid M, which is a simplicial polyhedral fan supported on the tropical linear space T(M) of M and is amenable to computational purposes. It slightly refines the nested set structure on T(M), and its rays are in bijection with flats of M which are either cyclic flats or singletons. We give a fast algorithm for calculating it, making some computational applications of tropical geometry now viable. Our C++ implementation, called TropLi, and a tool for computing vertices of Newton polytopes of A-discriminants, are both available online.