Coarse types of tropical matroid polytopes

TL;DR

This paper analyzes the combinatorial structure of tropical matroid polytopes, providing formulas for coarse types of maximal cells and describing minimal tropical halfspaces, linking tropical geometry with monomial ideal resolutions.

Contribution

It introduces formulas for the coarse types of maximal cells and characterizes minimal tropical halfspaces of tropical matroid polytopes, connecting tropical geometry with algebraic combinatorics.

Findings

Formulas for coarse types of maximal cells of tropical complexes

Complete description of minimal tropical halfspaces of tropical matroid polytopes

Connection established between tropical complexes and monomial ideal resolutions

Abstract

Describing the combinatorial structure of the tropical complex $C$ of a tropical matroid polytope, we obtain a formula for the coarse types of the maximal cells of $C$. Due to the connection between tropical complexes and resolutions of monomial ideals, this yields the generators for the corresponding coarse type ideal introduced in a recent paper of Dochtermann, Joswig and Sanyal (2010, preprint arXiv.org:1001.0237). Furthermore, a complete description of the minimal tropical halfspaces of the uniform tropical matroid polytopes, i.e. the tropical hypersimplices, is given.