Local tropical linear spaces

TL;DR

This paper introduces local tropical linear spaces, explores their structure and duality with polyhedral subdivisions, and proves properties including their homeomorphism to Euclidean space and bounds on their f-vectors.

Contribution

It defines local tropical linear spaces, establishes their duality with mixed subdivisions, and proves they satisfy Speyer's f-vector conjecture.

Findings

Local tropical linear spaces are homeomorphic to Euclidean space.

They are dual to mixed subdivisions of Minkowski sums of simplices.

Conical tropical linear spaces satisfy Speyer's f-vector conjecture.

Abstract

In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that make B a basis of maximal v-weight. The tropical linear space L can then be expressed as the union of all its local tropical linear spaces, which we prove are homeomorphic to Euclidean space. Local tropical linear spaces have a simple description in terms of polyhedral matroid subdivisions, and we prove that they are dual to mixed subdivisions of Minkowski sums of simplices. Using this duality we produce tight upper bounds for their f-vectors. We also study a certain class of tropical linear spaces that we call conical tropical linear spaces, and we give a simple proof that they satisfy Speyer's f-vector conjecture.