Complexity of deciding whether a tropical linear prevariety is a tropical variety

TL;DR

This paper presents a singly exponential algorithm to determine if a tropical linear prevariety is a tropical linear variety, based on a duality criterion involving tropical orthogonalization.

Contribution

It introduces a new algorithm with exponential complexity for this decision problem and provides a criterion based on duality between tropical orthogonalizations.

Findings

Algorithm with singly exponential complexity for the decision problem.

A criterion based on duality between $A^ot$ and $A^{ot ot}$.

An example of a family of tropical hyperplanes whose intersection is not a prevariety.

Abstract

We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization $A^\perp$ and the double tropical orthogonalization $A^{\perp \perp}$ of a subset $A$ of the vector space $({\mathbb R} \cup \{ \infty \})^n$. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.