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

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 ⊥ and the double tropical orthogonalization A ⊥⊥ of a subset A of the vector space (R ∪ {∞}) n . We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.

Entropy of radical ideal of a tropical prevariety

“…In [4] an example of intersection of an infinite number of tropical linear prevarieties was produced being not a tropical prevariety. Therefore, it is not clear whether W N is a tropical linear prevariety.…”