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

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.