On the frontiers of polynomial computations in tropical geometry

We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving $\mathcal{NP}$-hardness and #$\mathcal{P}$-hardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio

On a tropical dual Nullstellensatz

Since a tropical Nullstellensatz fails even for tropical univariate polynomials we study a conjecture on a tropical {\it dual} Nullstellensatz for tropical polynomial systems in terms of solvability of a tropical linear system with the Cayley matrix associated to the tropical polynomial system. The conjecture on a tropical effective dual Nullstellensatz is proved for tropical univariate polynomials

Tropical cryptography II: extensions by homomorphisms

We use extensions of tropical algebras as platforms for very efficient public key exchange protocols.Comment: 7 pages. arXiv admin note: text overlap with arXiv:1301.119

Tropical cryptography III: digital signatures

We use tropical algebras as platforms for a very efficient digital signature protocol. Security relies on computational hardness of factoring one-variable tropical polynomials; this problem is known to be NP-hard.Comment: 7 page