On the frontiers of polynomial computations in tropical geometry

Details On the frontiers of polynomial computations in tropical geometry On the frontiers of polynomial computations in tropical geometry

2004-10-31

arxiv.org/abs/math/0411012v2

Mathematics

Combinatorics

17 pages, 5 figures. To appear in Journal of Symbolic Computation

Scientific paper

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.

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