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On the frontiers of polynomial computations in tropical geometry. (English) Zbl 1121.14047
Summary: 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.

##### MSC:
 14P99 Real algebraic and real-analytic geometry 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 14Q15 Computational aspects of higher-dimensional varieties 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q25 Analysis of algorithms and problem complexity
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