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 \(A^\perp \) and the double tropical orthogonalization \(A^{\perp \perp }\) of a subset A of the vector space \(({{\mathbb {R}}}\cup \{ \infty \})^n\). We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.

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Acknowledgements

We thank M. Joswig, N. Kalinin, H. Markwig, and T. Theobald for useful discussions, and anonymous referees for constructive remarks and suggestions. Part of this research was carried out during our joint visit in September 2017 to the Hausdorff Research Institute for Mathematics at Bonn University, under the program Applied and Computational Algebraic Topology, to which we are very grateful. D. Grigoriev was partly supported by the RSF Grant 16-11-10075.

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Correspondence to Nicolai Vorobjov.

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Grigoriev, D., Vorobjov, N. Complexity of deciding whether a tropical linear prevariety is a tropical variety. AAECC 32, 157–174 (2021). https://doi.org/10.1007/s00200-019-00407-w

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Keywords

  • Tropical linear prevariety
  • Tropical variety
  • Complexity

Mathematics Subject Classification

  • 14T05