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

Dima Grigoriev; Nicolai Vorobjov

doi:10.1007/s00200-019-00407-w

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

Dima Grigoriev, Nicolai Vorobjov

Université de Lille

Research output: Contribution to journal › Article › peer-review

2 Citations (SciVal)

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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 ^⊥ 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.

Original language	English
Pages (from-to)	157-174
Number of pages	18
Journal	Applicable Algebra in Engineering Communication and Computing
Volume	32
Issue number	2
Early online date	29 Nov 2019
DOIs	https://doi.org/10.1007/s00200-019-00407-w
Publication status	Published - 31 Mar 2021

Keywords

Tropical algebra
Computer algebra
Complexity

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10.1007/s00200-019-00407-w

1803.01068.PDF
This is a post-peer-review, pre-copyedit version of an article published in Applicable Algebra in Engineering, Communication and Computing. The final authenticated version is available online at: https://doi.org/10.1007/s00200-019-00407-w
Accepted author manuscript, 239 KB

https://arxiv.org/pdf/1803.01068

Cite this

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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 ⊥ 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. ",

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AB - 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.

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KW - Computer algebra

KW - Complexity

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

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Keywords

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