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Tropical semimodules of dimension two


Author: Ya. Shitov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 2.
Journal: St. Petersburg Math. J. 26 (2015), 341-350
MSC (2010): Primary 15A03, 15A23, 15A80
DOI: https://doi.org/10.1090/S1061-0022-2015-01341-1
Published electronically: February 3, 2015
MathSciNet review: 3242042
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Abstract: The tropical arithmetic operations on $ \mathbb{R}$ are defined as $ a\oplus b=\min \{a,b\}$ and $ a\otimes b=a+b$. In the paper, the concept of a semimodule is discussed, which is rather ill-behaved in tropical mathematics. The semimodules $ S\subset \mathbb{R}^n$ having topological dimension two are studied and it is shown that any such $ S$ has a finite weak dimension not exceeding $ n$. For a fixed $ k$, a polynomial time algorithm is constructed that decides whether $ S$ is contained in some tropical semimodule of weak dimension $ k$ or not. This result provides a solution of a problem that has been open for eight years.


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Additional Information

Ya. Shitov
Affiliation: National Research University–Higher School of Economics, Myasnitskaya Ulitsa 20, Moscow 101000, Russia
Email: yaroslav-shitov@yandex.ru

DOI: https://doi.org/10.1090/S1061-0022-2015-01341-1
Keywords: Tropical mathematics, linear algebra, computational complexity
Received by editor(s): June 27, 2013
Published electronically: February 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society