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Orthogonal Tropical Linear Prevarieties

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Computer Algebra in Scientific Computing (CASC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11077))

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Abstract

We study the operation \(A^\perp \) of tropical orthogonalization, applied to a subset A of a vector space \(({\mathbb R}\cup \{ \infty \})^n\), and iterations of this operation. Main results include a criterion and an algorithm, deciding whether a tropical linear prevariety is a tropical linear variety formulated in terms of a duality between \(A^\perp \) and \(A^{\perp \perp }\). We give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.

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Acknowledgments

We thank M. Joswig, N. Kalinin, H. Markwig, and T. Theobald for useful discussions. 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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Authors and Affiliations

  1. CNRS, Mathématiques, Université de Lille, 59655, Villeneuve d’Ascq, France

    Dima Grigoriev

  2. Department of Computer Science, University of Bath, Bath, BA2 7AY, England, UK

    Nicolai Vorobjov

Authors
  1. Dima Grigoriev
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  2. Nicolai Vorobjov
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Correspondence to Nicolai Vorobjov .

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Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Grigoriev, D., Vorobjov, N. (2018). Orthogonal Tropical Linear Prevarieties. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2018. Lecture Notes in Computer Science(), vol 11077. Springer, Cham. https://doi.org/10.1007/978-3-319-99639-4_13

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  • DOI: https://doi.org/10.1007/978-3-319-99639-4_13

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  • Online ISBN: 978-3-319-99639-4

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