Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akian, M., Gaubert, S., Guterman, A.: Tropical polyhedra are equivalent to mean payoff games. Int. J. Algebra Comput. 22(1), 43 (2012)

    Article  MathSciNet  Google Scholar 

  2. Bogart, T., Jensen, A.N., Speyer, D., Sturmfels, B., Thomas, R.R.: Computing tropical varieties. J. Symb. Comput. 42(1–2), 54–73 (2007)

    Article  MathSciNet  Google Scholar 

  3. Butkovic, P., Hegedüs, G.: An elimination method for finding all solutions of the system of linear equations over an extremal algebra. Ekon.-Mat. Obzor 20, 203–214 (1984)

    MathSciNet  MATH  Google Scholar 

  4. Chistov, A.: An algorithm of polynomial complexity for factoring polynomials, and determination of the components of a variety in a subexponential time. J. Soviet Math. 34, 1838–1882 (1986)

    Article  Google Scholar 

  5. Chistov, A.: Polynomial complexity of Newton–Puiseux algorithm. Lect. Notes Comput. Sci. 233, 247–255 (1986)

    Article  MathSciNet  Google Scholar 

  6. Develin, M., Santos, F., Sturmfels, B.: On the rank of a tropical matrix. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, vol. 52. MSRI Publications (2005)

  7. Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Dress, A., Wenzel, W.: Algebraic, tropical, and fuzzy geometry. Beitr. Algebra Geom. 52(2), 431–461 (2011)

    Article  MathSciNet  Google Scholar 

  9. Gaubert, S., Katz, R.D.: Minimal half-spaces and external representation of tropical polyhedra. J. Algebr. Comb. 33(3), 325–348 (2011)

    Article  MathSciNet  Google Scholar 

  10. Görlach, P., Ren, Y., Sommars, J.: Detecting tropical defects of polynomial equations (2018). arXiv:1809.03350

  11. Grigoriev, D.: Polynomial factoring over a finite field and solving systems of algebraic equations. J. Soviet Math. 34, 1762–1803 (1986)

    Article  Google Scholar 

  12. Grigoriev, D.: Complexity of solving tropical linear systems. Comput. Complex. 22, 71–88 (2013)

    Article  MathSciNet  Google Scholar 

  13. Grigoriev, D.: Polynomial complexity recognizing a tropical linear variety. Lect. Notes Comput. Sci. 9301, 152–157 (2015)

    Article  MathSciNet  Google Scholar 

  14. Grigoriev, D., Podolskii, V.: Complexity of tropical and min-plus linear prevarieties. Comput. Complex. 24(1), 31–64 (2015)

    Article  MathSciNet  Google Scholar 

  15. Grigoriev, D., Vorobjov, N.: Orthogonal tropical linear prevarieties. In: CASC 2018, Lecture Notes in Computer Science, vol. 11077, pp. 187–196 (2018)

  16. Grigoriev, D., Vorobjov, N.: Upper bounds on Betti numbers of tropical prevarieties. Arnold Math. J. 4(1), 127–136 (2018)

    Article  MathSciNet  Google Scholar 

  17. Hept, K., Theobald, T.: Tropical bases by regular projections. Proc. Am. Math. Soc. 137(7), 2233–2241 (2009)

    Article  MathSciNet  Google Scholar 

  18. Jensen, A., Markwig, H., Markwig, T.: An algorithm for lifting points in a tropical variety. Collect. Math. 59(2), 129–165 (2008)

    Article  MathSciNet  Google Scholar 

  19. Jensen, A., Yu, J.: Stable intersections of tropical varieties. J. Algebr. Comb. 43(1), 101–128 (2016)

    Article  MathSciNet  Google Scholar 

  20. Joswig, M.: Essentials of Tropical Combinatorics. Springer, Berlin (2014)

    Google Scholar 

  21. Joswig, M.: The Cayley trick for tropical hypersurfaces with a view toward Ricardian economics. In: Homological and Computational Methods in Commutative Algebra, Springer INdAM Series, vol. 20. Springer (2017)

  22. Joswig, M., Schröter, B.: The degree of a tropical basis. Proc. Am. Math. Soc. 146(3), 961–970 (2018)

    Article  MathSciNet  Google Scholar 

  23. Kazarnovskii, Ya., Khovanskii, A.G.: Tropical noetherity and Gröbner bases. St. Petersb. Math. J. 26(5), 797–811 (2015)

    Article  Google Scholar 

  24. Lang, S.: Algebra. Springer, Berlin (2002)

    Book  Google Scholar 

  25. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. American Mathematical Society, Providence (2015)

    Book  Google Scholar 

  26. Markwig, T., Ren, Y.: Computing tropical varieties over fields with valuation. Found. Comput. Math. (2019). https://doi.org/10.1007/s10208-019-09430-2

  27. Murota, K., Tamura, A.: On circuit valuation of matroids. Adv. Appl. Math. 26, 192–225 (2001)

    Article  MathSciNet  Google Scholar 

  28. Osserman, B., Payne, S.: Lifting tropical intersections. Doc. Math. 18, 121–175 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Litvinov, G., Maslov, V. (eds.) Idempotent Mathematics and Mathematical Physics (Proceedings Vienna 2003), Contemporary Mathematics, vol. 377. American Mathematical Society, pp 289–317 (2005)

  30. Speyer, D.: Tropical linear spaces (2008). arXiv:0410455

  31. Yu, J., Yuster, D.S.: Representing tropical linear spaces by circuits. In: The 19th International Conference on Formal Power Series and Algebraic Combinatorics, 2007 (2006). arXiv:0611579

Download references

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.

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolai Vorobjov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolai Vorobjov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-019-00407-w

Keywords

Mathematics Subject Classification

Search

Navigation