Skip to main content

Complexity in Tropical Algebra (Invited Talk)

  • Conference paper
Computer Algebra in Scientific Computing (CASC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8136))

Included in the following conference series:

  • 914 Accesses

Abstract

We give a survey on complexity results in tropical algebra.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Butkovic, P.: Max-linear systems: theory and algorithms. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  2. Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical algebraic geometry. In: Oberwolfach Seminars, Birkhäuser, Basel (2009)

    Google Scholar 

  3. Akian, M., Gaubert, S., Guterman, A.: Tropical polyhedra are equivalent to mean payoff games. Int. J. Algebra Comput. 22, 1793–1835 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Davydow, A.: Upper and lower bounds for Grigoriev’s algorithm for solving integral tropical linear systems. Zap. Nauchn. Sem. POMI St.Petersbourg 402, 69–82 (2012)

    Google Scholar 

  6. Grigoriev, D., Podolskii, V.: Complexity of tropical and min-plus prevarieties. Preprint MPIM, 2013-23 (2012)

    Google Scholar 

  7. Develin, M., Santos, F., Sturmfels, B.: On the rank of a tropical matrix. Math. Sci. Res. Inst. Publ. 52, 213–242 (2005)

    MathSciNet  Google Scholar 

  8. Butkovic, P., Hevery, F.: A condition for the strong regularity of matrices in the minimax algebra. Discr. Appl. Math. 11, 209–222 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kim, K., Roush, F.: Factorization of polynomials in one variable over the tropical semiring. arXiv:math/050116/v2

    Google Scholar 

  10. Kim, K., Roush, F.: Kapranov rank vs. tropical rank. Proc. Amer. Math. Soc. 134, 2487–2494 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Izhakian, Z., Rowen, L.: The tropical rank of a tropical matrix. Communic. Algebra 37, 3912–3927 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Allamigeon, X., Gaubert, S., Katz, R.: Tropical polar cones, hypergraph transversals, and mean payoff games. Lin. Alg. and Its Appl. 435, 1549–1574 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Butkovic, P., Hegedus̈, G.: An elimination method for finding all solutions of the system of linear equations over an extremal algebra. Ekonom.-Mat. Obzor 20, 203–215 (1984)

    MathSciNet  Google Scholar 

  14. Bezem, M., Nieuwenhuis, R., Rodriguez-Carbonell, E.: Hard problem in max-algebra, control theory, hypergraphs and other areas. Inf. Procss. Lett. 110, 133–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Theobald, T.: On the frontiers of polynomial computations in tropical geometry. J. Symbolic Comput. 41, 1360–1375 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grigoriev, D., Shpilrain, V.: Tropical cryptography. Preprint MPIM, Bonn (2011)

    Google Scholar 

  17. Grigoriev, D.: On a tropical dual Nullstellensatz. Adv. Appl. Math. 48, 457–464 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Giusti, M., Heintz, J., Sabia, J.: On the efficiency of effective Nullstellensaetze. Comput. Complexity 3, 56–95 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kollár, J.: Sharp effective Nullstellensatz. J. Amer. Math. Soc. 1, 963–975 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grigoriev, D., Podolskii, V.: Tropical dual effective Nullstellensatz (in preparation)

    Google Scholar 

  21. Tabera, L.: Tropical resultants for curves and stable intersection. Rev. Mat. Iberoam. 24, 941–961 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Izhakian, Z.: Tropical algebraic sets, ideals and an algebraic Nullstellensatz. Internat. J. Algebra Comput. 18, 1067–1098 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shustin, E., Izhakian, Z.: A tropical Nullstellensatz. Proc. Amer. Math. Soc. 135, 3815–3821 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this paper

Cite this paper

Grigoriev, D. (2013). Complexity in Tropical Algebra (Invited Talk). In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2013. Lecture Notes in Computer Science, vol 8136. Springer, Cham. https://doi.org/10.1007/978-3-319-02297-0_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-02297-0_13

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02296-3

  • Online ISBN: 978-3-319-02297-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics