Skip to main content
SpringerLink
Log in

Tropical Computations in polymake

  • Chapter
  • First Online:
Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

Abstract

We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of tropical plane curves, tropical linear spaces and Grassmannians, lines on tropical cubic surfaces as well as intersection rings of matroids.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. X. Allamigeon, TPLib (Tropical Polyhedra Library) (2013), http://www.cmap.polytechnique.fr/~allamigeon/software/

    Google Scholar 

  2. L. Allermann, J. Rau, First steps in tropical intersection theory. Math. Z. 264(3), 633–670 (2010)

    Article  MathSciNet  Google Scholar 

  3. F. Ardila, C.J. Klivans, The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96(1), 38–49 (2006)

    Article  MathSciNet  Google Scholar 

  4. E. Baldwin, P. Klemperer, Understanding preferences: “Demand Types”, and the existence of equilibrium with indivisibilities, Nuffield College, Working Paper (2015)

    Google Scholar 

  5. A.L. Birkmeyer, A. Gathmann, Realizability of tropical curves in a plane in the non-constant coefficient case (2014, Preprint), arXiv:1412.3035

    Google Scholar 

  6. T. Bogart, E. Katz, Obstructions to lifting tropical curves in surfaces in 3-space. SIAM J. Discret. Math. 26(3), 1050–1067 (2012)

    Article  MathSciNet  Google Scholar 

  7. S. Brannetti, M. Melo, F. Viviani, On the tropical Torelli map. Adv. Math. 226(3), 2546–2586 (2011)

    Article  MathSciNet  Google Scholar 

  8. S. Brodsky, M. Joswig, R. Morrison, B. Sturmfels, Moduli of tropical plane curves. Res. Math. Sci. 2(4), 1–31 (2015)

    MathSciNet  MATH  Google Scholar 

  9. E. Brugallé, K. Shaw, Obstructions to approximating tropical curves in surfaces via intersection theory. Canad. J. Math. 67(3), 527–572 (2015)

    Article  MathSciNet  Google Scholar 

  10. W. Castryck, J. Voight, On nondegeneracy of curves. Algebra Number Theory 3(3), 255–281 (2009)

    Article  MathSciNet  Google Scholar 

  11. J.A. De Loera, J. Rambau, F. Santos, Triangulations. Structures for Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 25 (Springer, Berlin, 2010)

    Google Scholar 

  12. W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 4-1-0 – A computer algebra system for polynomial computations (2016), http://www.singular.uni-kl.de

    Google Scholar 

  13. M. Develin, B. Sturmfels, Tropical convexity. Doc. Math. 9, 1–27 (electronic) (2004). Correction: ibid., pp. 205–206

    Google Scholar 

  14. A.W.M. Dress, W. Wenzel, Valuated matroids. Adv. Math. 93(2), 214–250 (1992)

    Article  MathSciNet  Google Scholar 

  15. J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) (Gordon and Breach, New York, 1970), pp. 69–87

    Google Scholar 

  16. A. Fink, Tropical cycles and chow polytopes. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry 54(1), 13–40 (2013)

    Article  MathSciNet  Google Scholar 

  17. G. François, J. Rau, The diagonal of tropical matroid varieties and cycle intersections. Collect. Math. 64(2), 185–210 (2013)

    Article  MathSciNet  Google Scholar 

  18. W. Fulton, B. Sturmfels, Intersection theory on toric varieties. Topology 36(2), 335–353 (1997)

    Article  MathSciNet  Google Scholar 

  19. B. Ganter, Algorithmen zur formalen Begriffsanalyse, ed. by B. Ganter, R. Wille, K.E. Wolff. Beiträge zur Begriffsanalyse (Bibliographisches Institut, Mannheim, 1987), pp. 241–254

    Google Scholar 

  20. B. Ganter, K. Reuter, Finding all closed sets: a general approach. Order 8(3), 283–290 (1991)

    Article  MathSciNet  Google Scholar 

  21. E. Gawrilow, M. Joswig, Flexible object hierarchies in polymake, in Proceedings of the 2nd International Congress of Mathematical Software, Castro Urdiales, Spanien, 1–3 Sept 2006, ed. by A. Igelesias, N. Takayama (2006), pp. 219–221

    Google Scholar 

  22. I.M. Gel’fand, M. Goresky, R.D. MacPherson, V.V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(3), 301–316 (1987)

    Article  MathSciNet  Google Scholar 

  23. J. Giansiracusa, N. Giansiracusa, Equations of tropical varieties. Duke Math. J. 165(18), 3379–3433 (2016)

    Article  MathSciNet  Google Scholar 

  24. D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry (2017), available at http://www.math.uiuc.edu/Macaulay2/

    Google Scholar 

  25. W. Gubler, J. Rabinoff, A. Werner, Tropical skeletons (2015, preprint), arXiv:1508.01179

    Google Scholar 

  26. S. Hampe, a-tint: a polymake extension for algorithmic tropical intersection theory. Eur. J. Combin. 36, 579–607 (2014)

    Article  MathSciNet  Google Scholar 

  27. S. Hampe, The intersection ring of matroids. J. Comb. Theory Ser. B 122, 578–614 (2017)

    Article  MathSciNet  Google Scholar 

  28. S. Herrmann, On the facets of the secondary polytope. J. Comb. Theory Ser. A 118(2), 425–447 (2011)

    Article  MathSciNet  Google Scholar 

  29. S. Herrmann, A. Jensen, M. Joswig, B. Sturmfels, How to draw tropical planes. Electron. J. Comb. 16(2), Special volume in honor of Anders Björner, Research Paper 6, 26 (2009)

    Google Scholar 

  30. A.N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties, version 0.6, available at http://home.imf.au.dk/jensen/software/gfan/gfan.html (2017)

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

    Article  MathSciNet  Google Scholar 

  32. A.N. Jensen, H. Markwig, T. Markwig, Y. Ren, tropical.lib. a Singular 4-1-0 library for computations in tropical goemetry, Tech. report (2016)

    Google Scholar 

  33. M. Joswig, Tropical convex hull computations, in Tropical and Idempotent Mathematics, ed. by G.L. Litvinov, S.N. Sergeev. Contemporary Mathematics, vol. 495 (American Mathematical Society, Providence, RI, 2009)

    Google Scholar 

  34. M. Joswig, G. Loho, B. Lorenz, B. Schröter, Linear programs and convex hulls over fields of Puiseux fractions, in Proceedings of MACIS 2015, LNCS 9582, Berlin, 11–13 Nov 2015 (2016), pp. 429–445

    Chapter  Google Scholar 

  35. M.M. Kapranov, Chow Quotients of Grassmannians. I, ed. by I.M. Gel’ fand Seminar. Advances in Soviet Mathematics, vol. 16 (American Mathematical Society, Providence, RI, 1993), pp. 29–110

    Google Scholar 

  36. P. Klemperer, A new auction for substitutes: central bank liquidity auctions, the U.S. TARP, and variable product-mix auctions, University of Oxford, Working Paper (2008)

    Google Scholar 

  37. P. Klemperer, The product-mix auction: a new auction design for differentiated goods. J. Eur. Econ. Assoc. 8, 526–536 (2010)

    Article  Google Scholar 

  38. D. Maclagan, F. Rincón, Tropical schemes, tropical cycles, and valuated matroids (2014, preprint), arXiv:1401.4654

    Google Scholar 

  39. D. Maclagan, B. Sturmfels, Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161 (American Mathematical Society, Providence, RI, 2015)

    Google Scholar 

  40. G. Mikhalkin, Enumerative tropical algebraic geometry in \(\mathbb {R}^2\). J. Am. Math. Soc. 18(2), 313–377 (2005)

    Google Scholar 

  41. G. Mikhalkin, J. Rau, Tropical geometry, work in progress, available at https://www.math.uni-tuebingen.de/user/jora/downloads/main.pdf (2015)

  42. J. Oxley, Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. (Oxford University Press, Oxford, 2011)

    Google Scholar 

  43. Y. Ren, Computing tropical varieties over fields with valuation using classical standard basis techniques. ACM Commun. Comput. Algebra 49(4), 127–129 (2015)

    Article  MathSciNet  Google Scholar 

  44. Q. Ren, K. Shaw, B. Sturmfels, Tropicalization of del Pezzo surfaces. Adv. Math. 300, 156–189 (2016)

    Article  MathSciNet  Google Scholar 

  45. F. Rincón, Computing tropical linear spaces. J. Symb. Comput. 51, 86–98 (2013)

    Article  MathSciNet  Google Scholar 

  46. A. Schrijver, Combinatorial Optimization. Polyhedra and Efficiency. Vol. A. Algorithms and Combinatorics, vol. 24 (Springer, Berlin, 2003). Paths, flows, matchings, Chapters 1–38

    Google Scholar 

  47. K.M. Shaw, A tropical intersection product in matroidal fans. SIAM J. Discret. Math. 27(1), 459–491 (2013)

    Article  MathSciNet  Google Scholar 

  48. D.E. Speyer, Tropical linear spaces. SIAM J. Discret. Math. 22(4), 1527–1558 (2008)

    Article  MathSciNet  Google Scholar 

  49. D. Speyer, B. Sturmfels, The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)

    Article  MathSciNet  Google Scholar 

  50. B. Sturmfels, Algorithms in Invariant Theory. Texts and Monographs in Symbolic Computation, 2nd edn. (Springer, New York, 2008)

    Google Scholar 

  51. J.A. Thas, H. Van Maldeghem, Embeddings of small generalized polygons. Finite Fields Appl. 12(4), 565–594 (2006)

    Article  MathSciNet  Google Scholar 

  52. N.M. Tran, J. Yu, Product-mix auctions and tropical geometry (2015, preprint), arXiv:1505.05737

    Google Scholar 

  53. M.D. Vigeland, Tropical lines on smooth tropical surfaces (2007, preprint), arXiv:0708.3847

    Google Scholar 

  54. N. White (ed.), Theory of Matroids. Encyclopedia of Mathematics and Its Applications, vol. 26 (Cambridge University Press, Cambridge, 1986)

    Google Scholar 

  55. I. Zharkov, The Orlik-Solomon algebra and the Bergman fan of a matroid. J. Gökova Geom. Topol. GGT 7, 25–31 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Elizabeth Baldwin and Diane Maclagan for many helpful suggestions on improving this paper.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Sekretariat MA 6-2, Straß des 17. Juni 136, 10623, Berlin, Germany

    Simon Hampe & Michael Joswig

Authors
  1. Simon Hampe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Joswig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Joswig .

Editor information

Editors and Affiliations

  1. IWR, Heidelberg University, Heidelberg, Germany

    Gebhard Böckle

  2. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Wolfram Decker

  3. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Gunter Malle

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Hampe, S., Joswig, M. (2017). Tropical Computations in polymake . In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_14

Download citation

Publish with us

Policies and ethics

Search

Navigation