Tropical Computations in polymake
Chapter
First Online:
- 2 Citations
- 660 Downloads
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.
Keywords
Mathematical software Tropical hypersurfaces Tropical linear spacesSubject Classifications
14-04 (14T05 14Q99 52-04)Notes
Acknowledgements
We would like to thank Elizabeth Baldwin and Diane Maclagan for many helpful suggestions on improving this paper.
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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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:1412.3035Google 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)MathSciNetCrossRefGoogle Scholar
- 7.S. Brannetti, M. Melo, F. Viviani, On the tropical Torelli map. Adv. Math. 226(3), 2546–2586 (2011)MathSciNetCrossRefGoogle Scholar
- 8.S. Brodsky, M. Joswig, R. Morrison, B. Sturmfels, Moduli of tropical plane curves. Res. Math. Sci. 2(4), 1–31 (2015)MathSciNetzbMATHGoogle Scholar
- 9.E. Brugallé, K. Shaw, Obstructions to approximating tropical curves in surfaces via intersection theory. Canad. J. Math. 67(3), 527–572 (2015)MathSciNetCrossRefGoogle Scholar
- 10.W. Castryck, J. Voight, On nondegeneracy of curves. Algebra Number Theory 3(3), 255–281 (2009)MathSciNetCrossRefGoogle 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. 205–206Google Scholar
- 14.A.W.M. Dress, W. Wenzel, Valuated matroids. Adv. Math. 93(2), 214–250 (1992)MathSciNetCrossRefGoogle 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. 69–87Google Scholar
- 16.A. Fink, Tropical cycles and chow polytopes. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry 54(1), 13–40 (2013)MathSciNetCrossRefGoogle Scholar
- 17.G. François, J. Rau, The diagonal of tropical matroid varieties and cycle intersections. Collect. Math. 64(2), 185–210 (2013)MathSciNetCrossRefGoogle Scholar
- 18.W. Fulton, B. Sturmfels, Intersection theory on toric varieties. Topology 36(2), 335–353 (1997)MathSciNetCrossRefGoogle 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. 241–254Google Scholar
- 20.B. Ganter, K. Reuter, Finding all closed sets: a general approach. Order 8(3), 283–290 (1991)MathSciNetCrossRefGoogle 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. 219–221Google 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)MathSciNetCrossRefGoogle Scholar
- 23.J. Giansiracusa, N. Giansiracusa, Equations of tropical varieties. Duke Math. J. 165(18), 3379–3433 (2016)MathSciNetCrossRefGoogle 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:1508.01179Google Scholar
- 26.S. Hampe, a-tint: a polymake extension for algorithmic tropical intersection theory. Eur. J. Combin. 36, 579–607 (2014)MathSciNetCrossRefGoogle Scholar
- 27.S. Hampe, The intersection ring of matroids. J. Comb. Theory Ser. B 122, 578–614 (2017)MathSciNetCrossRefGoogle Scholar
- 28.S. Herrmann, On the facets of the secondary polytope. J. Comb. Theory Ser. A 118(2), 425–447 (2011)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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. 429–445CrossRefGoogle 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. 29–110Google 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)CrossRefGoogle Scholar
- 38.D. Maclagan, F. Rincón, Tropical schemes, tropical cycles, and valuated matroids (2014, preprint), arXiv:1401.4654Google 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)MathSciNetCrossRefGoogle Scholar
- 44.Q. Ren, K. Shaw, B. Sturmfels, Tropicalization of del Pezzo surfaces. Adv. Math. 300, 156–189 (2016)MathSciNetCrossRefGoogle Scholar
- 45.F. Rincón, Computing tropical linear spaces. J. Symb. Comput. 51, 86–98 (2013)MathSciNetCrossRefGoogle Scholar
- 46.A. Schrijver, Combinatorial Optimization. Polyhedra and Efficiency. Vol. A. Algorithms and Combinatorics, vol. 24 (Springer, Berlin, 2003). Paths, flows, matchings, Chapters 1–38Google Scholar
- 47.K.M. Shaw, A tropical intersection product in matroidal fans. SIAM J. Discret. Math. 27(1), 459–491 (2013)MathSciNetCrossRefGoogle Scholar
- 48.D.E. Speyer, Tropical linear spaces. SIAM J. Discret. Math. 22(4), 1527–1558 (2008)MathSciNetCrossRefGoogle Scholar
- 49.D. Speyer, B. Sturmfels, The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
- 52.N.M. Tran, J. Yu, Product-mix auctions and tropical geometry (2015, preprint), arXiv:1505.05737Google Scholar
- 53.M.D. Vigeland, Tropical lines on smooth tropical surfaces (2007, preprint), arXiv:0708.3847Google 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)MathSciNetzbMATHGoogle Scholar
Copyright information
© Springer International Publishing AG, part of Springer Nature 2017