Computation of Dressians by dimensional reduction
-
Madeline Brandt
and
David E. Speyer
Abstract
We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions.
Funding statement: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1752814. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. The second author was supported in part by NSF grants DMS-1855135 and DMS-1854225.
Acknowledgements
We thank Yue Ren and Paul Görlach for their assistance in computing tropical prevarieties. We thank Alex Fink, Felipe Rincón, and Mariel Supina for several useful conversations. We thank our referees for careful reading and helpful comments on earlier versions of this paper. Finally, we thank Bernd Sturmfels for his comments and suggestions.
Communicated by: M. Joswig
References
[1] D. Bendle, J. Boehm, Y. Ren, B. Schröter, Parallel computation of tropical varieties, their positive part, and tropical Grassmannians. Preprint 2020, arXiv:2003.13752 [math.AG]Search in Google Scholar
[2] M. Brandt, Matroids and their Dressians. Preprint 2019, arXiv:1902.05592Search in Google Scholar
[3] A. W. M. Dress, W. Wenzel, Geometric algebra for combinatorial geometries. Adv. Math. 77 (1989), 1–36. MR1014071 Zbl 0684.0501310.1016/0001-8708(89)90013-3Search in Google Scholar
[4] A. W. M. Dress, W. Wenzel, On combinatorial and projective geometry. Geom. Dedicata 34 (1990), 161–197. MR1061287 Zbl 0701.0501510.1007/BF00147324Search in Google Scholar
[5] A. W. M. Dress, W. Wenzel, Valuated matroids. Adv. Math. 93 (1992), 214–250. MR1164708 Zbl 0754.0502710.1016/0001-8708(92)90028-JSearch in Google Scholar
[6] E. M. Feichtner, B. Sturmfels, Matroid polytopes, nested sets and Bergman fans. Port. Math. N.S.) 62 (2005), 437–468. MR2191630 Zbl 1092.52006Search in Google Scholar
[7] E. Gawrilow, M. Joswig, polymake: a framework for analyzing convex polytopes. In: Polytopes—combinatorics and computationOberwolfach, 1997), volume 29 of DMV Sem., 43–73, Birkhäuser, Basel 2000. MR1785292 Zbl 0960.6818210.1007/978-3-0348-8438-9_2Search in Google Scholar
[8] B. Grünbaum, Configurations of points and lines, volume 103 of Graduate Studies in Mathematics. Amer. Math. Soc. 2009. MR2510707 Zbl 1205.5100310.1090/gsm/103Search in Google Scholar
[9] K. Hept, T. Theobald, Tropical bases by regular projections. Proc. Amer. Math. Soc. 137 (2009), 2233–2241. MR2495256 Zbl 1165.1301310.1090/S0002-9939-09-09843-8Search in Google Scholar
[10] K. Hept, T. Theobald, Projections of tropical varieties and their self-intersections. Adv. Geom. 12 (2012), 203–228. MR2911147 Zbl 1308.1406710.1515/advgeom.2011.042Search in Google Scholar
[11] S. Herrmann, A. Jensen, M. Joswig, B. Sturmfels, How to draw tropical planes. Electron. J. Combin. 16 (2009), Research Paper 6, 26 pages. MR2515769 Zbl 1195.1408010.37236/72Search in Google Scholar
[12] S. Herrmann, M. Joswig, D. E. Speyer, Dressians, tropical Grassmannians, and their rays. Forum Math. 26 (2014), 1853–1881. MR3334049 Zbl 1308.1406810.1515/forum-2012-0030Search in Google Scholar
[13] A. Jensen, J. Sommars, J. Verschelde, Computing tropical prevarieties in parallel. In: PASCO 2017—International Workshop on Parallel Symbolic Computation, Art. No. 9, 8 unnumbered pages, ACM, New York 2017. MR376708410.1145/3115936.3115945Search in Google Scholar
[14] A. N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties. http://home.imf.au.dk/jensen/software/gfan/gfan.html.Search in Google Scholar
[15] M. Joswig, B. Schröter, Matroids from hypersimplex splits. J. Combin. Theory Ser. A 151 (2017), 254–284. MR3663497 Zbl 1366.0502410.1016/j.jcta.2017.05.001Search in Google Scholar
[16] S. Kantor, Ueber eine Configuration (3, 3)10 und unicursale Curven 4. Ordnung. Math. Ann. 21 (1883), 299–303. MR1510198 JFM 15.0551.0110.1007/BF01442925Search in Google Scholar
[17] J. P. S. Kung, H. Q. Nguyen, Weak maps. In White [25]: Theory of matroids, volume 26 of Encyclopedia Math. Appl., 254–271, Cambridge Univ. Press 1986. MR849398 Zbl 0596.0501510.1017/CBO9780511629563.012Search in Google Scholar
[18] D. Maclagan, F. Rincón, Tropical ideals. Compos. Math. 154 (2018), 640–670. MR3778187 Zbl 1428.1409310.1112/S0010437X17008004Search in Google Scholar
[19] D. Maclagan, B. Sturmfels, Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. Amer. Math. Soc. 2015. MR3287221 Zbl 1321.1404810.1090/gsm/161Search in Google Scholar
[20] J. A. Olarte, M. Panizzut, B. Schröter, On local Dressians of matroids. In: Algebraic and geometric combinatorics on lattice polytopes, 309–329, World Sci. Publ., Hackensack, NJ 2019. MR3971702 Zbl 1419.0503910.1142/9789811200489_0020Search in Google Scholar
[21] J. Oxley, Matroid theory. Oxford Univ. Press 2011. MR2849819 Zbl 1254.0500210.1093/acprof:oso/9780198566946.001.0001Search in Google Scholar
[22] D. Speyer, B. Sturmfels, The tropical Grassmannian. Adv. Geom. 4 (2004), 389–411. MR2071813 Zbl 1065.1407110.1515/advg.2004.023Search in Google Scholar
[23] D. E. Speyer, Tropical geometry. PhD thesis, Univ. California, Berkeley 2005 http://www-personal.umich.edu/∼speyer/thesis.pdf.Search in Google Scholar
[24] D. E. Speyer, Tropical linear spaces. SIAM J. Discrete Math. 22 (2008), 1527–1558. MR2448909 Zbl 1191.1407610.1137/080716219Search in Google Scholar
[25] N. White, editor, Theory of matroids, volume 26 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 1986. MR849389 Zbl 0579.00001Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
