The degree of a tropical basis

Joswig, Michael; Schröter, Benjamin

doi:10.1090/proc/13787

The degree of a tropical basis

Authors: Michael Joswig and Benjamin Schröter
Journal: Proc. Amer. Math. Soc. 146 (2018), 961-970
MSC (2010): Primary 13P10, 14T05
DOI: https://doi.org/10.1090/proc/13787
Published electronically: October 5, 2017
MathSciNet review: 3750210
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. As an application -vectors of tropical varieties are discussed. Various examples illustrate differences between Gröbner and tropical bases.

[BG84] Robert Bieri and J. R. J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347 (1984), 168–195. MR 733052
[BJS$^+$07] T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas, Computing tropical varieties, J. Symbolic Comput. 42 (2007), no. 1-2, 54–73. MR 2284285, https://doi.org/10.1016/j.jsc.2006.02.004
[BM85] U. Betke and P. McMullen, Lattice points in lattice polytopes, Monatsh. Math. 99 (1985), no. 4, 253–265. MR 799674, https://doi.org/10.1007/BF01312545
[CM13] Andrew J. Chan and Diane Maclagan, Groebner bases over fields with valuations, 2013, Preprint arXiv:1303.0729.
[DLRS10] Jesús A. De Loera, Jörg Rambau, and Francisco Santos, Triangulations, Algorithms and Computation in Mathematics, vol. 25, Springer-Verlag, Berlin, 2010. Structures for algorithms and applications. MR 2743368
[Dub90] Thomas W. Dubé, The structure of polynomial ideals and Gröbner bases, SIAM J. Comput. 19 (1990), no. 4, 750–775. MR 1053942, https://doi.org/10.1137/0219053
[Eis95] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
[GJ00] Ewgenij Gawrilow and Michael Joswig, polymake: a framework for analyzing convex polytopes, Polytopes—combinatorics and computation (Oberwolfach, 1997) DMV Sem., vol. 29, Birkhäuser, Basel, 2000, pp. 43–73. MR 1785292
[Her26] Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736–788 (German). MR 1512302, https://doi.org/10.1007/BF01206635
[HJJS09] Sven Herrmann, Anders Jensen, Michael Joswig, and Bernd Sturmfels, How to draw tropical planes, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 6, 26. MR 2515769
[HT09] Kerstin Hept and Thorsten Theobald, Tropical bases by regular projections, Proc. Amer. Math. Soc. 137 (2009), no. 7, 2233–2241. MR 2495256, https://doi.org/10.1090/S0002-9939-09-09843-8
[Jen] Anders N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties, Available at http://home.imf.au.dk/jensen/software/gfan/gfan.html.
[LL91] Y. N. Lakshman and Daniel Lazard, On the complexity of zero-dimensional algebraic systems, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 217–225. MR 1106424
[MM82] Ernst W. Mayr and Albert R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in Math. 46 (1982), no. 3, 305–329. MR 683204, https://doi.org/10.1016/0001-8708(82)90048-2
[MR10] Ernst W. Mayr and Stephan Ritscher, Degree bounds for Gröbner bases of low-dimensional polynomial ideals, ISSAC 2010—Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2010, pp. 21–27. MR 2920532, https://doi.org/10.1145/1837934.1837945
[MR16] Thomas Markwig and Yue Ren, Computing tropical varieties over fields with valuation, 2016, Preprint arXiv:1612.01762.
[MS15] Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI, 2015. MR 3287221
[SS04] David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), no. 3, 389–411. MR 2071813, https://doi.org/10.1515/advg.2004.023
[Stu08] Bernd Sturmfels, Algorithms in invariant theory, 2nd ed., Texts and Monographs in Symbolic Computation, SpringerWienNewYork, Vienna, 2008. MR 2667486
[Zie00] Günter M. Ziegler, Lectures on 0/1-polytopes, Polytopes—combinatorics and computation (Oberwolfach, 1997) DMV Sem., vol. 29, Birkhäuser, Basel, 2000, pp. 1–41. MR 1785291