The degree of a tropical basis

Joswig, Michael; Schröter, Benjamin

doi:10.1090/proc/13787

The degree of a tropical basis
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by Michael Joswig and Benjamin Schröter PDF

Proc. Amer. Math. Soc. 146 (2018), 961-970 Request permission

Abstract:

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

References

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
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, DOI 10.1016/j.jsc.2006.02.004
U. Betke and P. McMullen, Lattice points in lattice polytopes, Monatsh. Math. 99 (1985), no. 4, 253–265. MR 799674, DOI 10.1007/BF01312545
Andrew J. Chan and Diane Maclagan, Groebner bases over fields with valuations, 2013, Preprint arXiv:1303.0729.
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, DOI 10.1007/978-3-642-12971-1
Thomas W. Dubé, The structure of polynomial ideals and Gröbner bases, SIAM J. Comput. 19 (1990), no. 4, 750–775. MR 1053942, DOI 10.1137/0219053
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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
Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736–788 (German). MR 1512302, DOI 10.1007/BF01206635
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
Kerstin Hept and Thorsten Theobald, Tropical bases by regular projections, Proc. Amer. Math. Soc. 137 (2009), no. 7, 2233–2241. MR 2495256, DOI 10.1090/S0002-9939-09-09843-8
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.
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
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, DOI 10.1016/0001-8708(82)90048-2
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, DOI 10.1145/1837934.1837945
Thomas Markwig and Yue Ren, Computing tropical varieties over fields with valuation, 2016, Preprint arXiv:1612.01762.
Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI, 2015. MR 3287221, DOI 10.1090/gsm/161
David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), no. 3, 389–411. MR 2071813, DOI 10.1515/advg.2004.023
Bernd Sturmfels, Algorithms in invariant theory, 2nd ed., Texts and Monographs in Symbolic Computation, SpringerWienNewYork, Vienna, 2008. MR 2667486
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