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Computing tropical varieties. (English) Zbl 1121.14051
Summary: The tropical variety of a \(d\)-dimensional prime ideal in a polynomial ring with complex coefficients is a pure \(d\)-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gröbner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.

14Q99 Computational aspects in algebraic geometry
14P99 Real algebraic and real-analytic geometry
68W30 Symbolic computation and algebraic computation
cdd; Gfan; gmp
Full Text: DOI arXiv
[1] Ardila, Federico; Klivans, Caroline, The Bergman complex of a matroid and phylogenetic trees, J. combin. theory ser. B, 96, 38-49, (2006) · Zbl 1082.05021
[2] Bergman, George, The logarithmic limit set of an algebraic variety, Trans. amer. math. soc., 157, 459-469, (1971) · Zbl 0197.17102
[3] Bieri, Robert; Groves, J.R.J., The geometry of the set of characters induced by valuations, J. reine angew. math., 347, 168-195, (1984) · Zbl 0526.13003
[4] Collart, Stéphane; Kalkbrener, Michael; Mall, Daniel, Converting bases with the Gröbner walk, J. symbolic comput., 24, 465-469, (1997) · Zbl 0908.13020
[5] Einsiedler, Manfred, Kapranov, Mikhail, Lind, Douglas, Non-archimedean amoebas and tropical varieties. J. Reine Angew. Math. (in press) · Zbl 1115.14051
[6] Eisenbud, David, Commutative algebra with a view toward algebraic geometry, Graduate texts in mathematics, vol. 150, (1995), Springer-Verlag New York · Zbl 0819.13001
[7] Fukuda, Komei, 2005. cddlib reference manual, cddlib Version 093b. EPFL Lausanne and ETH Zürich, http:www.ifor.math.ethz.ch/ fukuda/cdd_home/cdd.htm
[8] Granlund, Torbjorn, et al. september 2004. gnu multiple precision arithmetic library 4.1.4. Available from http://swox.com/gmp/
[9] Jensen, Anders, 2005. Gfan — a software system for Gröbner fans. Available athttp://home.imf.au.dk/ajensen/software/gfan/gfan.html
[10] Hartshorne, Robin, Algebraic geometry, () · Zbl 0367.14001
[11] Mikhalkin, Grigory, Enumerative tropical algebraic geometry in \(\mathbb{R}^2\), J. amer. math. soc., 18, 313-377, (2005) · Zbl 1092.14068
[12] Richter-Gebert, Jürgen; Sturmfels, Bernd; Theobald, Thorsten, First steps in tropical geometry, (), 289-317 · Zbl 1093.14080
[13] Mora, Teo; Robbiano, Lorenzo, The Gröbner Fan of an ideal, J. symbolic comput., 6, 183-208, (1988) · Zbl 0668.13017
[14] Saito, Mutsumi; Sturmfels, Bernd; Takayama, Nobuki, Gröbner deformations of hypergeometric differential equations, (2000), Springer-Verlag · Zbl 0946.13021
[15] Speyer, David; Sturmfels, Bernd, The tropical Grassmannian, Adv. geom., 4, 389-411, (2004) · Zbl 1065.14071
[16] Speyer, David, Sturmfels, Bernd, July 2004b. Tropical mathematics. Clay Institute lecture at Park City, Utah. math.CO/0408099
[17] Sturmfels, Bernd, ()
[18] Sturmfels, Bernd, ()
[19] Theobald, Thorsten, On the frontiers of polynomial computations in tropical geometry. J. Symbolic Comput. (in press). math.CO/0411012 · Zbl 1121.14047
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