The tropical Grassmannian. (English) Zbl 1065.14071

The paper makes a substantial step in the development of the tropical geometry, i.e., geometry over the tropical semiring, which is the reals equipped with operations of the ordinary addition and the minimum. This stuff has important links with applied mathematics problems as well as with the classical algebraic geometry. Tropical polynomials are real concave piece-wise linear functions, and tropical varieties are certain polyhedral complexes in real affine spaces. The tropical Grassmannian is defined as follows: one considers a classical Grassmannian, defined by Plücker equations in a torus over a field with a real non-Archimedean valuation, and takes the (closure of) the valuation projection to the respective real affine space (the so-called non-Archimedean amoeba). The authors establish basic properties of tropical Grassmannians, in particular, geometrically they are polyhedral fans, and they indeed parameterize the space of tropical linear subspaces of the respective dimension in a real space. A few examples are treated in details. So, the tropical Grassmannian parameterizing tropical lines is identified with the space of phylogenetic trees, some concrete Grassmannians are explicitly split into convex polyhedra. It is also shown that the tropical Grassmannian depends on the characteristic of the ground non-Archimedean field when considering spaces and subspaces of sufficiently high dimension.


14P99 Real algebraic and real-analytic geometry
55U10 Simplicial sets and complexes in algebraic topology
12J25 Non-Archimedean valued fields
15A39 Linear inequalities of matrices
16Y60 Semirings
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
13J30 Real algebra
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: DOI arXiv EuDML


[1] J.P. Barthelemy, A. Guenoche, Trees and proximity representations. Wiley-Interscience 1991.
[2] G., Trans. Amer. Math. Soc. 157 pp 459– (1971)
[3] Bernstein D., J. Algebraic Combin. 2 pp 111– (1993)
[4] Bieri R., J. Reine Angew. Math. 347 pp 168– (1984)
[5] Billera L. J., Adv. in Appl. Math. 27 pp 733– (2001)
[6] J. Combin. Theory Ser. 17 pp 48– (1974)
[7] M. Einsiedler, M. Kapranov, D. Lind, T. Ward, Non-archimedean amoebas.Preprint, 2003. · Zbl 1115.14051
[8] Fomin S., Foundations. J. Amer. Math. Soc. 15 pp 497– (2002)
[9] R. Math. Acad. Sci. Paris 336 pp 629– (2003)
[10] J.E. Pin, Tropical semirings.In: Idempotency (Bristol, 1994), volume 11 of Publ. Newton Inst., 50-69, Cambridge Univ. Press 1998.
[11] Robinson A., J. Pure Appl. Algebra 111 pp 245– (1996)
[12] B. Sturmfels, Algorithms in invariant theory. Springer 1993. · Zbl 0802.13002
[13] B. Sturmfels, Gr bner bases and convex polytopes, volume 8 of University Lecture Series. Amer. Math. Soc. 1996. · Zbl 0856.13020
[14] B. Sturmfels, Solving systems of polynomial equations, volume 97 of CBMS Regional Conference Series in Mathematics. Amer. Math. Soc. 2002. · Zbl 1101.13040
[15] Sturmfels B., Adv. Math. 98 pp 65– (1993)
[16] K. Vogtmann, Local structure of some Out Fn-complexes. Proc. Edinburgh Math. Soc. (2) 33 (1990), 367-379. · Zbl 0694.20021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.