The tropical Grassmannian

By David E. Speyer, Bernd Sturmfels. 2003

556 citations recorded by the source corpus.

In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Grobner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plucker relations. It parametrizes all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian G2; n equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians oer a natural generalization of the space of trees. Their faces correspond to monomial-free initial ideals of the Plucker ideal. The tropical Grassmannian G3; 6 is a simplicial complex glued from 1035 tetrahedra.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7649c85c9ff38dc13e1530331725ce2649e68174

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Citations in the corpus

On the frontiers of polynomial computations in tropical geometry (2004, background): “(However, we remark that Corollary 2.3 in [18], which claims that any universal Gröbner basis of I satisfies this condition, is not correct.”
The Tropical Totally Positive Grassmannian (2003, methodology): “In [ 18 ], it was shown that maximal cones of the fan Trop Gr2,n correspond to trivalent trees on n labelled leaves.”
On the tropical Torelli map (2009, background): “Though, the moduli space of n-pointed tropical rational curves M tr 0,n was constructed by different authors (see [56, 42,23,30]).”
Tropical and Ordinary Convexity Combined (2008, background): “More recently, polytropes occurred as the bounded intersections of apartments in Bruhat–Tits buildings of type Ãd, see Keel and Tevelev [16] or Joswig, Sturmfels, and Yu [15], as the inversion domains of Alessandrini [1], and as the max-plus definite closures of Sergeev [24].”
Moduli spaces of rational tropical curves (2007, unknown): “the related tropical Grassmannian picture in [7].”
Tropical Geometry and its applications (2006, unknown): “The semifield T is closely related to non-Archimedean fields K with (real) valuations val : K → T.”
Non-archimedean amoebas and tropical varieties (2004, background): “Then f t is a convex piecewise-linear function on R known as the tropicalization of f (see [21] and [24] for background).”
Cluster algebras (2014, unknown): “The tropical Grassmannian Trop Grk,n [29, 36, 57] is the space of realizable tropical linear spaces, obtained by applying the valuation map to elements of the Grassmannian Grk,n(K) over the field K = C{{t}} of Puiseux-series.”
Equations for Chow and Hilbert quotients (2007, methodology): “In [Speyer and Sturmfels 2004] it is shown that the tropical variety of AG0(2, n) is a (2n−3)dimensional fan in R( n 2) with lineality space L ⊗ R, and the simplicial complex corresponding to the fan structure on Trop(AG0(2, n)) is 1̃.”
Combinatorics of the tropical Torelli map (2010, background): “” Contrast this with the situation of moduli space M0,n of tropical rational curves with n marked points, constructed and studied in [Speyer and Sturmfels 2004; Mikhalkin 2007; Gathmann et al. 2009].”
Equations of tropical varieties (2013, background): “or if M is a (Laurent) polynomial algebra over F1 and f ∈M⊗Tis homogeneous of degree one, then by the preceding proposition bend (f)is a tropical hyperplane. Tropical hyperplanes were ﬁrst de ﬁned in [SS04] as the set-theoretic tropicalization of classical hyperplanes and later recast in terms of tr”
MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY (2014, background): “In [GM, Example 3.1], the locus of smooth curves M0,n is identified with the quotient of an open set of the Grassmannian, denoted G 0(2,n), by a an n − 1 dimensional torus.”
Analytification is the limit of all tropicalizations (2008, background): “The equivalence of (2)–(5) is standard; see [22] and [24].”
Compactifications of subvarieties of tori (2004, methodology): “If an auxiliary toric variety P is a projective space, Fvc is a tropical variety of Sturmfels and Speyer [ SS ], though not quite, as we explain below.”
Fibers of tropicalization (2007, background): “The problem of understanding fibers of tropicalization was discussed at a one-day workshop at MSRI, and I thank the organizers B. Sturmfels and E. Feichtner, as well as the other participants, for their shared insights and encouragement.”