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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.

MSC:

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
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