18 research outputs found
Isotropical Linear Spaces and Valuated Delta-Matroids
The spinor variety is cut out by the quadratic Wick relations among the
principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to
n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this
paper we tropicalize this picture, and we develop a combinatorial theory of
tropical Wick vectors and tropical linear spaces that are tropically isotropic.
We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid
polytopes, and we examine to what extent the Wick relations form a tropical
basis. Our theory generalizes several results for tropical linear spaces and
valuated matroids to the class of Coxeter matroids of type D
Isotropical Linear Spaces and Valuated Delta-Matroids
The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an skew-symmetric matrix. Its points correspond to -dimensional isotropic subspaces of a -dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type
Representability of orthogonal matroids over partial fields
Let be nonnegative integers, and let .
For a matroid of rank on the finite set and a partial field
in the sense of Semple--Whittle, it is known that the following are
equivalent: (a) is representable over ; (b) there is a point with support (meaning that of is the set of bases of )
satisfying the Grassmann-Pl\"ucker equations; and (c) there is a point with support satisfying just the 3-term
Grassmann-Pl\"ucker equations. Moreover, by a theorem of P. Nelson, almost all
matroids (meaning asymptotically 100%) are not representable over any partial
field. We prove analogues of these facts for Lagrangian orthogonal matroids in
the sense of Gelfand-Serganova, which are equivalent to even Delta-matroids in
the sense of Bouchet.Comment: 13 page
Local tropical linear spaces
In this paper we study general tropical linear spaces locally: For any basis
B of the matroid underlying a tropical linear space L, we define the local
tropical linear space L_B to be the subcomplex of L consisting of all vectors v
that make B a basis of maximal v-weight. The tropical linear space L can then
be expressed as the union of all its local tropical linear spaces, which we
prove are homeomorphic to Euclidean space. Local tropical linear spaces have a
simple description in terms of polyhedral matroid subdivisions, and we prove
that they are dual to mixed subdivisions of Minkowski sums of simplices. Using
this duality we produce tight upper bounds for their f-vectors. We also study a
certain class of tropical linear spaces that we call conical tropical linear
spaces, and we give a simple proof that they satisfy Speyer's f-vector
conjecture.Comment: 13 pages, 1 figure. Some results are stated in a bit more generality.
Minor corrections were also mad
Tropical ideals do not realise all Bergman fans
Every tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one
Tropical Ideals
International audienceWe introduce and study a special class of ideals over the semiring of tropical polynomials, which we calltropical ideals, with the goal of developing a useful and solid algebraic foundation for tropical geometry. We exploretheir rich combinatorial structure, and prove that they satisfy numerous properties analogous to classical ideals
Tropical Ideals
We introduce and study a special class of ideals over the semiring of tropical polynomials, which we calltropical ideals, with the goal of developing a useful and solid algebraic foundation for tropical geometry. We exploretheir rich combinatorial structure, and prove that they satisfy numerous properties analogous to classical ideals