Tropical cycles and Chow polytopes

The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety in a tropicalised toric variety. Several significant polyhedra associated to tropical varieties are special cases of our Chow polytope. The Chow polytope of a tropical variety $X$ is given by a simple combinatorial construction: its normal subdivision is the Minkowski sum of $X$ and a reflected skeleton of the fan of the ambient toric variety.Comment: 22 pp, 3 figs. Added discussion of arbitrary ambient toric varieties; several improvements suggested by Eric Katz; some rearrangemen

K-classes of matroids and equivariant localization

To every matroid, we associate a class in the K-theory of the Grassmannian. We study this class using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend results of the second author concerning the behavior of such classes under direct sum, series and parallel connection and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult.Comment: v2: added a starting point for combinatorialists in Section 2.4, + minor change

Valuations for matroid polytope subdivisions

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction

The binomial ideal of the intersection axiom for conditional probabilities

The binomial ideal associated with the intersection axiom of conditional probability is shown to be radical and is expressed as intersection of toric prime ideals. This resolves a conjecture in algebraic statistics due to Cartwright and Engstr\"om.Comment: 7 pages. Main proof rewritten to foreground equalities of Hilbert functions, for clarity; other minor change

Initial forms and a notion of basis for tropical differential equations

We show that solution sets of systems of tropical differential equations can be characterised in terms of monomial-freeness of an initial ideal. We discuss a candidate definition of tropical differential basis and give a nonexistence result for such bases in an example.Comment: 16 page