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Register a new userMinimal half-spaces and external representation of tropical polyhedra
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We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.
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The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the natural candidates to be the maximizing polyhedra, which are the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on the sign pattern, from which we deduce explicit bounds for the number of extreme points, showing in particular that the upper bound is asymptotically tight as the dimension tends to infinity, keeping the number of constraints fixed. When transposed to the classical case, the previous constructions yield some lattice path generalizations of Gales evenness criterion.
After endowing the space of diagrams of probability spaces with an entropy distance, we study its large-scale geometry by identifying the asymptotic cone as a closed convex cone in a Banach space. We call this cone the tropical cone, and its elements tropical diagrams of probability spaces. Given that the tropical cone has a rich structure, while tropical diagrams are rather flexible objects, we expect the theory of tropical diagrams to be useful for information optimization problems in information theory and artificial intelligence. In a companion article, we give a first application to derive a statement about the entropic cone.
This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.
A biconvex polytope is a convex polytope that is also tropically convex. It is well known that every bounded cell of a tropical linear space is a biconvex polytope, but its converse has been a conjecture. We classify biconvex polytopes, and prove the conjecture by constructing a matroid subdivision dual to a biconvex polytope. In particular, we construct matroids from bipartite graphs, and establish the relationship between bipartite graphs and faces of a biconvex polytope. We also show that there is a bijection between monomials and a maximal set of vertices of a biconvex polytope.
We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okni{n}ski that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank $n$ is satisfied by the monoid of $n times n$ upper triangular tropical matrices. In particular this implies that the variety generated by the $3 times 3$ upper triangular tropical matrices coincides with that generated by the plactic monoid of rank $3$, answering another question of Izhakian.
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