We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B'(K_n) of the graphical matroid of the complete graph K_n is homeomorphic to the space of phylogenetic trees T_n.
The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.
We give an explicit description of the poset of cells of Bergman complexes of Lattice Path Matroids and establish a criterion for its simpliciality, in terms of the shape of the bounding paths.
Bergman complexes are polyhedral complexes associated to matroids. Faces of these complexes are certain matroids, called matroid types, too. In order to understand the structure of these faces we decompose matroid types into direct summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be subdivided into the order complex of the proper part of its lattice of flats. Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be subdivided to the even coarser nested set complex. We will give a much shorter and more general proof of this fact. Generalizing formulas…
We study the positive Bergman complex B+(M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid. The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid M_I, the positive tropical variety associated to I is equal to the fan over B+(M_I). Our main result is that a certain "fine" subdivision of B+(M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of M. It follows that B+(M) is homeomorphic to a sphere. For the oriented matroid of…
The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex. It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation …
Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi, then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi) equals the nested set complex of A. In addition, we prove that for any orientable matroid M, one can find |mu(M)| different reorientations of …
Given a dissimilarity map $δ$ on finite set $X$, the set of ultrametrics (equidistant tree metrics) which are $l^\infty$-nearest to $δ$ is a tropical polytope. We give an internal description of this tropical polytope which we use to derive a polynomial-time checkable test for the condition that all ultrametrics $l^\infty$-nearest to $δ$ have the same tree structure. It was shown by Ardila and Klivans \cite{ardila-klivans2006} that the set of all ultrametrics on a finite set of size $n$ is the Bergman fan associated to the matroid underlying the complete graph on $n$ vertices. Therefore, we de…
Given a matroid $M$ one can define its Orlik-Solomon algebra $OS(M)$ and the Bergman fan $Σ_0(M)$. On the other hand to any rational polyhedral fan $Σ$ one can associate its tropical homology and cohomology groups $\F_\bullet(Σ)$, $\F^\bullet (Σ)$. We will show that the projective Orlik-Solomon algebra $OS_0(M)$ is canonically isomorphic to $\F^\bullet (Σ_0(M))$.
This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main theorems. First, we exhibit a matroid quotient involving geometric duals of networks embedded in surfaces with boundary. Second, we characterize the Bergman fans of Dirichlet matroids as subfans of graphic Bergman fans. Third, we prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the associated chromatic polynomial.
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman c…
Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters ar…
In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress towards a more general conjecture of Rota-Heron-Welsh. Our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety. The log-concavity then follows from an inequality of Hodge type.
A (pseudo-)metric $D$ on a finite set $X$ is said to be a `tree metric' if there is a finite tree with leaf set $X$ and non-negative edge weights so that, for all $x,y \in X$, $D(x,y)$ is the path distance in the tree between $x$ and $y$. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is -- up to canonical isomorphism -- uniquely determined by $D$, and one does not even need all of the distances in order to fully (re…
We study the augmented Bergman complex of a closure operator on a finite set, which interpolates between the order complex of proper flats and the independence complex of the operator. In 2020, Braden, Huh, Matherne, Proudfoot, and Wang showed that augmented Bergman complexes of matroids are always gallery-connected, and recently Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun, Tao, and Zhang strengthened "gallery-connected" to "shellable" by providing two classes of shelling orders: "flag-to-basis" shellings and "basis-to-flag" shellings. We show that augmented Bergman complexes of matroids ar…
Given a matroid M on the ground set E, the Bergman fan B(M), or space of M-ultrametrics, is a polyhedral complex in R^E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point w in R^E, we wish to find an M-ultrametric which is closest to it in the l_infty metric. The solution to this problem follows easily from the existence of the subdominant M-ultrametric: a componentwise maximum M-ultrametric which is componentwise smaller than w. A procedure fo…
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 we…
We investigate uniqueness issues that arise in $l^\infty$-optimization to linear spaces and Bergman fans of matroids. For linear spaces, we give a polyhedral decomposition of $\mathbb{R}^n$ based on the dimension of the set of $l^\infty$-nearest neighbors. This implies that the $l^\infty$-nearest neighbor in a linear space is unique if and only if the underlying matroid is uniform. For Bergman fans of matroids, we show that the set of $l^\infty$-nearest points is a tropical polytope and give an algorithm to compute its tropical vertices. A key ingredient here is a notion of topology that gener…
The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total positivity. Matroids capture the combinatorial essence that these different settings share. In recent years, the (classical, polyhedral, algebraic, and tropical) geometric roots of the field have grown much deeper, bearing new fruits. We survey some recent successes, stemming from three geometric models of a matroid: the matroid polytope, the Bergman fan, a…
To each prime ideal in a polynomial ring over a field we associate an algebraic matroid and show that it is preserved under tropicalization. This gives a necessary condition for a tropical variety to be set-theoretically realizable from a prime ideal. We also show that there are infinitely many Bergman fans that are not set-theoretically realizable as the tropicalization of any ideal.
Given a distance matrix consisting of pairwise distances between species, a distance-based phylogenetic reconstruction method returns a tree metric or equidistant tree metric (ultrametric) that best fits the data. We investigate distance-based phylogenetic reconstruction using the $l^\infty$-metric. In particular, we analyze the set of $l^\infty$-closest ultrametrics and tree metrics to an arbitrary dissimilarity map to determine its dimension and the tree topologies it represents. In the case of ultrametrics, we decompose the space of dissimilarity maps on 3 elements and on 4 elements relativ…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be representable. Still, one may apply linear algebraic constructions to non-representable matroids. There are a number of different definitions of matroids, a phenomenon known as cryptomorphism. In this survey, we begin by reviewing the classical definitions of matroids, develop operations in matroid theory, summarize some results in representability, and construc…
We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M. This allows us to express the h-vector of the broken circuit complex of M in terms of the intersection theory of the conormal fan of M. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. The Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M, when com…
We define and study the cyclic Bergman fan of a matroid M, which is a simplicial polyhedral fan supported on the tropical linear space T(M) of M and is amenable to computational purposes. It slightly refines the nested set structure on T(M), and its rays are in bijection with flats of M which are either cyclic flats or singletons. We give a fast algorithm for calculating it, making some computational applications of tropical geometry now viable. Our C++ implementation, called TropLi, and a tool for computing vertices of Newton polytopes of A-discriminants, are both available online.
