Cyclic Bergman Fan of a Matroid

• Bergman fan of a matroid is a rational, balanced, simplicial fan that encodes the combinatorial data of cyclic flats and singletons.
• Its cyclic refinement enhances computational methods, enabling the use of algorithms like TropLi for effective tropical and polyhedral analysis.
• The structure refines nested subdivisions and supports applications in tropical intersection theory, matroid invariants, and Newton polytope computations.

A Bergman fan of a matroid, and, in particular, its cyclic refinement, constitutes a central polyhedral structure in tropical geometry and matroid theory. For a matroid $M$ of rank $r$ on ground set $E=[n]$, the (cyclic) Bergman fan is a rational, simplicial, balanced fan supported on the tropical linear space $T(M)$, encoding the combinatorial data of the cyclic flats or singletons of $M$. Beyond its foundational combinatorial and geometric character, the cyclic Bergman fan enables effective computation in tropical intersection theory, matroid base polytopes, and the analysis of Newton polytopes of $A$-discriminants, with algorithmic realizations allowing large-scale implementation.

1. Tropical Linear Space and Fan Structures

Given a loopless, coloopless matroid $M$ of rank $r$ on $E=[n]$, its tropical linear space is

$$T(M)\;=\;\{\,w\in\Bbb R^n : \min_{i\in C} w_i \;\text{is attained at least twice for every circuit } C\subseteq E\,\}.$$

This locus is a pure, balanced polyhedral fan of dimension $r$, and is the tropicalization of the row space of any matrix realizing $M$ over $\mathbb{C}$ with the trivial valuation.

Several standard polyhedral structures are defined on $T(M)$ by refining the subdivision corresponding to chains (or nested sets) of flats:

• The Bergman fan is the coarsest such structure, with rays indexed by indicator vectors $e_F$ for flacets $F$, i.e., flats such that $M|F$ and $M/F$ are connected.
• The nested set fan refines this, with rays indexed by $e_F$ for all connected flats $F$.
• The fine subdivision distinguishes all flats and has cones described by explicit chains of flats. The cyclic Bergman fan $\Phi(M)$ refines the nested set fan slightly, with cones and rays reflecting the additional structure of the cyclic (circuit-union) flats (Rincón, 2011).

2. Cyclic Flats and the Cyclic Bergman Fan

A flat $F \subseteq E$ is cyclic if it is a union of circuits of $M$, equivalently $E\setminus F$ is a flat of the dual matroid $M^*$. The cyclic Bergman fan $\Phi(M)$ is a simplicial polyhedral fan supported on $T(M)$, whose rays are exactly the indicator vectors $e_F$ where $F$ is either a singleton or a cyclic flat.

Maximal cones of $\Phi(M)$ are indexed by regressive compatible pairs $(p,L)$ associated to a choice of basis $B$ of $M$. For each $k\in E\setminus B$, there is a fundamental circuit $C(k,B)$, and the regressive compatible pair encodes a function $p:E\setminus B \rightarrow B$ (with $p(k)\in C(k,B)\setminus\{k\}$, $p(k)<k$), together with a total order $L$ on $p(E\setminus B)$. The resulting maximal cones are spanned by indicator vectors for the appropriate flats, as prescribed by the compatible pair construction (Rincón, 2011).

3. Simplicial Structure, Balancing, and Algorithmic Computation

$\Phi(M)$ yields a simplicial, balanced fan structure covering all of $T(M)$. The balancing condition inherited from tropical geometry mandates that, at each codimension-one face, the sum of primitive integer normals of adjacent maximal cones vanishes; in $\Phi(M)$ all combinatorial weights are 1 (Rincón, 2011).

The computation of $\Phi(M)$ is optimized in the TropLi algorithm. The core steps are:

1. Basis Enumeration: All bases $B$ of $M$ of size $r$ are enumerated (reverse search and duality can be leveraged).
2. Fundamental Circuit Computation: For each basis and $k\in E\setminus B$, the fundamental circuit $C(k,B)$ is computed via row-reduction.
3. Compatible Pair Construction: All regressive compatible pairs $(p,L)$ are recursively built, with careful maintenance of local compatibility and regressivity.
4. Ray and Cone Output: For each maximal compatible pair, the extremal rays of the corresponding cone $\Gamma(p,L)$ are output (Rincón, 2011).

The complexity is roughly $O(\binom{n}{r} r^3 + \sum_{B}\#\{\text{regressive compatible pairs with$B$}\}\cdot r)$, greatly outperforming naïve enumeration of total flag orders.

4. Relationship to Other Fan Structures and Matroid Invariants

The cyclic Bergman fan occupies an intermediate refinement:

$$\text{Bergman fan} \;\prec\; \text{nested set fan} \;\prec\; \text{cyclic Bergman fan} \;\prec\; \text{fine subdivision}$$

The rays of each structure are in bijection with flacets, connected flats, cyclic flats and singletons, and all flats, respectively. Maximal cones correspond to maximal compatible sets or chains as dictated by the refinement.

$\Phi(M)$ is also directly relevant for the study of minimal tropical bases of the Bergman fan (i.e., minimal sets of circuits whose associated tropical hyperplanes cut out $B(M)$). In simple binary matroids, the set of non-"pasted" circuits forms the unique minimal tropical basis, confirming a prior conjecture (Nakajima, 2019).

5. Geometric and Polyhedral Applications

The cyclic Bergman fan is crucial in computational applications, most notably the determination of Newton polytopes of $A$-discriminants. For an integer matrix $A\in \mathbb{Z}^{m\times n}$ (with $(1,\dots,1)\in\operatorname{rowspan}(A)$), the tropicalization of the $A$-discriminant variety is $T(M(A^\perp)) + \operatorname{rowspace}(A)$, where $A^{\perp}$ is a Gale dual. The codimension-one skeleton is the normal fan of the Newton polytope $\mathrm{Newt}(\Delta_A)$, and ray-shooting over the cones of $T(M(A^\perp))$ recovers extreme points of $\mathrm{Newt}(\Delta_A)$. The combinatorial efficiency of $\Phi(M)$, via TropLi, renders these computations feasible for large instances (Rincón, 2011).

6. Illustrative Example: The Uniform Matroid $U_{2,4}$

For $M=U_{2,4}$, of rank 2 on $E = \{1,2,3,4\}$:

• All circuits are size 3, so $T(M)$ consists of points in $\mathbb{R}^4$ such that the minimum over each triple is attained at least twice.
• $T(M)$ is the tropical Grassmannian $\operatorname{TropGr}(2,4)$: a 2-dimensional fan with six rays and three 2-cones.
• Cyclic flats are the circuits (triples) and singletons.
• Rays of $\Phi(M)$ are $e_i$ (singletons) and $e_C$ (triple circuits).
• Maximal cones are indexed by bases (pairs) and regressive compatible pairs on the remaining two elements. Each maximal cone is spanned by $e_C$ for the two circuits containing the fixed pair, recreating the classical tropical Grassmannian decomposition (Rincón, 2011).

7. Summary and Significance

The cyclic Bergman fan $\Phi(M)$ refines the polyhedral structure of tropical linear spaces associated to matroids, provides a combinatorially explicit, simplicial, balanced fan structure, and supports fast algorithmic computation of matroidal and tropical-geometric invariants. Its construction and properties tightly encode the structure of cyclic flats and compatible pairs, making it a foundational object in computational tropical geometry, polyhedral combinatorics, and discriminantal computations (Rincón, 2011).