Chapter 8 — A Matroid Walks Into a Fan · Learn You a Tropical Geometry for Great Good!

Print edition for handwritten margin notes. Anchors in the left margin (§1.3 ¶2 style) match the web edition — notes digitised from this paper appear at the same anchors online. minplus.xyz

Part II counted curves; the interlude did linear algebra. Part IV puts the two together: what is a tropical linear space? The answer is one of the cleanest in the subject — a matroid is a tropical linear space, and its picture is a fan, the Bergman fan B(M).

A matroid is linear algebra with the vectors removed: a ground set E = {0, …, n−1} and a family of bases (the would-be bases of a vector configuration), governed by the basis-exchange axiom. Its flats are the would-be subspaces (closed sets), its circuits the minimal dependencies. Nothing in the definition mentions coordinates — which is the point: some matroids come from no vector configuration at all (a later chapter meets one), and tropical geometry believes them anyway.

Prediction, in the book’s usual order: if a purely combinatorial object really is a geometric one, then independent definitions of its geometry, written down by different people for different reasons, should land on the same set — and the geometry should be able to see the combinatorics. Both predictions are checked below, in code.

8.1 Three definitions, one fan

The Bergman fan B(M) ⊂ R^n / R·1 of a rank-r matroid has three classical descriptions, and the chapter program proves them equal by computation — not on samples, but on every weight order there is:

Initial matroids (Ardila–Klivans): w ∈ B(M) iff the matroid M^w of w-maximal bases is loop-free — no element is in no maximum-weight basis.
Chains of flats (Ardila–Klivans again — in their paper the fan’s fine subdivision is the order complex of the lattice of flats): to a chain ∅ ⊊ F₁ ⊊ … ⊊ F_k ⊊ E of flats assign the cone generated by the indicator vectors e_{F_i}; B(M) is the union of these cones (mod the line R·1).
Circuit hyperplanes (Nakajima): w ∈ B(M) iff on every circuit C, the minimum of w over C is attained at least twice — w lies on every tropical hyperplane the circuits define.

Honest Box — sign conventions are where fan code goes to be wrong quietly. This book is min-plus throughout (Chapter 1), and so is this fan: its circuit condition is Chapter 1’s own — the minimum of w over each circuit, attained at least twice — and its rays are the positive indicators e_F. The literature splits between min- and max-weight conventions; they are negatives of each other, and each pairs with exactly one circuit condition. The one place a max appears in this chapter is inside description 1: M^w is built from maximum-weight bases. That is bookkeeping, not a change of fan — the study’s first write-up paired maximum-weight bases with max-twice on circuits, the assertions killed it, and the pairing the code asserts is maximum-weight bases with min-twice. Let the assertions choose.

And “by computation” means exhaustive, not sampled. Whether w lies in the fan — under any of the three descriptions — depends only on the weak order of w: which coordinates are equal, and which are bigger. Each description visibly only ever compares coordinates (or selects bases by comparing weights), and there are only finitely many weak orders on n elements: 75 for n = 4, 541 for n = 5. So the program enumerates all of them — each at two different spacings, asserting the verdicts don’t move when the gaps do — and checks that the three tests agree on every single one. For U(3,5), exactly 36 of the 541 orders lie in the fan, one per chain of flats. A confession: the first shipped version of this chapter sampled random weights instead, and for U(3,5) its 150 samples only ever visited 13 of those 36 in-fan orders. Sampling is a sanity check; enumeration is the proof. (A referee caught the difference — the enumeration is the fix.)

8.2 Three matroids, three fans

U(2,4) — the uniform matroid (every 2-subset a basis): 6 flats, 4 circuits. Its fan is 4 rays e_0, …, e_3 in R⁴/R·1: a tropical line with 4 ends — Chapter 1’s tripod, with one more leg. Balanced at the origin: Σ e_i = (1,1,1,1) ≡ 0 mod 1.

The parallel-pair matroid (rank 2 on 4 elements, with 0 ∥ 1 — the pair {0,1} is a rank-1 flat): the fan has only 3 rays, e_0+e_1, e_2, e_3, still balancing to (1,1,1,1). Merging two parallel elements merged their rays. The fan remembers the matroid: the face poset of the fan is the chain poset of the flats, so a simple matroid can be read back off its fan’s shape.

U(3,4): 12 flats, 1 circuit ({0,1,2,3}). The fan is pure of dimension r − 1 = 2, with 10 rays — one e_F for every proper nonempty flat (e_i and e_i+e_j) — and 12 triangular 2-cones cone(e_i, e_i+e_j): the fine subdivision of the 6 coarse cones cone(e_i, e_j). Balanced at every ray, checked exactly. This is the tropical plane, and Chapter 9 will do Hodge theory on its Chow ring.

A fourth matroid, U(3,5) (17 flats, 5 circuits), runs as a larger cross-check: the three descriptions agree there too — on all 541 of its weight orders — and its fan has 15 rays and 20 two-dimensional cones — the ray count is the number of proper nonempty flats, as it is for every matroid here, and the finishing pass read it off the re-run rather than trusting a draft.

Matroid	rank	flats	circuits	rays of B(M)	dim
`U(2,4)`	2	6	4	4	1
parallel pair	2	5	3	3	1
`U(3,4)`	3	12	1	10	2
`U(3,5)`	3	17	5	15	2

Bergman fans: B(U(2,4)) with four rays, the parallel-pair fan with its merged ray, and B(U(3,4)) in 3D — the tropical plane

Figure 8.1 — A: B(U(2,4)), four rays; B: the parallel-pair fan, merged ray e_0+e_1 labelled; C: B(U(3,4)) in 3D, 4 singleton rays + 6 pair rays, 12 triangles = the tropical plane. The ray vectors drawn are read from the chapter program’s own summary, and every count in every panel was asserted before rendering.

Flats, circuits and rays for the four matroids of the chapter table, with the parallel pair’s merged ray annotated

Figure 8.2 — The table above, drawn. Rays = proper nonempty flats in every row (flats minus the empty flat and E itself); the parallel pair, on the same ground set and rank as U(2,4), has exactly one flat and one ray fewer — the merger, visible as a missing bar segment.

8.3 The whole chapter, as a program

Everything in this chapter is one runnable program: ch08_matroids.py — self-contained apart from numpy/scipy. The assertions are the chapter’s claims: the flat/circuit/ray counts in the table of §8.2 (including U(3,5)’s 15 rays and 20 two-cones, and U(3,4)’s 12 fine cones subdividing exactly 6 coarse ones); purity dim = r − 1 for every matroid; triple agreement (chains-of-flats LP membership ⇔ M^w loop-free ⇔ circuit min-twice) on every weak order of the ground set — 75 for the n = 4 matroids, 541 for U(3,5) and U(2,5), each at two spacings; balancing at the origin for the rank-2 fans and at every ray for U(3,4) and U(3,5); the parallel-pair ray merger (3 rays, the merged one e_0+e_1, with e_0 and e_1 absent as separate rays); and, asserted as the Try It answer key, B(U(2,5)) has 5 rays. Run it, and if it prints its final line, the claims are what we said they are. (This section used to print the listing inline; a screenful of code you can download is just a download with extra scrolling.)

8.4 Try It

How many rays does B(U(2,5)) have? (Count its proper nonempty flats first.)
The parallel-pair fan has 3 rays, not 4. What single fact about the matroid does the missing ray record?
What is the dimension of B(U(3,4)), and which formula gives it?

Answers

1. 5. U(2,5) has 7 flats — the empty set, the whole set, and the 5 singletons, which are exactly its proper nonempty flats, so the fan has 5 rays (a tropical line with 5 ends). The chapter program asserts this, triple agreement included.
2. That {0,1} is a single rank-1 flat: 0 and 1 are parallel — each closes to the same flat, so there is one ray e_0+e_1 where the uniform matroid has two. Here in rank 2 the proper nonempty flats are the rank-1 flats, so the ray set is exactly the rank-1 flats, mergers included — but that is a rank-2 accident, not the general rule: in general the rays are all proper nonempty flats, of every rank. U(3,4) above makes the point — its 10 rays are the 4 rank-1 singletons and the 6 rank-2 pairs.
3. 2, by dim B(M) = r − 1 with r = 3: the fan is pure of dimension 2 — a plane’s worth of cones, which is why panel C of Figure 8.1 is a surface, not a tripod.

What we learned

A matroid is a tropical linear space, three times over. The initial-matroid, chains-of-flats and circuit-hyperplane descriptions define the same fan — not as a cited theorem but as an exhaustive computation: every weak order of the ground set checked three ways, without a single disagreement.
The fan remembers the matroid. Rays are proper nonempty flats; merging parallel elements merges rays; the face poset of the fan is the chain poset of the flats. Geometry here is not a picture of the combinatorics — it is the combinatorics, drawn.
Fans are balanced. At the origin for the lines, at every ray for the plane: the neighbouring cone generators sum to zero mod the ray and the line R·1 — the same balancing condition as Chapter 1’s tripod, one dimension up.

The Bergman fan of U(3,4) — the tropical plane of panel C — has a Chow ring, and on that ring lives the mechanism behind June Huh’s Fields Medal work: Chapter 9 computes A*(U(3,4)), recovers its intersection numbers three independent ways, and verifies the Hodge–Riemann relations exactly. Hodge theory, with no Hodge theory underneath.

Source & provenance: Day 6 (2026-10-16) of the tropical geometry study — code hidden_files/code/day06_matroids.py (run log hidden_files/day06-run.log), figure files/figures/day06-bergman-fans.png, study log Day 6 entry. Sources: the initial-matroid and chains-of-flats descriptions are Ardila–Klivans, arXiv:math/0311370 (their subdivision of the Bergman complex is the order complex of the lattice of flats); the circuit-hyperplane description is Nakajima, arXiv:1901.06893. Two neighbours, cited for what they actually do: Fry, arXiv:1910.00627, meets this same fan from the curves side — for a graphic matroid it is the moduli fan of radially aligned tropical curves; Shaw, arXiv:1010.3967, builds an intersection product on cycles inside this fan — part of Chapter 9’s story, not a source for any description here. Huh hook: the Chow ring of this fan is Chapter 9’s subject (Baker, arXiv:1705.07960). The chapter program was re-run end-to-end on 2026-11-05 while finishing this chapter, and re-run again the same day after a referee pass replaced its sampled agreement check with the exhaustive weak-order enumeration of §8.1: all assertions passed, including the table’s counts, the U(3,5) ray count of 15, and the U(2,5) Try It count. Figures regenerated as SVG the same day: code hidden_files/code/ch08_extra_figures.py, which re-runs the chapter program, asserts every value it draws (the ray vectors themselves are read from the program’s summary), and only then renders.

Chapter code

The full program behind this chapter, as a download: ch08_matroids.py — the same file §8.3 points at. Its assertions are the chapter’s claims; run it.