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The external activity complex of a pair of matroids

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Pith's one-line read This paper proves the tropical f-vector conjecture by expressing the matroid invariant $\omega(M)$ as a sum of reduced homology dimensions of links in a new external activity complex for pairs of matroids.

desk verdict Berget and Fink resolve Speyer's 2005 tropical f-vector conjecture with a new pair-of-matroids external activity complex, but the last step to the conjecture is outsourced to a concurrent companion paper that needs independent checking. read the letter →

arxiv 2412.11759 v2 pith:E2M6XN2J submitted 2024-12-16 math.CO math.ACmath.AG

Andrew Berget , Alex Fink This is my paper · ORCID
classification math.COmath.ACmath.AG MSC 05B3514T0513D0205E45
keywords matroidsexternalactivitycomplextropicalf-vectorconjectureg-invariantCohen-MacaulaysimplicialcomplexestautologicalclassesofdiagonalDilworthtruncationK-polynomials
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves the tropical f-vector conjecture, a 2005 bound on how many faces of each dimension can appear when a hypersimplex is subdivided into matroid base polytopes; such subdivisions describe tropical linear spaces, so the bound controls their combinatorial complexity. The proof works for all matroids, realizable or not, by introducing a simplicial complex attached to a pair of matroids: the external activity complex. The central result is a formula expressing the matroid invariant $\omega(M)$, the top coefficient of the $g$-invariant, as a sum of dimensions of reduced homology groups of links in this complex, making nonnegativity manifest. Because nonnegativity of $\omega$ on all minors is known to force nonnegativity of all coefficients of the $g$-invariant, the face-count conjecture follows.

What carries the argument

The external activity complex of a pair of matroids $(M_1,M_2)$ is the simplicial complex whose facets are the monomials $x^{B\cup E_1(B)}y^{B\cup E_2(B)}$ for each basis $B$ of the diagonal Dilworth truncation $D(M_1,M_2)$, the matroid whose circuits are the minimal nonempty subsets $C$ with $\operatorname{rank}_{M_1}(C)+\operatorname{rank}_{M_2}(C)=|C|$; the sets $E_1(B)$ and $E_2(B)$ record external activity with respect to a weight vector $w$. This complex generalizes the external activity complex of a single matroid and arises as a Gr\"obner degeneration of the Schubert variety of a pair of linear spaces. The paper proves that $\Delta_w(M_1,M_2)$ is Cohen-Macaulay, that its finely graded $K$-polynomial is bivaluative in the two matroids, and that this $K$-polynomial equals $\sum_{i,j}\chi(\wedge^i[Q^\vee_{M_1}]\cdot\wedge^j[Q^\vee_{M_2}])(-U_1)^i(-U_2)^j$. Hochster's formula then identifies the coefficients of total degree $n$ with sums of reduced homology dimensions of links inside the complex.

What would settle it

Compute both sides of Equation (1.1) for a connected non-realizable matroid such as the non-Fano matroid: if the sum of reduced homology dimensions of the links in $\Delta_w(M,M)$ differs from $\omega(M)$ computed from the exterior-power definition of the $g$-invariant, the central formula is false.

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Extended reading notes

Core claim

The paper establishes that for every matroid $M$ on $[n]$ of rank $r$, the invariant $\omega(M)$ is a sum of nonnegative integers: $\omega(M)=\sum_{B\in\binom{[n]}{r}}\dim \widetilde{H}_{2r-2}(\operatorname{link}_{\Delta_w(M,M)}(x_{[n]\setminus B}\,y_B))$. Since nonnegativity of $\omega$ on all minors implies every coefficient of the $g$-invariant is nonnegative, this proves the tropical f-vector conjecture. The central construction is the external activity complex $\Delta_w(M_1,M_2)$ of a pair of matroids, defined combinatorially by facets $x^{B\cup E_1(B)}y^{B\cup E_2(B)}$ for each basis $B$ of the diagonal Dilworth truncation $D(M_1,M_2)$, together with the proof that this complex is Cohen-Macaulay for every pair. Its finely graded $K$-polynomial is shown to be bivaluative and equal to an Euler-characteristic expression in exterior powers of dual tautological quotient classes, and Hochster's formula converts the top-degree coefficients of that polynomial into the homology sums above.

Load-bearing premise

The load-bearing premise is the theorem, proved in [FSS24], that nonnegativity of $\omega$ on all minors forces every coefficient of the $g$-invariant to be nonnegative; if that theorem failed, the paper would still prove $\omega(M)\geq 0$ but not the full tropical f-vector conjecture.

Editorial extensions

If this is right

  • The number of $(n-i)$-dimensional interior faces in any subdivision of $\Sigma(r,n)$ into matroid base polytopes is at most $(n-i-1)!/((r-i)!(n-r-i)!(i-1)!)$.
  • Every matroid $M$ satisfies $\omega(M)\geq 0$, with $\omega(M)$ realized as a sum of Betti numbers of links in $\Delta_w(M,M)$.
  • All coefficients of the $g$-invariant are nonnegative for every matroid, not only for matroids realizable over $\mathbb{C}$.
  • For pairs with $D(M_1,M_2)$ of expected rank, the sequence counting bases of $D(M_1,M_2)$ by external 1-activity is log-concave.
  • The $K$-polynomial identity yields the positivity statement $(-1)^{\operatorname{rank}D}\chi(\wedge^p[Q^\vee_{M_1}]\cdot\wedge^q[Q^\vee_{M_2}])\geq 0$ whenever $p+q=n$.
  • The Cohen-Macaulay property and bivaluativity of the external activity complex give a matroidal formula for the higher cohomology of exterior powers of dual tautological bundles when the pair is realizable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof suggests a general recipe: matroid invariants expressible as Euler characteristics of tautological classes might be proved nonnegative by finding Cohen-Macaulay initial degenerations whose links compute them; the same bivaluativity bridge could be used for other coefficients of the $g$-invariant, not just the top one.
  • The authors conjecture that $\Delta_w(M_1,M_2)$ is shellable; if true, the homology groups in the formula for $\omega(M)$ would have integer coefficients and likely explicit bases, connecting to the external-activity bases used for single matroids.
  • The log-concavity of external activity counts in the expected-rank case has combinatorial content independent of the tropical f-vector conjecture, and may extend to the full bivariate activity distribution rather than just one marginal.
  • The tropical cell complex dual to $\Delta_w(M_1,M_2)$, built from intersections of translated Chern class fans, appears to carry enough structure to encode the full minimal free resolution of the Stanley-Reisner ring, which would give finer homological information than the top-coefficient formula.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 6 minor

Summary. The paper introduces two related objects: the Schubert variety of a pair of linear subspaces of C^n and the external activity complex Δ_w(M1,M2) of a pair of matroids. The authors prove that this complex is Cohen-Macaulay, that its finely graded K-polynomial is bivaluative, and that its Z2-graded K-polynomial matches an Euler-characteristic formula involving exterior powers of tautological quotient classes. From these results they derive a nonnegative formula for the Fink-Shaw-Speyer invariant ω(M) as a sum of dimensions of reduced homology groups of links in Δ_w(M,M). Combining this with a theorem of Fink-Shaw-Speyer, they conclude that Speyer's 2005 tropical f-vector conjecture holds.

Significance. If correct, this resolves a long-standing conjecture and provides a new, explicitly homological mechanism for the nonnegativity of ω(M). The main technical engine—the external activity complex, the diagonal Dilworth truncation, and the tropical cell-complex description of Chern class products—is substantial and likely to be influential. The paper also contains several structurally interesting intermediate results, such as bivaluativity of the K-polynomial and the Cohen-Macaulay property, which are established by a novel combination of Gröbner degeneration, Kempf collapsing, and tropical intersection theory. The chief caveat is that the final step to the full f-vector conjecture is outsourced to a concurrent, overlapping-authorship preprint.

major comments (1)
  1. [Section 1.4, proof of Theorem E] The inference from ω(M) ≥ 0 to nonnegativity of all coefficients of g_M(t) is stated as "Theorem (Fink–Shaw–Speyer [FSS24])" but is not proved in this manuscript; it is imported from the concurrent preprint arXiv:2411.19521, which shares an author with the present paper. This inference is load-bearing for the resolution of Conjecture 1.1, because the paper's internal results establish only nonnegativity of ω(M) (Equation (1.1)), not nonnegativity of the other coefficients of g_M(t). The authors should either include a self-contained proof of the reduction theorem or explicitly mark Theorem E and the resolution of Speyer's conjecture as conditional on independent verification of [FSS24].
minor comments (6)
  1. [Section 1, first paragraph] There is a typo: "linear susbspaces" should be "linear subspaces".
  2. [Section 1, item (3) of Theorem A] The phrase "Gröbner" is typeset as "Gr¨obner" in the running text; please use the correct umlaut rendering consistently.
  3. [Section 5, paragraph before Definition 5.1] The text "when proving proving Theorem A" contains a duplicated word; it should read "when proving Theorem A".
  4. [Section 3.2 heading] The heading "tatutological bundles" should be "tautological bundles".
  5. [Theorem 7.2 proof] The word "uneffected" should be "unaffected".
  6. [Example 4.20] The table of circuit decompositions is dense; adding one sentence pointing out that the row for circuit 123567 illustrates the difference between Δ_w(F,F) and Δ_w(F-,F-) would improve readability.
Assumptions & free parameters 0 free parameters · 6 assumptions · 4 invented entities

No free parameters are fitted to data; the weight vector w is a generic auxiliary choice, and the final statistics are independent of it (Corollary 6.18). The proof is a pure derivation resting on standard commutative algebra and tropical geometry background, plus several heavy cited theorems; the only notable external and concurrent dependency is [FSS24]. The new objects are all explicitly constructed and cross-checked in examples.

assumptions (6)
  • standard math Proposition 2.9 (Derksen-Fink, [DF10, Theorem 4.2]): every matroid base polytope indicator expands as an integral combination of Schubert matroid base polytopes; hence a bivaluation is determined by its values on Schubert matroids (Corollary 2.10).
    Used to extend K-polynomial and Euler characteristic formulas from realizable matroids to all matroids in Section 7 (Theorems 7.2 and 7.4).
  • domain assumption Theorem of Fink-Shaw-Speyer [FSS24]: if ω(N) ≥ 0 for all minors N of M, then all coefficients of Speyer's g_M(t) are non-negative.
    Load-bearing for the conclusion of Conjecture 1.1 in Theorem E; not proved here, by overlapping authors, appeared as arXiv:2411.19521 one month before this preprint.
  • standard math Weyman's geometric method: for a birational collapsing of a vector bundle E to a normal variety with rational singularities, the minimal free resolution is given by cohomology of exterior powers of the quotient bundle ([Wey03, Theorem 5.1.3]).
    Underpins the K-polynomial formula in Theorem A(3) and Theorem 5.14.
  • standard math Cartwright-Sturmfels* property: CS* ideals have the same Zn-graded Betti numbers as their monomial counterparts and admit universal Gröbner bases ([CDNG20, Propositions 1.9, 1.12]).
    Used in Sections 5 and 7 to transfer Betti numbers from the Schubert variety ideal to the initial ideal and to prove Proposition 7.3.
  • standard math Provan-Billera: the independent set complex of a matroid is vertex-decomposable, hence Cohen-Macaulay (Proposition 2.6, [PB80]).
    Core of the proof of Theorem 7.2 that Δ_w(M1,M2) is Cohen-Macaulay via the CS* reduction to A/I(D).
  • domain assumption Standing assumption: all varieties are over C; geometric results use characteristic-zero rational singularities (stated in Section 2.1 and used in Section 5).
    The main theorems are over C; the paper notes it cannot recover Eur's positive-characteristic cohomology results (Section 1.5).
invented entities (4)
  • External activity complex Δ_w(M1,M2) independent evidence
    purpose: Central combinatorial object whose K-polynomial and homology encode matroid tautological classes and ω(M).
    Defined purely combinatorially in Definition 4.18 for arbitrary matroids; exemplified for Fano/non-Fano pairs in Example 4.20 and computed in Figures 2-3.
  • Diagonal Dilworth truncation D(M1,M2) independent evidence
    purpose: Auxiliary matroid whose circuits index the universal Gröbner basis and whose bases index the facets of Δ_w.
    Defined by an explicit rank formula (Definition 4.1), shown to be a matroid (Proposition 4.3), and used in Examples 4.7-4.8.
  • Schubert variety of a pair of linear spaces Y_{L1,L2} (and affine version) independent evidence
    purpose: Geometric model for realizable pairs; its K-polynomial is matched by Δ_w via Gröbner degeneration.
    Defined as a Zariski closure in Definition 5.1; its properties are established geometrically (Theorem A), giving an independent check of the combinatorial formulas for realizable pairs.
  • Tropical cell complex σ_∞(m) with graphs G(p) and holsters h_{i,k} independent evidence
    purpose: Computes products of Chern classes c_i(Q_{M1})c_j(Q_{M2}) and proves bivaluativity of the finely graded K-polynomial.
    Constructed in Section 6.1-6.3 from matroid Chern classes; its compactly supported Euler characteristics yield the K-polynomial formula in Theorem 6.23.

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Pith. "Pith review of The external activity complex of a pair of matroids." pith.science (2026). https://pith.science/paper/E2M6XN2J

@misc{pith2026241211759,
  author       = {Pith},
  title        = {Pith review of: The external activity complex of a pair of matroids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E2M6XN2J}},
  note         = {Machine review of arXiv:2412.11759}
}
abstract

We introduce the Schubert variety of a pair of linear subspaces in $\mathbf{C}^n$ and the external activity complex of a pair of not necessarily realizable matroids. Both of these generalize constructions of Ardila et al., which occur when one of the linear spaces is one-dimensional. We prove that our external activity complex is Cohen-Macaulay and deduce a formula for its $K$-polynomial in terms of exterior powers of the dual tautological quotient classes of matroids. As a consequence, we deduce a non-negative formula for the matroid invariant $\omega(M)$ of Fink, Shaw, and Speyer in terms of certain homology groups of links within an external activity complex, proving the 2005 tropical $f$-vector conjecture of Speyer.

Figures

Figures reproduced from arXiv: 2412.11759 by the authors.

Figure 1
Figure 1. The graphs G(p) of Example 6.1. graph are drawn so that the coordinates of p − wx and p − wy increase when read left-to-right, and we have labeled each vertex according to which edges ei emanate from it in order to enhance the readability of these figures. We will maintain these conventions in future examples. One imagines that as the last entry of p increases from t = a to t = c the edge e4 is dragged to the right,… view at source ↗
Figure 2
Figure 2. , after choosing coordinates as done in Example 3.13. Here D(M1, M2) = U2,3. We have labeled the 2-dimensional cells by their indexing monomial. The monomials of other cells are the least common multiple of their bounding faces. The 0-dimensional intersection (c1(QM1 ) + wx) ∩ (c1(QM2 ) + wy) consists of the points F and G, whose associated monomials are x123y23 and x123y13, respectively. The 0-dimensional intersect… view at source ↗
Figure 3
Figure 3. The compactified tropical cell complex encoding ∆w(U2,3, U1,3). The six newly added boundary points • and • belong to the 1-dimensional cell they bound, and the connected components of the boundary minus these points belong to the 2-dimensional cell they bound. For example, σ∞(x123y3) is the red cell, including the point c, and the other previously unbounded 1-dimensional cells become half-open line segments. As ano… view at source ↗

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