Simplicial Dollar Game

David Perkinson; Jesse Kim

Simplicial Dollar Game

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every high-degree chain wins the simplicial dollar game

desk verdict This is a genuine higher-dimensional generalization of the graph dollar game, and the main theorems—large-degree winnability and the forest/torsion-free characterization—are proven carefully from a new Hilbert-basis degree; it deserves a serious referee. read the letter →

arxiv 1908.09350 v3 pith:DWTCTJNW submitted 2019-08-25 math.CO

Jesse Kim , David Perkinson This is my paper · ORCID

classification math.CO MSC 05E45

keywords chip-firingdollargamesimplicialcomplexcriticalgroupHilbertbasisLaplacianwinnabilityspanningforest

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The dollar game on a graph asks whether an integer assignment of dollars to vertices can be transformed by lending and borrowing moves into a position with no debts. This paper lifts that game to simplicial complexes of any dimension, playing on $i$-faces with firing moves governed by the combinatorial Laplacian $L_i$. Because the total amount of money is not conserved in higher dimensions, the authors replace scalar degree with a vector-valued degree $\deg(\sigma)$ computed by the Hilbert basis of the nonnegative kernel of $L_i$. Their main theorem, Theorem 18, says that for every dimension $i$ there is a realizable degree $\delta$ such that every $i$-chain with degree at least $\delta$ is winnable, generalizing the graph statement that divisors of degree at least the genus can be won. A second result, Corollary 34, characterizes the simplicial complexes on which every degree-zero chain is winnable: exactly those whose $i$-skeleton is a spanning forest that is torsion-free in codimension one.

What carries the argument

The load-bearing object is the Hilbert basis of the nonnegative kernel of the Laplacian. Concretely, $\mathcal H_i$ is the unique minimal set such that every nonnegative integer $i$-chain in $\ker L_i$ is a nonnegative integer combination of elements of $\mathcal H_i$, and the degree of a chain is the vector of inner products with these basis elements. A second mechanism is Lemma 10, which guarantees a strictly positive element in $\ker L_i$; that positivity forces the nonnegative kernel to span the whole kernel, so degree-zero chains can be identified with torsion classes of the critical group and Theorem 18 follows from a finite decomposition argument. For orientable pseudomanifolds, the Hilbert basis in codimension one is combinatorially explicit: it consists of the incidence vectors of simple directed cycles of the $\gamma$-incidence graph.

What would settle it

For any fixed simplicial complex and dimension $i$, compute the realizable degrees and test whether every chain of degree at least some $\delta$ is winnable; if no such $\delta$ exists, Theorem 18 fails, and independently, finding a complex where $\ker L_i$ contains no strictly positive integer vector would falsify Lemma 10, the step on which the theorem rests.

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Extended reading notes

Core claim

The paper establishes that the graph-theoretic dichotomy 'large enough degree wins, and zero degree wins exactly on trees' survives in higher dimensions once degree is redefined. Two $i$-chains are linearly equivalent if they differ by an element in the image of $L_i$, and the degree of a chain is the vector of dot products with the elements of the Hilbert basis $\mathcal H_i$ of $\ker^+ L_i$, the monoid of nonnegative integer chains in the kernel of the Laplacian. This degree is invariant under firing moves and nonnegative on effective chains, while the naive sum of coefficients is neither. Theorem 18 states that for each $i$ there exists a realizable degree $\delta$ such that every chain of degree at least $\delta$ is winnable; Theorem 13 identifies degree-zero classes modulo firing with the torsion subgroup of the critical group $K_i(\Delta)$; and Corollary 34 shows that all $(i-1)$-chains of degree zero are winnable in $\Delta$ if and only if the $i$-skeleton is a spanning $i$-forest with $\widetilde H_{i-1}(\Delta)$ torsion-free.

Load-bearing premise

The load-bearing premise is Lemma 10, the assertion that in every dimension the Laplacian kernel contains a chain whose coefficients are all strictly positive, since without such a positive kernel element the identification of degree-zero classes with critical-group torsion, and hence the high-degree winnability theorem, would not follow.

Editorial extensions

If this is right

If the theorem is right, a finite computation at one sufficiently large degree settles winnability for all larger degrees, because degree-zero classes form a finite torsion group.
The tree characterization carries over: the complexes with universally winnable degree-zero chains are exactly the spanning forests torsion-free in codimension one, so winnability at degree zero detects a homological property.
On orientable pseudomanifolds, the Hilbert basis is encoded by simple directed cycles in a gamma-incidence graph, making degree calculations combinatorial rather than algebraic.
The set of minimal winning degrees is a finite antichain of vectors, replacing the single graph genus as the threshold invariant; the paper computes one such antichain for the hollow tetrahedron.

Reading between the lines

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

When the Hilbert basis consists of 0-1 vectors, Corollary 20 upgrades the theorem to an exact threshold: winnability at a single degree forces winnability at all larger degrees, so the minimal winning set is particularly clean for such complexes.
Because degree-zero classes are torsion of the critical group, tabulating critical-group torsion over families of complexes would automatically answer whether the zero-degree dollar game is universally winnable.
The documented failure of the greedy algorithm and q-reduction in higher dimensions suggests that deciding winnability for unfixed dimension could be harder than in graphs, though the paper only poses this as an open question.
The paper's open problem about minimal winning degrees for (d-2)-chains on the d-simplex could serve as a concrete benchmark for how the vector threshold grows with dimension.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know before you read it. First, the paper really does generalize the graph dollar game to all dimensions, not just relabel the old theorems. The new object is a vector-valued degree defined by dot products with the Hilbert basis of the nonnegative Laplacian kernel, and the two headline results—Theorem 18 (large enough degree implies winnable) and Corollary 34 (all degree-zero chains winnable iff the appropriate forest number is one)—are genuine extensions of the Baker–Norine facts. Second, the proofs hold up. I traced the chain behind Theorem 18: Lemma 10's lexicographic maximality argument is sound, Lemma 17's cone decomposition is standard, and the deduction from DKM's Theorem 33 is legitimate. No circularity: the degree invariant is defined without reference to winnability.

The strongest parts are the clean proof of Theorem 18 and the pseudomanifold result, Theorem 22, where the Hilbert basis is described as incidence vectors of simple directed cycles in the γ-incidence graph. That is a concrete and useful payoff. The paper also improves DKM's reduced Laplacian theorem by loosening the hypotheses, and the appendix proofs are careful. Proposition 21 for orientable pseudomanifolds is a nice complement to DKM.

Soft spots are minor. The Sage/PyNormaliz computations in the examples are not attached as code or data, so those examples are not independently reproducible without re-doing the computations; this affects examples, not the main theorems. There is a stray definition of C = {v : L_i v ≥ 0, v ≥ 0} just before Definition 4 that is never used—a harmless typo. The degree is a vector in Z^{|H|}, so Theorem 18's δ is existential and nonconstructive; the paper is upfront about this and lists it under further work. None of this undermines the central argument.

The reader's ACCEPT with moderate confidence matches my own read. This is a solid, important-for-the-subfield extension, not a breakthrough, and it will be cited by people working on combinatorial divisor theory and chip-firing. I would send it to a serious referee: the main results deserve verification and the computational examples could be checked. I'd also bring it to a reading group if anyone in the group works on chip-firing, because the Hilbert-basis degree is worth understanding in its own right.

Referee Report

0 major / 5 minor

Summary. The paper introduces a higher-dimensional analogue of the Baker--Norine dollar game on simplicial complexes, using Duval--Klivans--Martin's chip-firing theory. It defines a degree vector for i-chains via the Hilbert basis of the nonnegative kernel of the i-th up-down Laplacian. The main results are Theorem 18, stating that every i-chain of sufficiently large degree is winnable, and Corollary 34, characterizing when all (i-1)-chains of degree zero are winnable in terms of spanning i-forests and torsion-free (i-1)-homology, thereby generalizing the tree case for graphs. Additional contributions include the identification of degree-zero chains modulo firing with the torsion of the critical group (Theorem 13), a combinatorial description of the Hilbert basis in codimension one for orientable pseudomanifolds (Theorem 22), a generalization of the reduced Laplacian isomorphism (Theorem 30), and several worked examples including minimal winning degrees.

Significance. The paper gives a natural invariant notion of degree for simplicial chip-firing and establishes a higher-dimensional analogue of the classical winnability threshold. The proof strategy is clean: the degree is defined independently of winnability, the key technical Lemma 10 is proved directly, and Theorem 18 follows from Lemma 17 together with finiteness of the torsion of the critical group. The identification of degree-zero chains modulo firing with torsion (Theorem 13) and the pseudomanifold Hilbert-basis theorem (Theorem 22) are valuable structural results. The paper also provides explicit computations and examples showing that higher-dimensional behavior differs from graphs, which is informative. I find no circularity: external dependencies, such as Duval--Klivans--Martin's Theorem 33, are prior results by other authors, and the authors' own arguments are not used to prove the main theorems in a circular manner.

minor comments (5)

[Section 4, opening paragraph] The displayed definition "C := {v in R^{f_i} : L_i v >= 0 and v >= 0}" is inconsistent with Definition 4, which correctly defines the degree using the Hilbert basis of ker_+ L_i. The cone C is never used afterward, and a reader could momentarily mistake it for the intended kernel; it should be removed or replaced.
[Lemma 10 and Theorem 18] The phrase "for each integer i" should be restricted to i with 0 <= i <= dim(Delta), or the empty-face case should be handled explicitly, since the proof of Lemma 10 chooses a lexicographically smallest i-face and does not apply when Delta_i is empty.
[Section 2.2, Polyhedral cones] The term "integrally generated" is used without definition; because the Hilbert basis property depends on it, a one-sentence definition would help the reader.
[Examples 36 and 37] The Sage computations are reported but the input data are not included; for reproducibility, the authors might include a short appendix or reference to the code used.
[Proof of Proposition 32] The phrase "unique maximal linearly independent subset" should be read as "unique basis of the column space" or "unique maximum-cardinality linearly independent subset"; the current wording is slightly ambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the degree invariant is defined independently of winnability, and Theorem 18 is proven from the cone decomposition and torsion identification without fitting or renaming.

full rationale

The derivation chain is self-contained. Definition 4 defines the i-th degree as the vector of dot products of a chain with the Hilbert basis of the nonnegative kernel ker+ L_i; this definition does not presuppose winnability. Proposition 6 proves degree invariance under firing by orthogonality of ker L_i to im L_i, and Corollary 7 (winnability implies degree at least 0) is a consequence, not an input. Lemma 10 is proven directly by a lexicographic maximality argument using the star of a face, with no dependence on the paper's conclusions. Corollary 12 follows from Lemma 10 and shows that the Z-span of ker+ L_i is all of ker L_i, so degree-zero chains are exactly (ker L_i)^⊥; Theorem 13 is then an explicit cokernel/torsion identification, not a restatement of the degree definition. Lemma 17 is a rational-polyhedral-cone decomposition with a bounded correction set P_i, and Theorem 18 uses only finiteness of torsion representatives and P_i to choose an existential dominating chain omega; the resulting degree delta = deg(omega) is not fitted to any winnability data. Later results, including Corollary 34, rely on Duval-Klivans-Martin's Theorem 33, which is an external result by other authors. The paper's self-citations (e.g., the Corry-Perkinson textbook and Perkinson et al.) are not load-bearing for the main theorems, and the unspecified cone C appearing just before Definition 4 is unused and appears to be a harmless typographical relic. No step reduces one of the paper's claims to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central construction is the vector-valued degree defined from the Hilbert basis of the nonnegative kernel of the Laplacian; this is a definition, not a free parameter or an added entity. Theorems 18 and Corollary 34 are proven from this definition using standard polyhedral geometry and cited results of Duval, Klivans, and Martin. No constants are fitted to data, and no new physical or combinatorial objects are postulated beyond the definitions.

assumptions (7)

standard math Finite simplicial complex Delta with fixed vertex order; chain boundary maps satisfy partial_i partial_{i+1} = 0 and L_i = partial_{i+1} partial_{i+1}^t is symmetric.
Used throughout, Sections 2.1 and 3.
standard math The cone ker_+ L_i = {v >= 0 : L_i v = 0} is a pointed rational polyhedral cone, so it has a unique and finite Hilbert basis.
Definition 4 and Remark 5; standard result from polyhedral geometry cited to [14] and [18].
standard math Each rational polyhedral cone Q satisfies Q = Q_Z + Pi for a fundamental parallelepiped Pi, with Q intersect Z^n = Q_Z + (Pi intersect Z^n).
Used in Lemma 17 to decompose degree-nonnegative chains.
domain assumption Theorem 33: |T(K_{i-1}(Delta))| = tau_i(Delta), from Duval, Klivans, and Martin [9, Theorem 8.1].
Relied on in Corollary 34 and Proposition 39 to connect winnability to spanning forests.
domain assumption Proposition 29 from [8]: any two of the three conditions defining a spanning i-forest imply the third.
Used in Section 6 to work with spanning forests and trees.
domain assumption For orientable pseudomanifolds, H_d(Delta, partial Delta) is isomorphic to Z and H_{d-1}(Delta, partial Delta) is torsion-free.
Used in Section 5 to define pseudomanifold orientation and in Proposition 21.
domain assumption Definition of X_i(Delta) introduced by Corry and Keenan in private communication.
Used in Section 6.1; the paper proves the conjecture, so it does not assume its truth.

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Cite this review

Pith. "Pith review of Simplicial Dollar Game." pith.science (2026). https://pith.science/paper/DWTCTJNW

@misc{pith2026190809350,
  author       = {Pith},
  title        = {Pith review of: Simplicial Dollar Game},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DWTCTJNW}},
  note         = {Machine review of arXiv:1908.09350}
}

read the original abstract

The dollar game is a chip-firing game introduced by Baker and Norine (2007) as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these is Theorem 18, generalizing the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second is Corollary 34, generalizing the fact that trees (graphs of genus 0) are exactly the graphs on which every divisor of degree 0, interpreted as an instance of the dollar game, is winnable.

Figures

Figures reproduced from arXiv: 1908.09350 by the authors.

**Figure 1.** Winning the dollar game σ = −12+2·13−3·23+2·24−34 on the 2-dimensional simplicial complex with facets 123 and 234. Example 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Two dollar games on the edges of a 2-simplex. Only the first is winnable. Example 3 (Graphs). Let ∆ = G be a connected, undirected graph as in the introduction. In that case, the dollar game for 0-chains on ∆ we just defined is the same as the dollar game for graphs from [2]. If the vertices of G are vi = i for i = 1, . . . , n, then the 0-th Laplacian is the usual discrete Laplacian for a graph: L0 = diag(degG(v1),… view at source ↗

**Figure 3.** The hollow tetrahedron and its γ-incidence graph (cf. Example 24). hence the elements of the Hilbert basis for ker+ L1, are listed as rows in the table below: 12 13 14 23 24 34 1 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 1 0 . Example 25 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: A triangulated annulus and its γ-incidence graph (cf. Example 25). 12 13 14 15 23 25 26 34 36 45 46 56 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 Two elements in the Hilbert basis for ker+ L1. Example 26. The condition of being orientable as a pseudomanifold is ne…

**Figure 5.** Figure 5: Triangulation of a Klein bottle (cf. Example 26). Example 27 (Computing minimal winning degrees). Let ∆ be the hollow tetrahedron in Example 24, and use lexicographic ordering of the edges of ∆ to identify C1(∆) with Z 6 , as usual. For the purpose of computing degrees…

**Figure 6.** Figure 6: illustrates a two-dimensional complex P which is a triangulation of the real projective plane. We have He0(P) = He2(P) = 0, and He1(P) ≈ Z/2Z. Therefore, P is a spanning tree with tree number τ2(P) = 4. The cycle σ := 12 + 23− 13 is a 1-chain in the image of ∂2 and hen…

**Figure 7.** Figure 7: A simplicial complex with facets 123, 124, and 34 (cf. Example 40). 7. Further work There is still much to be learned about winnability of the dollar game on a simplicial complex. Here, we will present three general open areas of investigation: computation of minimal w…

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Problems from the AIMS Chip-Firing Workshop , https://aimath.org/WWN/chipfiring/aim_chip-firing_problems.pdf, July 2013

work page 2013

[2] [2]

Matthew Baker and Serguei Norine, Riemann-Roch and Abel-Jacobi theory on a ﬁnite graph , Adv. Math. 215 (2007), no. 2, 766–788

work page 2007

[3] [3]

Matthew Baker and Farbod Shokrieh, Chip-ﬁring games, potential theory on graphs, and spanning trees, J. Combin. Theory Ser. A 120 (2013), no. 1, 164–182

work page 2013

[4] [4]

N. L. Biggs, Chip-ﬁring and the critical group of a graph , J. Algebraic Combin. 9 (1999), no. 1, 25–45

work page 1999

[5] [5]

Sarah Brauner, Forrest Glebe, and David Perkinson, Enumerating linear systems on graphs , https://arxiv.org/abs/1906.04768, 2019

work page Pith review arXiv 1906

[6] [6]

Scott Corry and Liam Keenan, private communication, 2017

work page 2017

[7] [7]

Scott Corry and David Perkinson, Divisors and Sandpiles , American Mathematical Society, Providence, RI, 2018, An introduction to chip-ﬁring

work page 2018

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Duval, Caroline J

Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, Critical groups of simplicial complexes , Ann. Comb. 17 (2013), no. 1, 53–70

work page 2013