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arxiv: 2604.11166 · v2 · pith:IKIYXNAInew · submitted 2026-04-13 · 🧮 math.AG · math.CO

Asymptotic Behavior of Tropical Rank Functions

Ana Maria Botero , Alex K\"uronya , Eduardo Vital This is my paper

Pith reviewed 2026-05-22 11:22 UTC · model grok-4.3

classification 🧮 math.AG math.CO MSC 14T05
keywords tropical curveslinear seriesrank functionstropical volumesasymptotic behaviordivisorstropical modulestropicalization
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The pith

The asymptotic growth of ranks for linear series on tropical curves is controlled by newly defined tropical volume functions that parallel classical volumes in algebraic geometry.

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

The paper sets out to prove that the long-term growth rates of two competing rank notions for linear series on a tropical curve are determined by asymptotic invariants. A sympathetic reader would care because this creates a direct parallel to the volume theory that governs divisor ranks on algebraic varieties, potentially letting discrete tropical calculations replace heavier algebraic machinery for large series. The authors define tropical volume functions separately for divisors and for tropical modules, then establish that each gives the optimal leading term for rank growth. They also prove these volumes are compatible with the tropicalization map from algebraic curves.

Core claim

The asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants. The authors introduce and study tropical notions of volume associated to both divisors and tropical modules, prove optimal asymptotic results for each case, and show that the tropical volume is compatible with the tropicalization of curves.

What carries the argument

tropical volume functions defined for divisors and for tropical modules that govern the asymptotic growth of ranks

If this is right

  • Optimal leading-term asymptotics hold for both the divisor-based and module-based notions of rank.
  • The tropical volumes supply explicit invariants that determine the growth rate exactly in the large-degree limit.
  • Tropical volumes obtained directly on the curve agree with those arising from tropicalizing an algebraic linear series.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same volume constructions might simplify asymptotic calculations for ranks in related discrete settings such as graphs or matroids.
  • Compatibility with tropicalization suggests a route to transfer volume formulas from algebraic geometry to their tropical counterparts.
  • If the volumes remain well-behaved under degeneration, they could serve as a tool for studying limits of algebraic linear series.

Load-bearing premise

The newly introduced tropical volume functions for divisors and modules are well-defined and capture the asymptotic rank growth for the standard notions of linear series on tropical curves without further restrictions on the curve or the series.

What would settle it

A concrete tropical curve and linear series where the observed asymptotic rank growth rate fails to match the value computed from the corresponding tropical volume function.

Figures

Figures reproduced from arXiv: 2604.11166 by Alex K\"uronya, Ana Maria Botero, Eduardo Vital.

Figure 1
Figure 1. Figure 1: Asymptotic behavior of rpℓDq{ℓ. A special case of Inequalities (1) is when degpDq “ rpDq, which implies volT GpDq “ degpDq. We illustrate this case in the following Example. Example 2.5. Consider the multigraph G in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A multigraph G, where rpDq “ degpDq. say v, in G to identify PicpGq “ Zv. Thus, for any divisor D with degpDq ě ´1, it follows that rpDq “ degpDq. Indeed, given D P Divd pGq, it follows that D „ dv. By Inequalities (1), it follows that volT GpDq “ degpDq [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A path in G, from p to q. If ev1,N0 ă valpv1q, then there exists ℓ1 ď valpv1q ´ ev1,N0 such that after ℓ1 firings from FpEN0 q—which includes firing from p P NN0—we obtain ev1,N0`ℓ1 ě valpv1q. This means that: 1) ev1,ℓ ě 0 for all ℓ ě N0 ` ℓ1, and; 2) to increase ev2,N0 we need at most valpv1q ´ev1,N0 `1 firings. Thus, after finitely many increases of ev2,N0 , say ℓ 1 2 , we have [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 4
Figure 4. Figure 4: Graph G. CG “ 2p4 ` 1qp4 ` 1q “ 50. Let D, D1 P DivpGq be divisors given by D “ 90v0 ´ 8v1 ´ 8v2 ´ 8v3 and D1 “ 10v0 ` 2v1 ` 2v2 ` 2v3. As in (the proof of) Proposition 2.6, degpDq ě CG. Now, define E :“ D ´ D1 “ 80v0 ´ 10v1 ´ 10v2 ´ 10v3. Note that FpEq “ tv0u and N pEq “ tv1, v2, v3u. Firing from FpEq, repeatedly, 12 times, we obtain the divisor E12 “ 56v0 ` 2v1 ´ 10v2 ` 2v3. Our new partition of V pGq i… view at source ↗
Figure 5
Figure 5. Figure 5: The function fe on e, which is constant in C ∖ teu. (CD1) degpF1|VpCq q ě degpF|VpCq q, and the inequality is strict if there exists e1 P EpCq with degpF `|e ˝ 1 q ě 2; (CD2) degpF ´ 1 q ě degpF ´q; (CD3) deppr2, v2q “ depp2, v2q ´ s1. As our only condition is degpF `|e ˝ q ě 2, we repeat this procedure until we reach Fn „ F such that degpFn|e ˝ q ď 1 for all edges e. We say that Fn has degree concentrated… view at source ↗
Figure 6
Figure 6. Figure 6: The global rational function fv in a sv-neighborhood of the vertex v, with valpvq “ 3. Note that, degpF ´ 1 q ě degpF ´q, and with strict inequality if pi,v P SupppF ´q for some v P FpFq and i P t1, . . . , valpvqu. If necessary, concentrate the degree of F1 to obtain a [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A minimal path in C that connects p to q. Since p P VpCq ∖ VN0 it follows that ep,ℓ ě valppq for all ℓ P N. Suppose ev1,N0 ă valpv1q. As p P FpEN0 q, either: 1) ev1,N0`1 ě ev1,N0 ` 1 or 2) there exists p1 P α ˝ 1 such that ep1,N0`1 ě ep1,N0 ` 1. Let s1 :“ dα1 pp, p1q. Now, we can see that either: 1) ev1,N0`2 ě ev1,N0`1 ` 1 or 2) there exists p2 P α ˝ 1 such that ep2,N0`2 ě ep2,N0`1 ` 1 and dα1 pp2, v1q “ d… view at source ↗
Figure 8
Figure 8. Figure 8: Compact tropical curve that consists of one edge. Now, let D “ v1 P DivpΓq, in this case RpDq “ xφ0, φ1y where φ0 “ 0 and φ1pxq “ x. As rindpDq “ 2 and rBNpDq “ 1, it follows that RpDq is a tropical linear series. Let Σ Ď RpDq be the following T-submodule Σ :“ tφ P RpDq | φpv2q ´ φpv1q ă 1u Y t8u. Since there does not exist φ P Σ such that D `divpφq ě v2, it follows that rBNpΣq “ 0. As φ0 and 1 2 d φ0 ‘ φ1… view at source ↗
Figure 9
Figure 9. Figure 9: The compact tropical curve Γ. Let KΓ “ v1 ` v2 be its canonical divisor. Directly from the definition it follows that rBNpKΓq “ 1. On the other hand, let φ1, φ2, φ3 P PLpΓq be such that each φi is constant equal to φipvj q in each loop, and φi has constant slope equal to i ´ 2 in the edge, for each i “ 1, 2, 3. Thus, it is clear that φi P RpKΓq and also that φ1, φ2 and φ3 are tropically linearly independen… view at source ↗
Figure 10
Figure 10. Figure 10: The curve Γ and the functions φx,y. Let φx,y be a function with slope 1 from v :“ 0 to x and from v to y, as in [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The “lollipop” tropical curve Γ. Let KΓ “ v be the canonical divisor of Γ. Let ℓ P N and consider ℓKΓ. Analyzing the possible slopes on the edge, for a function φ P RpℓKΓq, which ranges from 0 to ℓ, we can see that rindpℓKΓq “ ℓ ` 1. On the other hand, it follows that rBNpℓKΓq “ ℓ ´ 1, for all ℓ ě 1. Thus, it follows that rindpℓKΓq ą rBNpℓKΓq ` 1, for each ℓ ě 1. This shows that RpℓKΓq is not a tropical l… view at source ↗
Figure 12
Figure 12. Figure 12: The family of conics C{R with smooth generic fiber CK, and special fiber Ck, a nodal curve of compact type with two components (on the left), and the dual multigraph Γ of the special fiber Ck (on the right). The dual graph Γ of Ck has two vertices vx, vy, that correspond to the components V pxq, V pyq Ă Ck respectively, and one edge e, which corresponds to the nodal singu￾larity. See the graph Γ in [PITH… view at source ↗
Figure 13
Figure 13. Figure 13: to visualize the tropical curve troppXrq. x y p1, 0q p0, 1q p´1, ´1q w x y p1, 0q p0, 1q p´1, ´1q w [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The family C{R with smooth generic fiber CK, and special fiber Ck – V pxyzq, a union of three P 1 k ’s meeting pairwise (on the left), and the dual multigraph Γ of the special fiber Ck (on the right). Let ρ : DivpCKq Ñ DivpΓq be the specialization map. Note that ρpDKq “ 2w ` vz; see [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
read the original abstract

We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and study tropical notions of volume associated to both divisors and tropical modules. We prove optimal asymptotic results for each case. In addition, we show that the tropical volume is compatible with the tropicalization of curves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript shows that the asymptotic behavior of the two main competing notions of rank (Baker-Norine and module-theoretic) of a linear series on a tropical curve is governed by newly introduced tropical volume functions associated to divisors (via a limiting process on ranks of multiples) and to tropical modules (via a filtration-based construction on graded pieces). Optimal asymptotic results with matching upper and lower bounds are proved for each, and compatibility with algebraic tropicalization is established via a continuity argument under degeneration. The volumes are shown to be well-defined on arbitrary tropical curves without additional regularity assumptions.

Significance. This work strengthens the dictionary between tropical and algebraic geometry by providing a direct tropical analogue of volume theory, with the central strengths being the well-definedness of the tropical volumes on arbitrary curves, the matching asymptotic bounds for both rank notions, and the degeneration-continuity argument for compatibility. These features make the results potentially useful for transferring asymptotic information between the two settings.

minor comments (2)
  1. [Introduction] The introduction would benefit from a brief explicit comparison of the two rank notions (Baker-Norine versus module-theoretic) with a short example on a simple tropical curve to orient readers.
  2. [Definition of tropical volume for divisors] In the section defining the tropical volume for divisors, the limiting process could be illustrated with one concrete low-genus computation to make the construction more accessible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the recognition of its significance in strengthening the tropical-algebraic dictionary, and the recommendation for minor revision. We appreciate the emphasis on the well-definedness of the tropical volumes, the optimal asymptotic bounds, and the degeneration-continuity argument.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation defines tropical volume functions independently via a limiting process on ranks of multiples for divisors and a filtration on graded pieces for modules. These are proven well-defined on arbitrary tropical curves, after which the paper establishes matching asymptotic bounds for the two rank notions and compatibility under tropicalization via continuity. No step reduces by construction to its inputs, no self-citation chain carries the central claim, and the proofs rely on external tropical geometry tools without renaming or smuggling ansatzes. The chain from definitions to asymptotics is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on the standard framework of tropical curves and linear series together with the newly introduced volume functions; no explicit free parameters or invented entities beyond the volumes are visible in the abstract.

axioms (2)
  • domain assumption Existence and relevance of the two main competing notions of rank for linear series on tropical curves
    The abstract refers to these notions as established in the field without re-deriving them.
  • domain assumption Tropicalization preserves the relevant volume data for curves
    The compatibility statement is presented as a result but presupposes the standard tropicalization functor.
invented entities (2)
  • Tropical volume associated to divisors no independent evidence
    purpose: To control asymptotic rank behavior
    Newly defined in the paper to parallel algebraic volumes.
  • Tropical volume associated to tropical modules no independent evidence
    purpose: To control asymptotic rank behavior
    Newly defined in the paper to parallel algebraic volumes.

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Works this paper leans on

10 extracted references · 10 canonical work pages

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