Tropical Linear Series Theory

• Tropical linear series are combinatorial structures on metric graphs that mimic algebraic linear series using Baker–Norine rank, tropical independence, and slope conditions.
• They employ tropical submodules and valuated matroids to precisely define dimensions and dependencies, ensuring pure dimensionality and effective tropicalization.
• Their framework supports finite generation, restriction theorems, and lifting techniques, addressing realizability challenges and linking tropical geometry with matroid theory.

Tropical linear series are combinatorial structures on a metric graph $\Gamma$ designed to capture the behavior of linear series on algebraic curves after tropicalization. In the formulation introduced by Jensen and Payne, they isolate the simultaneous roles of Baker–Norine rank and tropical independence, and they are intended to model tropicalizations of possibly incomplete linear series rather than only complete systems $R(D)$ (Jensen et al., 2022). Subsequent work has recast the subject in terms of independence rank, pure dimensionality, and matroidal local and global structure, so the phrase “tropical linear series” now encompasses a family of closely related but not identical formalisms (Chang et al., 27 Aug 2025).

1. Foundational setting on metric graphs

Let $\Gamma$ be a metric graph. A divisor on $\Gamma$ is a formal integer sum

$D=\sum a_v\cdot v,$

and a piecewise-linear function $\varphi\in PL(\Gamma)$ with integer slopes defines a principal divisor

$$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$

where the sum runs over tangent directions $\zeta$ at $v$. Two divisors are linearly equivalent if they differ by a principal divisor. The Baker–Norine rank of a divisor $D$ is the largest integer $R(D)$0 such that for every effective divisor $R(D)$1 of degree $R(D)$2, there exists $R(D)$3 with $R(D)$4; equivalently,

$R(D)$5

If $R(D)$6 is the canonical divisor and $R(D)$7 is the genus, then tropical Riemann–Roch takes the form

$R(D)$8

These notions provide the ambient linear-series theory on which tropical linear series are built (Jensen et al., 2022).

The tropical semiring is $R(D)$9 with $\Gamma$0 and $\Gamma$1. A tropical submodule $\Gamma$2 is closed under tropical linear combinations $\Gamma$3. For a divisor $\Gamma$4, the complete tropical linear series is

$\Gamma$5

which is a tropical submodule. Haase–Musiker–Yu proved that $\Gamma$6 is finitely generated as a tropical module, but its polyhedral dimension and number of generators typically differ from $\Gamma$7, so $\Gamma$8 by itself is not an adequate proxy for the tropicalization of an algebraic linear series (Jensen et al., 2022).

A later rank-theoretic formulation extends this picture from complete systems to arbitrary tropical submodules. For a subsemimodule $\Gamma$9, the tropical rank is the maximal size of a tropically independent family in $\Gamma$0, and the topological dimension of $\Gamma$1 is defined by the associated linear system $\Gamma$2. Amini, Gaubert, and Gierczak proved that for every subsemimodule $\Gamma$3,

$\Gamma$4

and for finitely generated $\Gamma$5, divisorial rank equals tropical rank minus one exactly when the linear system is pure-dimensional (Amini et al., 3 Apr 2025).

2. Tropical independence and the Jensen–Payne definition

For functions $\Gamma$6, a tropical linear combination is

$\Gamma$7

The set is tropically dependent if there exist real numbers $\Gamma$8 such that at every point $\Gamma$9, the minimum in $D=\sum a_v\cdot v,$0 is achieved at least twice. Equivalently, no term $D=\sum a_v\cdot v,$1 achieves the pointwise minimum uniquely anywhere on $D=\sum a_v\cdot v,$2. A useful certificate of independence is the converse condition: if there are constants $D=\sum a_v\cdot v,$3 such that for each $D=\sum a_v\cdot v,$4 some point $D=\sum a_v\cdot v,$5 exists where $D=\sum a_v\cdot v,$6 uniquely attains the minimum, then the family is tropically independent. Jensen and Payne show that such certificates always exist for independent families, via a KKM/Brouwer-type argument (Jensen et al., 2022).

In this framework, a tropical linear series of rank $D=\sum a_v\cdot v,$7 is a tropical submodule $D=\sum a_v\cdot v,$8 satisfying three conditions. First, the rank condition: for every effective divisor $D=\sum a_v\cdot v,$9 of degree $\varphi\in PL(\Gamma)$0, there exists $\varphi\in PL(\Gamma)$1 such that $\varphi\in PL(\Gamma)$2. Second, the tropical independence bound: every set of $\varphi\in PL(\Gamma)$3 functions in $\varphi\in PL(\Gamma)$4 is tropically dependent. Third, the slope-subseries condition: for each tangent vector $\varphi\in PL(\Gamma)$5, if the distinct slopes realized by functions in $\varphi\in PL(\Gamma)$6 are

$\varphi\in PL(\Gamma)$7

then for each $\varphi\in PL(\Gamma)$8, the subset

$\varphi\in PL(\Gamma)$9

contains a tropical linear series of rank $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$0. The definition is recursive, and rank $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$1 tropical linear series are principal submodules $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$2 (Jensen et al., 2022).

A central structural consequence is the slope-count lemma: if $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$3 is a tropical linear series of rank $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$4, then for each tangent direction $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$5 there are exactly $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$6 slopes realized by functions in $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$7 along $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$8. This is one of the main constraints distinguishing tropical linear series from arbitrary tropical submodules of $$\operatorname{div}(\varphi)=\sum \operatorname{ord}_v(\varphi)\cdot v, \qquad \operatorname{ord}_v(\varphi)=-\sum_\zeta s_\zeta(\varphi),$$9.

Jensen and Payne also define a stronger notion. A tropical linear series $\zeta$0 of rank $\zeta$1 is strong if its projectivization $\zeta$2 is a closed, definable subset of $\zeta$3, and if there exists a valuated matroid $\zeta$4 of rank $\zeta$5 on $\zeta$6 such that any valuated circuit produces a tropical dependence

$\zeta$7

This “strong” condition packages the topological and matroidal properties expected from tropicalizations of algebraic linear series (Jensen et al., 2022).

3. Tropicalization from algebraic curves and the restriction theorem

Let $\zeta$8 be a curve over a valued field with skeleton $\zeta$9, and let $v$0 be a linear series of rank $v$1, possibly incomplete. Its tropicalization is

$v$2

Jensen and Payne prove that $v$3 is a strong tropical linear series of rank $v$4. The Baker–Norine rank condition follows because for any effective $v$5 of degree $v$6 on $v$7, there exists $v$8 with $v$9. Every $D$0 functions from $D$1 are tropically dependent because $D$2 has dimension $D$3. The slope-subseries condition is compatible with vanishing and vanishing sequences in metrized complexes. Moreover, $D$4 is closed and definable, and the associated valuated matroid may be chosen realizable over $D$5 (Jensen et al., 2022).

This produces a precise specialization principle. Algebraic linear dependencies in $D$6 specialize to tropical dependencies in $D$7, while tropical independence in $D$8 certifies algebraic linear independence of lifts. In that sense, tropical independence is not merely an analogy with linear algebra: it is a specialization-theoretic obstruction and certificate.

One of the most distinctive results of the 2022 theory is the restriction property. If $D$9 is a connected metric subgraph and $R(D)$00 is a tropical linear series of rank $R(D)$01, then one defines a divisor $R(D)$02 on $R(D)$03 by

$R(D)$04

where the sum runs over tangent vectors $R(D)$05 at $R(D)$06 pointing into $R(D)$07. The restricted module

$R(D)$08

is again a tropical linear series of rank $R(D)$09 on $R(D)$10. A similar statement holds for strong tropical linear series. This invariance under restriction sharply contrasts with the behavior of complete linear systems $R(D)$11, whose local structure often contains extraneous degrees of freedom (Jensen et al., 2022).

4. Finite generation in rank $R(D)$12 and characteristic examples

The main theorem of Jensen and Payne states that every tropical linear series $R(D)$13 of rank $R(D)$14 is finitely generated as a tropical module. More precisely, after choosing a finite vertex set containing $R(D)$15 and all non-$R(D)$16-valent points so that slope data are constant along oriented edges, one constructs edge-local generators $R(D)$17 with prescribed endpoint slopes. A finite set $R(D)$18 controls the exceptional locus: for $R(D)$19, there is a unique divisor $R(D)$20 whose support contains $R(D)$21, and the corresponding function lies in the tropical span of the two edge generators on the edge containing $R(D)$22. From this, one deduces global finite generation (Jensen et al., 2022).

Several consequences are specific to rank $R(D)$23. Minimal finite generating sets are unique up to tropical scaling, and every rank $R(D)$24 tropical linear series is strong. Its projectivization $R(D)$25 is compact and of pure dimension $R(D)$26. The interval classification makes the theorem concrete: for $R(D)$27 and $R(D)$28 of degree $R(D)$29, a rank $R(D)$30 subseries is generated either by two functions $R(D)$31 adapted to endpoint slopes, or by three generators $R(D)$32 when a distinguished midpoint $R(D)$33 produces a “middle bend.” On a genus-$R(D)$34 circle, a rank $R(D)$35 subseries can be generated by two functions whose slopes alternate appropriately along the circle.

These results also clarify what tropical linear series are not. On a lollipop graph with divisor $R(D)$36 at the junction, the complete system $R(D)$37 is not a tropical linear series because the number of slopes realized along the stem differs from the number realized on the loop. On the genus-$R(D)$38 barbell graph, $R(D)$39 is not a tropical linear series, but it contains a unique rank $R(D)$40 tropical linear series. On an interval of length $R(D)$41 with $R(D)$42 at the midpoint, one can construct a non-finitely-generated submodule of $R(D)$43 consisting of “V-shaped” functions $R(D)$44 with $R(D)$45; it satisfies the rank and slope-count properties but fails the tropical dependence bound. There are also divisors of Baker–Norine rank $R(D)$46 for which no rank-$R(D)$47 tropical linear series exists inside $R(D)$48: Luo’s genus-$R(D)$49 “loop with 3 spines” example has this property because the forced functions are tropically independent (Jensen et al., 2022).

Higher-rank examples already reveal realizability obstructions. For a simple rank-$R(D)$50 matroid $R(D)$51, the Levi graph $R(D)$52 and divisor $R(D)$53 admit a rank-$R(D)$54 tropical linear series $R(D)$55; if $R(D)$56 tropicalizes an algebraic series over $R(D)$57, then $R(D)$58 must be realizable over $R(D)$59. This places matroid realizability directly inside the lifting theory of tropical linear series (Jensen et al., 2022).

5. Rank, dimension, and matroidal geometry

A later formulation, due to Jensen and Ulirsch, starts from two ranks attached to a tropical submodule $R(D)$60. The Baker–Norine rank is

$R(D)$61

and the independence rank $R(D)$62 is the largest size of a tropically independent subset of $R(D)$63. For finitely generated or polyhedral $R(D)$64, the projectivization $R(D)$65 satisfies

$R(D)$66

A tropical linear series of dimension $R(D)$67 is then defined as a finitely generated pair $R(D)$68 such that

$R(D)$69

In this formalism, dimension is built into the definition rather than recovered recursively from slope subseries (Chang et al., 27 Aug 2025).

This reformulation yields strong geometric consequences. If $R(D)$70 is a tropical linear series, then $R(D)$71 has pure dimension $R(D)$72. At a nondegenerate divisor $R(D)$73, write $R(D)$74 for the set of connected components of $R(D)$75, and for $R(D)$76 define

$R(D)$77

Then the collection $R(D)$78 is the lattice of flats of a matroid $R(D)$79 of rank $R(D)$80, and the star of $R(D)$81 identifies with the support of the Bergman fan $R(D)$82. If $R(D)$83 comes from tropicalizing an algebraic linear series, then $R(D)$84 is the image of a realizable valuated matroid under a surjective homomorphism of tropical modules, and every local matroid at a nondegenerate divisor is realizable. Conversely, every loopless matroid appears as the local matroid of some tropical linear series, even on an interval or a loop (Chang et al., 27 Aug 2025).

The rank-dimension correspondence has also been established in a broader semimodule setting. Amini, Gaubert, and Gierczak prove that for any subsemimodule $R(D)$85, tropical rank equals topological dimension, and for finitely generated $R(D)$86 the equality

$R(D)$87

is equivalent to pure dimensionality of $R(D)$88. This result explains why the equality between rank and projective dimension is natural in the tropical setting and why failures of pure dimensionality account for discrepancies between complete systems and genuine tropical linear series (Amini et al., 3 Apr 2025).

6. Competing definitions, realizability, and open problems

The relation between different definitions of tropical linear series has become a subject in its own right. Burkholder proves that tropical linear series in the Farkas–Jensen–Payne sense are combinatorial limit linear series in the Amini–Gierczak sense: one constructs local slope-index arrays $R(D)$89 from a tropical linear series, shows that each $R(D)$90 is the redundant closure of a rank-$R(D)$91 permutation array, and then obtains an $R(D)$92-slope structure with the required Baker–Norine and local-rank properties. The converse fails in rank $R(D)$93: using the Vámos matroid on an interval, one gets a finitely generated admissible tropical submodule that is a combinatorial limit linear series but violates the FJP recursion axiom. The converse holds trivially for $R(D)$94, fails for $R(D)$95, likely fails for $R(D)$96, and remains open for $R(D)$97 (Burkholder, 19 Jun 2025).

Realizability questions are especially sharp for canonical divisors. In the framework of enhanced level graphs, if $R(D)$98 is an effective tropical canonical divisor, then $R(D)$99 is realizable if and only if every inconvenient vertex lies in a simple cycle $\Gamma$00 with $\Gamma$01 and every horizontal edge lies in a simple cycle $\Gamma$02 with $\Gamma$03. The realizable locus $\Gamma$04 is tropically convex, definable, and closed, hence an abstract polyhedral complex; it contains maximal cells of dimension $\Gamma$05. If $\Gamma$06 has no disjoint cycles, then every canonical divisor is realizable, whereas disjoint cycles can produce non-realizable canonical divisors (Dupraz, 26 Jun 2025).

Several open problems organize the current research agenda. Jensen and Payne ask whether tropicalizations of algebraic linear series are always finitely generated as tropical modules, whether all tropical linear series—or at least all strong ones—are finitely generated, whether every tropical linear series is strong, and whether the proposed rank

$\Gamma$07

satisfies a Riemann–Roch formula. Jensen and Ulirsch add questions about whether every tropical linear series is matroidal, whether all tropical linear series on an interval or loop arise from explicit valuated-matroid constructions, how to describe local structure at degenerate divisors, how multiplication maps behave, and whether one can construct moduli spaces of tropical or matroidal linear series with workable dimension theory and local-matroid stratifications (Jensen et al., 2022, Chang et al., 27 Aug 2025).

A common misconception is that tropical linear series are simply complete linear systems on graphs. The examples above show otherwise: $\Gamma$08 may have the wrong dimension, the wrong slope structure, or the wrong dependence behavior. The modern theory treats tropical linear series as a narrower class of tropical submodules whose rank, dimension, and local combinatorics mimic algebraic linear series closely enough to support restriction theorems, lifting criteria, and matroidal obstruction theory.