q-Reduced Divisors on Metric and Tropical Graphs

• q-reduced divisors are canonical representatives in a divisor class on graphs or metric graphs, defined by being effective away from a chosen base point and immune to further chip-firing moves.
• Chip-firing algorithms and saturated cut conditions play a central role in ensuring the uniqueness and stability of q-reduced divisors across different linear systems.
• Tropical projection methods extend the notion of q-reduced divisors to broader tropically convex sets, linking them to harmonic morphisms and the geometry of tropical trees.

Searching arXiv for recent and foundational papers on q-reduced divisors, tropical reduced divisors, and related divisor reduction notions. A $q$-reduced divisor is a divisor-theoretic normal form determined by a chosen base point $q$ on a graph or metric graph. In the Baker–Norine framework and its metric-graph extensions, it is the unique divisor in a linear equivalence class that is effective away from $q$ and admits no effective chip-firing move supported in a closed subset disjoint from $q$. This uniqueness makes $q$-reduced divisors a canonical choice of representatives for divisor classes, while later work reinterprets them as tropical projections and extends the construction from complete linear systems to arbitrary compact tropically convex linear systems (Amini, 2010, Luo, 2018).

1. Classical definition on graphs and metric graphs

On a metric graph $\Gamma$, a divisor is a finite formal linear combination

$D=\sum_{p\in\Gamma} m_p\,(p),$

with integer coefficients in $\Div(\Gamma)$ and real coefficients in $\RDiv(\Gamma)$. Its degree is $\deg D=\sum m_p$, and it is effective when all coefficients are nonnegative. Linear equivalence is defined by principal divisors $q$0, where $q$1 is a continuous piecewise-affine function with integral slopes; two divisors $q$2 are linearly equivalent when

$q$3

The complete linear system of $q$4 is

$q$5

where $q$6 (Luo, 2018).

Fix a base point $q$7. An integer divisor $q$8 is $q$9-reduced if it satisfies two conditions. First, it is effective away from $q$0: $q$1 for all $q$2. Second, for every nonempty closed subset $q$3, there exists a boundary point $q$4 such that

$q$5

where $q$6 counts the open segments of $q$7 incident to $q$8. Equivalently, no effective chip-firing move can be performed inside a closed subset disjoint from $q$9 (Luo, 2018).

Amini’s metric-graph formulation replaces discrete vertex subsets by closed connected subsets, or cuts. If $q$0 is a closed connected subset and $q$1, then $q$2 is saturated with respect to $q$3 and $q$4 when $q$5, and $q$6 is saturated when all boundary points are saturated. In this language, a divisor is $q$7-reduced if it is effective away from $q$8 and no cut $q$9 with $q$0 is saturated (Amini, 2010).

The foundational structural statement is existence and uniqueness: for any divisor $q$1 and any point $q$2, there exists a unique divisor $q$3 that is $q$4-reduced. If $q$5 is effective, then this representative lies in $q$6 (Luo, 2018). This canonicality is what makes $q$7-reduced divisors central in chip-firing, tropical Brill–Noether theory, and the study of linear systems on metric graphs.

2. Chip-firing, saturated cuts, and uniqueness

The condition defining $q$8-reducedness is best understood through chip-firing. Principal divisors encode chip-firing moves, and the inequality

$q$9

at some boundary point of every closed subset $\Gamma$0 means that every attempted firing away from $\Gamma$1 fails somewhere on the boundary. In the discrete language, one cannot perform an effective chip-firing move supported inside any closed subset disjoint from $\Gamma$2 (Luo, 2018). In the metric-graph language, no cut avoiding $\Gamma$3 is saturated (Amini, 2010).

This criterion explains the uniqueness theorem. If two linearly equivalent divisors were both $\Gamma$4-reduced, their difference would be $\Gamma$5 for a nonconstant rational function $\Gamma$6. Taking a component of the maximum locus of $\Gamma$7 not containing $\Gamma$8 produces a cut whose boundary behavior forces negativity away from $\Gamma$9, contradicting effectivity away from the base point (Amini, 2010). The uniqueness is therefore not merely combinatorial; it is a potential-theoretic equilibrium statement.

The same viewpoint clarifies the role of Dhar’s burning algorithm and its metric analogues. In finite graphs, burning tests whether a divisor is $D=\sum_{p\in\Gamma} m_p\,(p),$0-reduced by propagating a fire from $D=\sum_{p\in\Gamma} m_p\,(p),$1; in metric graphs, the corresponding obstruction is saturation of a closed connected subset disjoint from $D=\sum_{p\in\Gamma} m_p\,(p),$2. This suggests that $D=\sum_{p\in\Gamma} m_p\,(p),$3-reduced divisors are canonical stopping configurations for chip-firing dynamics. The literature cited here does not formulate this as a variational principle at first, but later work makes that interpretation explicit.

3. Reduced-divisor maps and infinitesimal variation

When $D=\sum_{p\in\Gamma} m_p\,(p),$4 has nonnegative rank, the unique $D=\sum_{p\in\Gamma} m_p\,(p),$5-reduced divisor $D=\sum_{p\in\Gamma} m_p\,(p),$6 linearly equivalent to $D=\sum_{p\in\Gamma} m_p\,(p),$7 defines a map

$D=\sum_{p\in\Gamma} m_p\,(p),$8

If $D=\sum_{p\in\Gamma} m_p\,(p),$9, this reduced-divisor map is integral affine, hence continuous (Amini, 2010). The image lies in the $\Div(\Gamma)$0-skeleton of the polyhedral complex $\Div(\Gamma)$1 (Amini, 2010).

Amini’s key contribution is an explicit infinitesimal description of how $\Div(\Gamma)$2 changes when the base point moves along an edge. Starting from $\Div(\Gamma)$3, one studies a tangent direction $\Div(\Gamma)$4 at $\Div(\Gamma)$5. Either $\Div(\Gamma)$6 remains reduced for nearby points, in which case the map is locally constant, or there is a maximal cut $\Div(\Gamma)$7 saturated with respect to $\Div(\Gamma)$8 such that $\Div(\Gamma)$9 meets the chosen small segment only at $\RDiv(\Gamma)$0. One then defines the excess

$\RDiv(\Gamma)$1

and constructs a rational function $\RDiv(\Gamma)$2 whose divisor moves chips along the selected segment and along the outgoing directions from $\RDiv(\Gamma)$3. For sufficiently small $\RDiv(\Gamma)$4, the divisor

$\RDiv(\Gamma)$5

is exactly the $\RDiv(\Gamma)$6-reduced divisor (Amini, 2010).

This local model shows that the reduced-divisor map is piecewise integral affine because each chip either remains fixed or moves with integer slope determined by the excess and the geometry of $\RDiv(\Gamma)$7. It also explains why the image of $\RDiv(\Gamma)$8 in $\RDiv(\Gamma)$9 is effectively graph-like. A plausible implication is that $\deg D=\sum m_p$0-reduced divisors provide a tropical analog of moving frames in a linear system: the base point moves continuously, while the canonical representative changes by controlled chip-firing events.

The reduced-divisor map has several geometric applications. A divisor $\deg D=\sum m_p$1 is very ample exactly when $\deg D=\sum m_p$2 is injective (Amini, 2010). This criterion yields a classification of tropical curves with non-very-ample canonical divisor and implies that if $\deg D=\sum m_p$3, then $\deg D=\sum m_p$4 is very ample (Amini, 2010). The same framework also gives the existence of Weierstrass points on tropical curves of genus at least two and yields a streamlined proof of Luo’s theorem on rank-determining sets (Amini, 2010).

4. Tropical projections and generalized $\deg D=\sum m_p$5-reduced divisors

Luo reinterprets reduced divisors through idempotent analysis and tropical convexity. For a compact tropically convex set $\deg D=\sum m_p$6 and an effective divisor $\deg D=\sum m_p$7 of degree $\deg D=\sum m_p$8, there is a unique divisor $\deg D=\sum m_p$9 minimizing the $q$00-pseudonorm function

$q$01

for all $q$02 (Luo, 2018). In divisor-theoretic terms, the minimizer is characterized by

$q$03

for all $q$04, equivalently by the condition

$q$05

for all $q$06 (Luo, 2018).

Taking $q$07 and $q$08, Luo proves that the classical $q$09-reduced divisor is exactly the tropical projection of $q$10 onto $q$11. More precisely, for $q$12, the following are equivalent: $q$13 is $q$14-reduced in the classical sense; for each $q$15, $q$16; and

$q$17

Thus classical $q$18-reduced divisors are precisely tropical projections (Luo, 2018).

This gives a canonical extension to arbitrary compact tropically convex sets. If $q$19 is compact and tropically convex, the $q$20-reduced divisor in $q$21 is defined to be

$q$22

When $q$23 is a complete linear system, this recovers the Baker–Norine notion; when $q$24 is a proper tropical polytope, it yields a distinguished divisor even though the classical definition had no canonical representative in such a subsystem (Luo, 2018).

The same framework produces the reduced divisor map

$q$25

which extends Amini’s reduced-divisor map from complete linear systems to all compact tropically convex linear systems. The map is continuous and piecewise-linear, and projections are compatible with nested tropical convex subsets: $q$26 whenever $q$27 is compact and tropically convex (Luo, 2018).

5. Tropical trees, harmonic morphisms, and rank

The tropical projection viewpoint makes $q$28-reduced divisors part of a broader geometry of linear systems. A linear system on $q$29 is defined as a tropical polytope $q$30 containing $q$31, and a tropical tree is a $q$32-dimensional linear system of the form

$q$33

Its extremals are the leaves, and it is dominant when

$q$34

(Luo, 2018).

For any compact tropically convex $q$35, the reduced divisor map $q$36 assembles the $q$37-reduced divisors into a global map from $q$38 to divisor space. When $q$39 is a dominant tropical tree of degree $q$40, this map is a continuous surjective finite map that is piecewise linear with integral slopes. Luo proves that $q$41 is a pseudo-harmonic morphism and extends, after a suitable modification $q$42, to a harmonic morphism

$q$43

of degree $q$44 (Luo, 2018).

Conversely, a pseudo-harmonic morphism to a metric tree that extends to a harmonic morphism of degree $q$45 reconstructs a dominant tropical tree $q$46, and the given map coincides with $q$47 (Luo, 2018). This establishes a one-to-one correspondence, up to natural modifications, between dominant tropical trees of degree $q$48 and degree $q$49 harmonic morphisms from modifications of $q$50 to metric trees (Luo, 2018).

This correspondence feeds into Luo’s geometric rank. For a linear system $q$51, the geometric rank $q$52 is defined using continuous rank-$q$53 maps

$q$54

whose images are tropically convex and dominate the input divisors. Luo proves that the following are equivalent: $q$55 is stably $q$56-gonal, $q$57 admits a degree $q$58 dominant tropical tree, and there exists a divisor of degree $q$59 with geometric rank $q$60 (Luo, 2018). This resolves the discrepancy between divisorial gonality and stable gonality by showing that the correct tropical-linear notion matching harmonic morphisms is geometric rank rather than the conventional Baker–Norine rank alone (Luo, 2018).

A common misconception is that $q$61-reduced divisors are relevant only to complete linear systems. Luo’s theory shows that the classical complete-system case is only one instance of a projection principle on tropical convex sets. Another misconception is that reduced-divisor maps are merely bookkeeping devices; in the dominant tropical tree setting, they are exactly the divisor-space avatars of harmonic morphisms to trees.

6. Related reduction notions beyond metric graphs

The term $q$62-reduced divisor is not used uniformly across all divisor theories. In the Kowalevski-top setting, the relevant notion is reduction of divisor classes on elliptic curves. There, one works with genus-$q$63 curves, and each divisor class has a unique effective divisor of minimal degree in its linear equivalence class. On an elliptic curve this means a unique degree-$q$64 representative, so reduced divisors correspond to single points, with semi-reduced divisors appearing as degree-$q$65 intermediates. The paper explicitly states that it does not use the term “$q$66-reduced divisor”; rather, the reduction is with respect to the point at infinity, which in general language would be reduction with base point $q$67 (Tsiganov, 2020).

This genus-$q$68 picture is formally analogous to $q$69-reduction on graphs in that a chosen base point determines a canonical representative, but the mechanics are different. On elliptic curves the reduction is expressed through the Abel–Jacobi group law and geometric intersection constructions such as passing from $q$70 to the third intersection point $q$71 on a cubic or quartic model (Tsiganov, 2020). A plausible implication is that “$q$72-reduced” language in algebraic geometry should be interpreted relative to a chosen base point for the Abel map, rather than through chip-firing.

Arakelov theory supplies another distinct generalization. Tran defines strongly $q$73-reduced divisors of a number field by requiring that $q$74 be primitive in a fractional ideal $q$75 and satisfy

$q$76

while Ho generalizes this further to $q$77-reduced divisors in real quadratic fields, with testing algorithms polynomial in $q$78 (Tran, 2015, Tran, 2014). These are parameterized relaxations of Arakelov reducedness rather than base-point reductions. The data explicitly present them as conceptually relevant to possible $q$79-type generalizations, but not as instances of the Baker–Norine $q$80-reduced notion itself (Tran, 2015, Tran, 2014).

The modern landscape therefore contains at least three distinct uses of “reduction.” On metric graphs, $q$81-reduction is a base-point-normalization inside a chip-firing equivalence class. On elliptic curves, reduced divisors are minimal-degree representatives in divisor classes, typically relative to the point at infinity. In Arakelov theory, $q$82-reduced or strongly $q$83-reduced divisors are approximate shortest-vector conditions in ideal lattices. The common thread is the selection of canonical representatives; the underlying geometry, however, differs substantially across the tropical, algebraic, and arithmetic settings.