The Tropical Moduli Space of Degree-3 Rational Maps

Abstract: We construct and study the tropical moduli space (\mathcal{M}_3^{\mathrm{trop}}) of degree-$3$ tropical rational maps (\mathbb{T}\PP^1 \to \mathbb{T}\PP^1) up to post-composition. Using a combinatorial description in terms of slope sequences, we classify all such maps and show that there are exactly ten combinatorial types. This yields a polyhedral model of (\mathcal{M}_3^{\mathrm{trop}}) parametrized by gap lengths between break points. We determine the automorphism groups and obtain a stratification by explicit linear conditions. We also relate the construction to tropical Hurwitz theory and describe a natural compactification via degenerations of the parameters.

• The paper constructs a four-dimensional open polyhedral complex for degree-3 tropical rational maps, enumerating ten combinatorial types and specifying their sign-compatible degeneration rules.
• The paper proves that map symmetries are limited to the trivial group or Z/2Z, with nontrivial symmetry characterized by palindromic slopes and the linear condition ℓ₁ = ℓ₃.
• The paper recovers the tropical Hurwitz number h₃ = 9 through a nine-sheeted branch map and proposes a rigid compactification with six boundary components.

Overview and main results

This paper constructs an explicit combinatorial model for the tropical moduli space $M_3^{trop}$ of degree-3 tropical rational maps $\varphi: T^1 \to T^1$, considered up to post-composition by automorphisms of the target. The approach is entirely piecewise-linear: a map is determined, up to additive constant, by its ordered break points and its slope sequence $(s_0,\dots,s_k)$ with $s_0 = s_k = 3$. The tropical Riemann–Hurwitz constraint $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$, together with positivity of slopes (forced by properness), reduces the classification to a finite enumeration of integer sequences $(\delta_i)$ with $\sum |\delta_i| = 4$ and $\sum \delta_i = 0$.

The paper's central results are:

• Enumeration: exactly ten combinatorial types of degree-3 tropical rational maps exist up to source reversal — five generic types with four simple break points ($k=4$), three types with one double and two simple ramification points ($k=3$), and two fully degenerate two-break-point types ($\varphi: T^1 \to T^1$0). The case $\varphi: T^1 \to T^1$1 is impossible.
• Moduli construction: $\varphi: T^1 \to T^1$2 is assembled as a polyhedral complex whose maximal cells are cones $\varphi: T^1 \to T^1$3 parametrized by gap lengths between break points plus a translation parameter; gluing occurs along faces where gap lengths vanish, but only when adjacent slope jumps share the same sign. Degenerations with opposite-signed jumps decrease total variation and exit the moduli space, so $\varphi: T^1 \to T^1$4 is naturally an open polyhedral complex of dimension $\varphi: T^1 \to T^1$5.
• Stratification: automorphism groups are severely restricted — always trivial or $\varphi: T^1 \to T^1$6 — yielding a stratification by explicit linear conditions on gap parameters.
• Hurwitz count: the branch map over configurations of four labeled branch points is a 9-sheeted cover, recovering the classical/tropical Hurwitz number $\varphi: T^1 \to T^1$7.
• Compactification: adjoining boundary strata at $\varphi: T^1 \to T^1$8 and $\varphi: T^1 \to T^1$9 produces a compact polyhedral complex of pure dimension 3 (after source quotienting).

Enumeration of combinatorial types

The classification rests on a sharp bound: since $(s_0,\dots,s_k)$0 and total variation is 4, every intermediate slope satisfies $(s_0,\dots,s_k)$1. Enumerating integer sequences $(s_0,\dots,s_k)$2 with $(s_0,\dots,s_k)$3 and $(s_0,\dots,s_k)$4 then yields, case by case:

Break points	Ramification	Slope sequences
$(s_0,\dots,s_k)$5	four simple	$(s_0,\dots,s_k)$6, $(s_0,\dots,s_k)$7, $(s_0,\dots,s_k)$8, $(s_0,\dots,s_k)$9, $s_0 = s_k = 3$0
$s_0 = s_k = 3$1	one double, two simple	$s_0 = s_k = 3$2, $s_0 = s_k = 3$3, $s_0 = s_k = 3$4
$s_0 = s_k = 3$5	two double	$s_0 = s_k = 3$6, $s_0 = s_k = 3$7

Each type determines a weighted abstract tropical curve whose vertices are the break points, whose inner edges have lengths $s_0 = s_k = 3$8, and whose edge multiplicities are the local ramification weights $s_0 = s_k = 3$9. This curve carries the automorphism group that governs the symmetry stratification discussed below.

Polyhedral structure of the moduli space

For fixed combinatorial type $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$0 with $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$1 break points, the parameter space is $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$2: the positive coordinates are gap lengths, and the residual translation reflects that post-composition quotients out target translations but not source translations. Maximal cells therefore have dimension $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$3, so $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$4.

The gluing mechanism is the key structural feature. When $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$5, adjacent slope jumps merge into $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$6. If the signs agree, total variation is preserved and the limit is a valid degree-3 map of lower complexity, gluing cone $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$7 to $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$8. If the signs disagree, total variation strictly decreases and the limit violates Riemann–Hurwitz, leaving the moduli space. Consequently some boundary faces are included while others are not — a point the authors state plainly, noting that $\sum_i |s_i - s_{i-1}| = 2d - 2 = 4$9 must be regarded as open and that a compactification requires adjoining the missing strata.

Automorphism groups and symmetry stratification

The ambient group $(\delta_i)$0 admits only the trivial group and $(\delta_i)$1 as finite subgroups, and surjectivity of $(\delta_i)$2 forces any automorphism pair $(\delta_i)$3 to be determined by its source component. The result is a rigidity theorem: $(\delta_i)$4 for all degrees, and no tropical rational map admits a Klein four-group of symmetries. This contrasts sharply with the classical setting, where $(\delta_i)$5 contains both $(\delta_i)$6 and $(\delta_i)$7 strata defined by polynomial invariant relations; in the tropical picture the $(\delta_i)$8 stratum disappears entirely and the remaining condition becomes linear.

The explicit criterion is as follows. Nontrivial symmetry requires a palindromic slope sequence. For palindromic types, $(\delta_i)$9 if and only if $\sum |\delta_i| = 4$0 (equivalently $\sum |\delta_i| = 4$1); otherwise the automorphism group is trivial. Non-palindromic types always have trivial automorphism group, and the two-break-point types $\sum |\delta_i| = 4$2 and $\sum |\delta_i| = 4$3 are automatically symmetric. The resulting stratification has three tiers:

Stratum	Dimension	Condition
Generic	4	trivial automorphism group, open dense
Symmetric	3	palindromic type with $\sum |\delta_i| = 4$4
Symmetric boundary	2	two-break-point types

The replacement of invariant-theoretic equations by the single linear relation $\sum |\delta_i| = 4$5 exemplifies the general principle that tropicalization converts algebraic conditions into piecewise-linear ones.

Tropical Hurwitz theory

To align with the tropical Hurwitz space, which parametrizes covers up to isomorphism of the source, the authors form the fully quotiented space $\sum |\delta_i| = 4$6, reducing dimension from 4 to 3. The branch map sends a cover to the configuration of its four labeled branch points modulo translations, landing in a 3-dimensional configuration space — notably distinct from $\sum |\delta_i| = 4$7, which is one-dimensional, a discrepancy the authors attribute to the reduced size of $\sum |\delta_i| = 4$8 relative to $\sum |\delta_i| = 4$9.

The main enumerative result is that over the locus of four distinct branch points, the branch map is a 9-sheeted cover counted with multiplicity, recovering the Cavalieri–Johnson–Markwig correspondence value $\sum \delta_i = 0$0. The count is distributed across the five generic types as follows: type $\sum \delta_i = 0$1 contributes 1 admissible assignment, while each of $\sum \delta_i = 0$2, $\sum \delta_i = 0$3, $\sum \delta_i = 0$4, and $\sum \delta_i = 0$5 contributes 2, summing to 9. Monotonicity of segments constrains which assignments of critical points to ordered branch points are compatible with each slope sequence; generic covers have trivial automorphism group, so each assignment carries multiplicity 1. This gives a concrete combinatorial realization of the correspondence theorem in degree 3.

Compactification

The compactification $\sum \delta_i = 0$6 extends each gap coordinate to $\sum \delta_i = 0$7, producing a compact polyhedral complex of pure dimension 3. Its codimension-1 boundary consists of six components: three collision divisors $\sum \delta_i = 0$8 where $\sum \delta_i = 0$9, and three components at infinity where $k=4$0, recording configurations in which adjacent break points separate indefinitely. Higher-codimension strata arise from simultaneous collisions or mixed collision/infinity conditions.

Two caveats are stated explicitly. First, this compactification is rigid: the source curve $k=4$1 is held fixed, unlike the standard stable-map compactification of Gathmann–Kerber–Markwig, which allows the source to degenerate into additional components. Second, a full intersection theory in the sense of weighted balanced complexes would require assigning multiplicities and incorporating stable degenerations; the authors leave this refinement open, providing only the underlying compact polyhedral framework.

Relation to the classical moduli space

Over a non-Archimedean field $k=4$2, tropicalization of a rational function $k=4$3 via coordinate-wise valuation induces a well-defined map $k=4$4 compatible with post-composition. Every combinatorial type is realizable by appropriate choices of valuations, and varying valuations realizes arbitrary gap lengths, so the image meets every maximal cone and is dense. Thus $k=4$5 serves as a polyhedral model for $k=4$6.

The invariant-theoretic description of $k=4$7 via weighted invariants $k=4$8 tropicalizes to piecewise-linear functions on $k=4$9, whose corner loci reproduce the automorphism stratification; the linear condition $k=3$0 appears as the "tropical shadow" of the corresponding invariant relations. The authors also outline a map from $k=3$1 into the tropical weighted projective space $k=3$2, mirroring the classical embedding, though this is presented at the level of a construction sketch rather than a developed theory.

Connection to ReLU networks

A recurring theme is that degree-3 tropical rational functions coincide with continuous piecewise-linear functions realized by shallow feedforward ReLU networks with four activation thresholds. The sorted thresholds play the role of break points, gap lengths are edge lengths of the associated weighted tropical curve, and the automorphism group of that curve acts on $k=3$3 to induce the symmetry stratification. Invoking vanishing results for cohomology on tropical curves (tropical Hilbert Theorem 90), the authors argue that canonical $k=3$4-invariant representatives correspond exactly to functional equivalence classes of ReLU networks independent of concrete weights, and they suggest that boundary strata encode neuron-death/pruning events. This connection is asserted rather than developed: no loss-landscape analysis or pruning algorithm appears in the paper, and the cited applications are deferred to companion work on higher-dimensional piecewise-linear models.

Limitations and open questions

Several limitations are acknowledged within the paper itself. The compactification is rigid rather than stable, so it does not capture degenerations of the source curve, and the promised intersection-theoretic structure (multiplicities, balanced weights) is left undeveloped. The Hurwitz count is established only for the fully simple ramification profile in degree 3; extension to higher degree, where slope-sequence combinatorics grow richer, remains open. The tropicalization map is shown to be dense but no statement about surjectivity onto specific strata, nor about compatibility with the full invariant-theoretic embedding, is proved. Finally, the ReLU-network interpretation — including pruning criteria and symmetry-induced saddle points — is proposed as motivation without formal results, and its substantiation is deferred to future work.

Conclusion

This paper provides a complete, explicitly computable model of the degree-3 tropical rational map moduli problem: ten combinatorial types, a four-dimensional open polyhedral complex glued along sign-compatible degenerations, a rigidity theorem confining symmetries to $k=3$5 and $k=3$6 with the single linear symmetry condition $k=3$7, a combinatorial recovery of the Hurwitz number $k=3$8, and a compact cube-like compactification with six boundary divisors. The degree-3 case demonstrates concretely how invariant-theoretic conditions on $k=3$9 tropicalize to linear geometry, and it supplies the foundational example for the announced higher-dimensional extensions.