I‑AIMS Institute for Artificial Intelligence
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The Tropical Moduli Space of Degree-3 Rational Maps

Tanush Shaska, Mohammad-Reza Siadat

I-AIMS preprint sh-117 · 2026

We study tropical rational functions on the tropical projective line with finite zero and pole divisors, modulo target automorphisms. In every degree d, the principal divisor identifies the moduli space with unordered pairs of disjoint effective divisors of degree d; the space has dimension 2d, exactly 1/2C(2d, d) contractible connected components, and an explicit cell enumerator and orbit count in terms of signed compositions and run decompositions. We work out degree three completely: there are 53 cells and ten connected components, and the source–target reflection loci are defined by linear gap equations. Allowing common support and points at infinity gives a divisor-pair compactification. Over an algebraically closed non-Archimedean field with value group ℝ, root–pole tropicalization is surjective from valuation-separated cubic rational functions modulo the target torus normalizer. Explicit counterexamples show that the valuation of the conjugacy invariant J₂/I₃ does not descend to this tropical moduli space. Generic cubic functions admit minimal six-unit shallow ReLU representations with three positive and three negative unit output weights.

MSC classes: Primary 14T05; Secondary 14D22, 14H10

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