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PreprintarXiv licencearXiv

The Tropical Moduli Space of Degree-3 Rational Maps

  • Tony Shaska
  • Mohammad-Reza Siadat
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Abstract

We study tropical rational functions on the tropical projective line with three zeros and three poles in the torus, counted with multiplicity, modulo target automorphisms. The principal divisor identifies their moduli space with unordered pairs of disjoint effective divisors of degree three. We classify its 53 cells and prove that it is six-dimensional with ten contractible connected components. Source--target automorphism groups are trivial or of order two, and 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 โ„, 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 functions admit minimal six-unit shallow ReLU representations with three positive and three negative unit output weights.

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Cite

BibTeXRIS
BibTeX
@misc{shaska2026tropical,
  title = {{The Tropical Moduli Space of Degree-3 Rational Maps}},
  author = {Tony Shaska and Mohammad-Reza Siadat},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.15347},
  eprint = {2605.15347},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.15347},
}
RIS
TY  - UNPB
TI  - The Tropical Moduli Space of Degree-3 Rational Maps
AU  - Tony Shaska
AU  - Mohammad-Reza Siadat
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2605.15347
UR  - https://arxiv.org/abs/2605.15347
AB  - We study tropical rational functions on the tropical projective line with three zeros and three poles in the torus, counted with multiplicity, modulo target automorphisms. The principal divisor identifies their moduli space with unordered pairs of disjoint effective divisors of degree three. We classify its 53 cells and prove that it is six-dimensional with ten contractible connected components. Source--target automorphism groups are trivial or of order two, and 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 โ„, 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 J^2/I^3 does not descend to this tropical moduli space. Generic functions admit minimal six-unit shallow ReLU representations with three positive and three negative unit output weights.
ER  -