Pith Number

pith:NUHSAMEQ

pith:2026:NUHSAMEQEOALIOYAYCL4KOQYNL

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The Tropical Moduli Space of Degree-3 Rational Maps

Mohammad-Reza Siadat, Tony Shaska

Degree-3 tropical rational maps fall into exactly ten combinatorial types classified by slope sequences.

arxiv:2605.15347 v1 · 2026-05-14 · math.AG · math.CO

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Claims

C1strongest claim

Using a combinatorial description in terms of slope sequences, we classify all such maps and show that there are exactly ten combinatorial types.

C2weakest assumption

That every degree-3 tropical rational map is completely captured by its sequence of slopes and the gap lengths between breakpoints, with no additional continuous moduli or hidden constraints that would merge or split the ten types.

C3one line summary

The authors classify all degree-3 tropical rational maps into exactly ten combinatorial types and build a polyhedral model of their moduli space parametrized by gap lengths between breakpoints.

References

18 extracted · 18 resolved · 3 Pith anchors

[1] T. Shaska and M.-R. Siadat, Symmetry Detection and Functional Equivalence in ReLU Networks, preprint, Oakland University, 2026 2026

[2] Shaska, Graded N eural N etworks , Int 2025

[3] E. Badr, E. Shaska, and T. Shaska, Rational Functions on the Projective Line from a Computational Viewpoint , arXiv:2503.10835 [math.AG], 2025 2025

[4] Automorphism Groups on Tropical Curves: Some Cohomology Calculations 2010 · arXiv:1006.4869

[5] Moduli spaces of rational tropical curves 2007 · arXiv:0704.0839

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:53.696788Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6d0f2030902380b43b00c097c53a186afb60b6466fd67b0f4972d66b82ca7178

Aliases

arxiv: 2605.15347 · arxiv_version: 2605.15347v1 · doi: 10.48550/arxiv.2605.15347 · pith_short_12: NUHSAMEQEOAL · pith_short_16: NUHSAMEQEOALIOYA · pith_short_8: NUHSAMEQ

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/NUHSAMEQEOALIOYAYCL4KOQYNL \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 6d0f2030902380b43b00c097c53a186afb60b6466fd67b0f4972d66b82ca7178

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "15e695487b04de841cba5b77d330c0396dd756cdf10ab6ee45c014f8eb628d19",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-05-14T19:16:27Z",
    "title_canon_sha256": "751a7a499649ccfd6d66ed614eb467be5ac71c8adf7ff24886b78db2cb1c93f1"
  },
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  "source": {
    "id": "2605.15347",
    "kind": "arxiv",
    "version": 1
  }
}