Pith Number

pith:LT3MDIT2

pith:2026:LT3MDIT2FZ47YJLMBFO6GLIYGJ

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Classification of Rational Functions of Degree Three over Finite Fields

Shujun Zhao, Siyu Peng, Xiang-dong Hou, Yongyu Qiang

Rational functions of degree three over finite fields in odd characteristic are fully classified under PGL-equivalence.

arxiv:2604.18954 v2 · 2026-04-21 · math.NT

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\pithnumber{LT3MDIT2FZ47YJLMBFO6GLIYGJ}

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Claims

C1strongest claim

In the present paper, we classify all rational functions of degree three over finite fields in odd characteristic.

C2weakest assumption

The approach based on careful analyses of the value frequencies and the ramification points of the degree three rational functions, together with the explicit formula for the number of equivalence classes recently obtained by the first author, is sufficient to produce a complete classification without omissions.

C3one line summary

All degree-three rational functions over finite fields in odd characteristic are classified under PGL-equivalence via value frequencies, ramification points, and an explicit count formula.

Receipt and verification

First computed	2026-05-20T02:05:43.337714Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5cf6c1a27a2e79fc256c095de32d18327d68f1504a99ddf4a45e82c5a0af028b

Aliases

arxiv: 2604.18954 · arxiv_version: 2604.18954v2 · doi: 10.48550/arxiv.2604.18954 · pith_short_12: LT3MDIT2FZ47 · pith_short_16: LT3MDIT2FZ47YJLM · pith_short_8: LT3MDIT2

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LT3MDIT2FZ47YJLMBFO6GLIYGJ \
  | 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: 5cf6c1a27a2e79fc256c095de32d18327d68f1504a99ddf4a45e82c5a0af028b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "982c2614c58bf6c04ac71f85f4d9cb040fc13ecb2375103ccc3d282e77fe2cbe",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-21T01:05:17Z",
    "title_canon_sha256": "7aa3dc2407ef2d482eaf7edaf83cf04d3d39695647a26d619ef99987b71dd35c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.18954",
    "kind": "arxiv",
    "version": 2
  }
}