Pith Number

pith:TJUJ2JMI

pith:2026:TJUJ2JMI5I5J6XIYXKOC5KGYEF

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The Isomorphism Classes of the Surfaces $x_1^{a_1} + x_2^{a_2} + x_3^{a_3} + 1 = 0$

Buddhadev Hajra, Michael Chitayat

The affine surfaces x₁^{a₁} + x₂^{a₂} + x₃^{a₃} + 1 = 0 are isomorphic over the complex numbers precisely when the exponent triples agree up to permutation.

arxiv:2605.07617 v2 · 2026-05-08 · math.AG · math.AC

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\pithnumber{TJUJ2JMI5I5J6XIYXKOC5KGYEF}

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We prove that the surfaces V(f) subset A^3 and V(g) subset A^3 are isomorphic if and only if (a1,a2,a3) = (b1,b2,b3) up to a permutation of the entries.

C2weakest assumption

The exponents a1,a2,a3,b1,b2,b3 are integers greater than or equal to 2 and the base field is the complex numbers; the surfaces are considered as affine hypersurfaces in A^3.

C3one line summary

The surfaces V(x1^{a1} + x2^{a2} + x3^{a3} + 1 = 0) in affine 3-space are isomorphic if and only if the exponent triples (a1,a2,a3) are permutations of each other, for all ai >= 2.

Receipt and verification

First computed	2026-06-29T01:15:05.045375Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

9a689d2588ea3a9f5d18ba9c2ea8d821751fb00f9798e25c3fdef1a1e9eb114c

Aliases

arxiv: 2605.07617 · arxiv_version: 2605.07617v2 · doi: 10.48550/arxiv.2605.07617 · pith_short_12: TJUJ2JMI5I5J · pith_short_16: TJUJ2JMI5I5J6XIY · pith_short_8: TJUJ2JMI

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/TJUJ2JMI5I5J6XIYXKOC5KGYEF \
  | 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: 9a689d2588ea3a9f5d18ba9c2ea8d821751fb00f9798e25c3fdef1a1e9eb114c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6755f1c914f051a3f99c7a8122d2e52a51710e4326b06f806df7cb27ce91c220",
    "cross_cats_sorted": [
      "math.AC"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-05-08T11:45:16Z",
    "title_canon_sha256": "55837b9ca8bb570a7cc105e6717a55d3e41c3615d7c66a705a34f4f434959e70"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.07617",
    "kind": "arxiv",
    "version": 2
  }
}