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We prove that the surfaces $V(f) \\subset \\mathbb{A}^3$ and $V(g) \\subset \\mathbb{A}^3$ are isomorphic if and only if $(a_1,a_2,a_3) = (b_1,b_2,b_3)$ up to a permutation of the entries.","authors_text":"Buddhadev Hajra, Michael Chitayat","cross_cats":["math.AC"],"headline":"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-05-08T11:45:16Z","title":"The Isomorphism Classes of the Surfaces $x_1^{a_1} + x_2^{a_2} + x_3^{a_3} + 1 = 0$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.07617","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-11T02:36:45.096751Z","id":"81c05029-c155-4403-960c-f0565e952049","model_set":{"reader":"grok-4.3"},"one_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.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"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.","strongest_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.","weakest_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."}},"verdict_id":"81c05029-c155-4403-960c-f0565e952049"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:16778b57629d0f5e48f0903221e462ad10363eeaf940a649f67378b703e4991a","target":"record","created_at":"2026-06-29T01:15:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"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}},"canonical_sha256":"9a689d2588ea3a9f5d18ba9c2ea8d821751fb00f9798e25c3fdef1a1e9eb114c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a689d2588ea3a9f5d18ba9c2ea8d821751fb00f9798e25c3fdef1a1e9eb114c","first_computed_at":"2026-06-29T01:15:05.045375Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-29T01:15:05.045375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OmVC9YAr3GfZlDYJeHpQ30gG24D0pZXSGxEKd19VjdRyB2AycpTs9df2dxjMhtp7VFF3uOUpPIQsBd5kHvVFAQ==","signature_status":"signed_v1","signed_at":"2026-06-29T01:15:05.045844Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.07617","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16778b57629d0f5e48f0903221e462ad10363eeaf940a649f67378b703e4991a","sha256:44b31570011aecb2dea0e8802d1290f89e92abead705cabd3af8ebeb319c84fc"],"state_sha256":"01ff45223815b38304dd7e8fd813f8855f1d4079eb141dad6fe56e09a84008c1"}