{"paper":{"title":"The Isomorphism Classes of the Surfaces $x_1^{a_1} + x_2^{a_2} + x_3^{a_3} + 1 = 0$","license":"http://creativecommons.org/licenses/by/4.0/","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.","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Buddhadev Hajra, Michael Chitayat","submitted_at":"2026-05-08T11:45:16Z","abstract_excerpt":"Let $f = x_1^{a_1} + x_2^{a_2} + x_3^{a_3} + 1 \\in \\mathbb{C}[x_1,x_2,x_3]$ and let $g = y_1^{b_1} + y_2^{b_2} + y_3^{b_3} + 1 \\in \\mathbb{C}[y_1,y_2,y_3]$ where $a_1,a_2,a_3,b_1,b_2,b_3 \\geq 2$. 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."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"29ec5a317b1c4b454ca63a7b11a6cf99225d975a08dfa1d85e970a7491f710a4"},"source":{"id":"2605.07617","kind":"arxiv","version":2},"verdict":{"id":"81c05029-c155-4403-960c-f0565e952049","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-11T02:36:45.096751Z","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.","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","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.","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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.07617/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T10:42:02.788551Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-20T05:38:09.973780Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T16:01:18.925637Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T11:40:25.364643Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b245141b1b2498954936e8da39db40e2fa1fc630ffb10cfc556f6f877adee213"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}