{"paper":{"title":"Commutative Semifields from bijections of the Desarguesian plane","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"Semiquadratic homogeneous bijections of the Desarguesian plane produce large families of commutative semifields that are neither fields nor twisted fields.","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Faruk G\\\"olo\\u{g}lu, Lukas K\\\"olsch","submitted_at":"2026-05-13T18:14:07Z","abstract_excerpt":"The Menichetti-Kaplansky theorem states that a finite semifield that is three-dimensional over its center is either a field or a twisted field of Albert. This implies that a quadratic homogeneous bijection of $\\mathbb{P}^2(\\mathbb{F}_q)$ is equivalent to a Dembowski-Ostrom monomial. In this paper, we give a large class of semiquadratic homogeneous bijections of $\\mathbb{P}^2(\\mathbb{F}_q)$ that are inequivalent to Dembowski-Ostrom monomials. Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, which in turn give"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we give a large class of semiquadratic homogeneous bijections of P^2(F_q) that are inequivalent to Dembowski-Ostrom monomials. Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, which in turn give rise to a large family of non-Desarguesian commutative semifield planes.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructed maps are indeed bijections and the resulting multiplication defines a semifield (i.e., the algebraic identities hold for the chosen parameters), which must be verified explicitly for each member of the family.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A large family of commutative semifields non-isotopic to fields or Albert twisted fields is obtained from new semiquadratic bijections on P^2(F_q).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Semiquadratic homogeneous bijections of the Desarguesian plane produce large families of commutative semifields that are neither fields nor twisted fields.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"178242a8d972cdfacbda793c67948ede0c1e5e6f8c0e31b3abd7866906541b97"},"source":{"id":"2605.14009","kind":"arxiv","version":1},"verdict":{"id":"2afca6f8-fd93-4c5f-a192-aa4f8ce4a083","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:52:43.624715Z","strongest_claim":"we give a large class of semiquadratic homogeneous bijections of P^2(F_q) that are inequivalent to Dembowski-Ostrom monomials. Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, which in turn give rise to a large family of non-Desarguesian commutative semifield planes.","one_line_summary":"A large family of commutative semifields non-isotopic to fields or Albert twisted fields is obtained from new semiquadratic bijections on P^2(F_q).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructed maps are indeed bijections and the resulting multiplication defines a semifield (i.e., the algebraic identities hold for the chosen parameters), which must be verified explicitly for each member of the family.","pith_extraction_headline":"Semiquadratic homogeneous bijections of the Desarguesian plane produce large families of commutative semifields that are neither fields nor twisted fields."},"references":{"count":25,"sample":[{"doi":"","year":1997,"title":"Shreeram Abhyankar,Projective polynomials, Proceedings of the American Mathematical Society125(1997), no. 6, 1643–1650","work_id":"81b54528-e6ca-4bd6-8950-8e623fe072b3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1960,"title":"A. A. Albert,Finite division algebras and finite planes, Proc. Sympos. Appl. Math., Vol. 10, American Math- ematical Society, Providence, R.I., 1960, pp. 53–70. MR 0116036","work_id":"9f920ca4-2ede-4061-a741-2448d0ab6f7a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"J¨ urgen Bierbrauer,New semifields, PN and APN functions, Des. Codes Cryptogr.54(2010), no. 3, 189–200. MR 2584973","work_id":"96eee5da-5840-4707-bb28-d41f527ae4bb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"Mauro Biliotti, Vikram Jha, and Norman L Johnson,The collineation groups of generalized twisted field planes, Geometriae Dedicata76(1999), 97–126","work_id":"5e2ba953-f440-4fdc-94a7-eb81700ecc74","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"Coulter and Marie Henderson,Commutative presemifields and semifields, Adv","work_id":"4373282f-add8-44ad-9b46-ac22556c356a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":25,"snapshot_sha256":"1ce6676c4482e18b0248aec7e8511ba924612ea26a499b633907f6eec5a2ed64","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"}