{"paper":{"title":"Factorization of Additive Polynomials and van der Geer--van der Vlugt curves in characteristic 2","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Factorization of additive polynomials yields a simpler formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daichi Takeuchi, Takahiro Tsushima, Tetsushi Ito","submitted_at":"2026-05-16T00:42:22Z","abstract_excerpt":"In our previous work, we gave a formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2 by considering suitable quotients of the curve. Although the formula is explicit, it depends on many choices, which makes the formula complicated. In this article, we take a different approach using a factorization of additive polynomials, and prove a new formula. The resulting formula is simpler and is useful for explicit computations. As applications, we provide a method for constructing maximal and minimal van der Geer--van der Vlugt curves, and show that every suc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove a new formula for the Frobenius eigenvalues using factorization of additive polynomials. The resulting formula is simpler and is useful for explicit computations. As applications, we provide a method for constructing maximal and minimal van der Geer--van der Vlugt curves, and show that every such curve arises from this construction.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Factorization of the relevant additive polynomials yields a uniform description of the Frobenius eigenvalues that does not reintroduce the many auxiliary choices present in the quotient approach.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"New simpler formula for Frobenius eigenvalues of vdGV curves in char 2 via additive polynomial factorization, enabling construction of all maximal and minimal curves.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Factorization of additive polynomials yields a simpler formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fb1f17979df481a269f68a8609c19bba83f1adc792d4d45b45604c31c371cc95"},"source":{"id":"2605.16729","kind":"arxiv","version":1},"verdict":{"id":"648129c4-9402-424e-b1cd-a4848ccfe1b0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:28:01.249559Z","strongest_claim":"We prove a new formula for the Frobenius eigenvalues using factorization of additive polynomials. The resulting formula is simpler and is useful for explicit computations. As applications, we provide a method for constructing maximal and minimal van der Geer--van der Vlugt curves, and show that every such curve arises from this construction.","one_line_summary":"New simpler formula for Frobenius eigenvalues of vdGV curves in char 2 via additive polynomial factorization, enabling construction of all maximal and minimal curves.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Factorization of the relevant additive polynomials yields a uniform description of the Frobenius eigenvalues that does not reintroduce the many auxiliary choices present in the quotient approach.","pith_extraction_headline":"Factorization of additive polynomials yields a simpler formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16729/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T21:01:19.276442Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:40:55.697694Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:01:56.344705Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.470580Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"0d340ab206370e26d1a4be89030facdb56033bf87676c570b90b9c0126b77fd6"},"references":{"count":13,"sample":[{"doi":"","year":2023,"title":"R. Blache and T. Pierre,Zeta functions of quadratic Artin–Schreier curves in characteristic two, Acta Arith.207(2023), No. 1, 39–56","work_id":"bbf1655f-6361-436a-a723-386a6886a06a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"D. W. Bump,Automorphic forms and representations, Cambridge Studies in Advanced Mathe- matics, 55, Cambridge Univ. Press, Cambridge, 1997","work_id":"b5f31f0f-a23d-49cf-bb59-80d4338285a7","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"R. S. Coulter,The number of rational points of a class of Artin–Schreier curves, Finite Fields Appl.8(2002), No. 4, 397–413","work_id":"d738969b-8256-480c-ab44-66e7608d2df2","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"Goss,Basic Structures of Function Field Arithmetic, Springer, 1996","work_id":"9ad22fed-9526-4320-8276-05c3d2ea285d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1992,"title":"van der Geer and M","work_id":"876d5fcb-46e2-4d48-99d4-46e117071f0b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"db384795c1fc4f614b645f51d24bd614e0bdd5a403ee40c6c83dbc4a23b350a1","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7bc6fadec6e96a2dc8c631f2d6f23ee4566e52e1d298475774fadc88fe494191"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}