{"paper":{"title":"Weierstrass semigroups at totally ramified places of degree one on Kummer extensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Weierstrass semigroup at totally ramified degree-one places on Kummer extensions y^m = f(x) admits an explicit unified description.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Chang-An Zhao, Huachao Zhang","submitted_at":"2026-05-14T08:54:10Z","abstract_excerpt":"We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation\n  $y^m = f(x)$ over $K$, an algebraic extension of $\\mathbb{F}_q$, where $f(x)\\in K(x)$.\n  Our description takes a unified form for distinct totally ramified places of degree one.\n  We then provide a necessary and sufficient condition for the Weierstrass semigroup at a totally ramified place of degree one to be symmetric.\n  Furthermore, we investigate the minimal generating set of the Weierstrass semigroups at many totally ramified "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation y^m = f(x) over K, an algebraic extension of F_q, where f(x) in K(x). Our description takes a unified form for distinct totally ramified places of degree one.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that the place is totally ramified of degree one on the Kummer extension y^m = f(x), with the description holding uniformly without additional restrictions on the ramification or the polynomial f beyond the Kummer setup.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit descriptions of Weierstrass semigroups and gaps at totally ramified places on Kummer extensions, plus symmetry conditions and minimal generators, with applications to GGS and BM curves.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Weierstrass semigroup at totally ramified degree-one places on Kummer extensions y^m = f(x) admits an explicit unified description.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4725b5724ff3d025cedd5ce290bb33f275cef045eb0e5baeb7b2e699b76f76e7"},"source":{"id":"2605.14583","kind":"arxiv","version":1},"verdict":{"id":"f7df0384-448d-4c9a-b3b0-87cbf578e42e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:30:43.826896Z","strongest_claim":"We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation y^m = f(x) over K, an algebraic extension of F_q, where f(x) in K(x). Our description takes a unified form for distinct totally ramified places of degree one.","one_line_summary":"Explicit descriptions of Weierstrass semigroups and gaps at totally ramified places on Kummer extensions, plus symmetry conditions and minimal generators, with applications to GGS and BM curves.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the place is totally ramified of degree one on the Kummer extension y^m = f(x), with the description holding uniformly without additional restrictions on the ramification or the polynomial f beyond the Kummer setup.","pith_extraction_headline":"The Weierstrass semigroup at totally ramified degree-one places on Kummer extensions y^m = f(x) admits an explicit unified description."},"references":{"count":44,"sample":[{"doi":"10.1007/s10623-023-01339-w","year":2024,"title":"A. S. Castellanos, E. A. R. Mendoza, L. Quoos, Weierstrass semigroups, pure gaps and codes on function fields, Designs, Codes and Cryptography 92 (5) (2024) 1219–1242.doi:10.1007/s10623-023-01339-w","work_id":"2005396b-31d7-4a3b-9dfa-f81054b501bb","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.ffa.2023.102209","year":2023,"title":"E. A. Mendoza, On Kummer extensions with one place at infinity, Finite Fields and Their Applications 89 (2023) 102209.doi:10.1016/j.ffa.2023.102209","work_id":"0ee77e15-0b2e-42f2-b766-7ee41bdf6c79","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1515/advgeom-2018-0021","year":2019,"title":"M. Abdón, H. Borges, L. Quoos, Weierstrass points on Kummer extensions, Ad- vances in Geometry 19 (3) (2019) 323–333.doi:10.1515/advgeom-2018-0021","work_id":"d5b8df91-216b-49b4-87e9-b682644602a2","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"P. Beelen, L. Landi, M. Montanucci, Weierstrass semigroups on the Skabelund maximal curve, Finite Fields and Their Applications 72 (2021) 101811.doi:10. 1016/j.ffa.2021.101811","work_id":"1a4da7de-f1fc-4c00-89bb-414e0f3be055","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"D. Bartoli, M. Montanucci, G. Zini, Weierstrass semigroups at every point of the Suzuki curve, Acta Arithmetica 197 (1) (2021) 1–20.doi:10.4064/ aa181203-24-2","work_id":"d734309f-1697-4472-b77f-9cb92b049cc6","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":44,"snapshot_sha256":"fd592532026747be91def64ad10a2f385797143aa248327ae8a4190cb16f5a39","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"}