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For a totally ramified place $Q$ of degree one in $F/K(x)$, we give a unified description of the set $G(Q)$ of gaps at $Q$.\n  As a consequence, we explicitly provide a system of generators, the multiplicity, and the Frobenius number of the Weierstrass semigroup $H(Q)$. Moreover, we give a necessary and sufficient condition for $H(Q)$ to be symmetric.\n  Then we investigate the minimal generating set of the Weierstrass semigroups at several totally ramified places of degree one.\n  We not only"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.29311","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-28T03:43:13Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"4e7c3a576fc5dc4a7f00f94ee6f4f2c620f2a24bc5ee1e977cbc53f02ec12aa6","abstract_canon_sha256":"3f7407b587b89f4c46f2800b37cdc8976781bf749064ac4f4e3eef3579e6476a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T01:05:32.006897Z","signature_b64":"PjjFLMNWoErtkaG12H22P+WHq4MJJ76txN9tV2OO9tC4KDjgozi3MzfBQYjs7gLMza5nrgvdSGaljlUMPlpSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93abcb473ce686f780a3db08af7bafb0717e63b50ae0200c4714b3e15d7ff7f2","last_reissued_at":"2026-05-29T01:05:32.006348Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T01:05:32.006348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weierstrass semigroups at totally ramified places of degree one on linearized function fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Chang-An Zhao, Huachao Zhang","submitted_at":"2026-05-28T03:43:13Z","abstract_excerpt":"A linearized function field $F$ can be viewed as a Galois extension of a rational function field $K(x)$. 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Moreover, we give a necessary and sufficient condition for $H(Q)$ to be symmetric.\n  Then we investigate the minimal generating set of the Weierstrass semigroups at several totally ramified places of degree one.\n  We not only"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.29311","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.29311/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.29311","created_at":"2026-05-29T01:05:32.006434+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.29311v1","created_at":"2026-05-29T01:05:32.006434+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.29311","created_at":"2026-05-29T01:05:32.006434+00:00"},{"alias_kind":"pith_short_12","alias_value":"SOV4WRZ442DP","created_at":"2026-05-29T01:05:32.006434+00:00"},{"alias_kind":"pith_short_16","alias_value":"SOV4WRZ442DPPAFD","created_at":"2026-05-29T01:05:32.006434+00:00"},{"alias_kind":"pith_short_8","alias_value":"SOV4WRZ4","created_at":"2026-05-29T01:05:32.006434+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB","json":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB.json","graph_json":"https://pith.science/api/pith-number/SOV4WRZ442DPPAFD3MEK665PWB/graph.json","events_json":"https://pith.science/api/pith-number/SOV4WRZ442DPPAFD3MEK665PWB/events.json","paper":"https://pith.science/paper/SOV4WRZ4"},"agent_actions":{"view_html":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB","download_json":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB.json","view_paper":"https://pith.science/paper/SOV4WRZ4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.29311&json=true","fetch_graph":"https://pith.science/api/pith-number/SOV4WRZ442DPPAFD3MEK665PWB/graph.json","fetch_events":"https://pith.science/api/pith-number/SOV4WRZ442DPPAFD3MEK665PWB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB/action/storage_attestation","attest_author":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB/action/author_attestation","sign_citation":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB/action/citation_signature","submit_replication":"https://pith.science/pith/SOV4WRZ442DPPAFD3MEK665PWB/action/replication_record"}},"created_at":"2026-05-29T01:05:32.006434+00:00","updated_at":"2026-05-29T01:05:32.006434+00:00"}