{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:2TZFJEWBW52D5GBJVCIFLTBA4Q","short_pith_number":"pith:2TZFJEWB","schema_version":"1.0","canonical_sha256":"d4f25492c1b7743e9829a89055cc20e43aec8273197291f02f41bbdef461ee87","source":{"kind":"arxiv","id":"2107.03154","version":2},"attestation_state":"computed","paper":{"title":"Dependence over subgroups of free groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Amnon Rosenmann, Enric Ventura Capell","submitted_at":"2021-07-07T11:30:24Z","abstract_excerpt":"Given a finitely generated subgroup $H$ of a free group $F$, we present an algorithm which computes $g_1,\\ldots,g_m\\in F$, such that the set of elements $g\\in F$, for which there exists a non-trivial $H$-equation having $g$ as a solution, is, precisely, the disjoint union of the double cosets $H\\sqcup Hg_1H\\sqcup \\cdots \\sqcup Hg_mH$. Moreover, we present an algorithm which, given a finitely generated subgroup $H\\leqslant F$ and an element $g\\in F$, computes a finite set of elements of $H * \\langle x \\rangle$ that generate (as a normal subgroup) the ``ideal\" $I_H(g) \\unlhd H * \\langle x \\rangl"},"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":"2107.03154","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2021-07-07T11:30:24Z","cross_cats_sorted":[],"title_canon_sha256":"bb01ab9a99cfc0a960db5d97a91232f9c02d7ad876a7ada977d5c429467ead93","abstract_canon_sha256":"19da03a836565968a21559396d26cfc3a27ed5a828404028fc53362eb10502f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:09:03.413343Z","signature_b64":"Qd2S2vKKiWKoyVW5zOBrkBFHwASmS/aaOJRH1O++0ur43Ega+yeO3hkHXqymWHfWvgayrj624/tZW2peI9dVAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4f25492c1b7743e9829a89055cc20e43aec8273197291f02f41bbdef461ee87","last_reissued_at":"2026-07-05T06:09:03.412902Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:09:03.412902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dependence over subgroups of free groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Amnon Rosenmann, Enric Ventura Capell","submitted_at":"2021-07-07T11:30:24Z","abstract_excerpt":"Given a finitely generated subgroup $H$ of a free group $F$, we present an algorithm which computes $g_1,\\ldots,g_m\\in F$, such that the set of elements $g\\in F$, for which there exists a non-trivial $H$-equation having $g$ as a solution, is, precisely, the disjoint union of the double cosets $H\\sqcup Hg_1H\\sqcup \\cdots \\sqcup Hg_mH$. Moreover, we present an algorithm which, given a finitely generated subgroup $H\\leqslant F$ and an element $g\\in F$, computes a finite set of elements of $H * \\langle x \\rangle$ that generate (as a normal subgroup) the ``ideal\" $I_H(g) \\unlhd H * \\langle x \\rangl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.03154","kind":"arxiv","version":2},"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/2107.03154/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":"2107.03154","created_at":"2026-07-05T06:09:03.412960+00:00"},{"alias_kind":"arxiv_version","alias_value":"2107.03154v2","created_at":"2026-07-05T06:09:03.412960+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.03154","created_at":"2026-07-05T06:09:03.412960+00:00"},{"alias_kind":"pith_short_12","alias_value":"2TZFJEWBW52D","created_at":"2026-07-05T06:09:03.412960+00:00"},{"alias_kind":"pith_short_16","alias_value":"2TZFJEWBW52D5GBJ","created_at":"2026-07-05T06:09:03.412960+00:00"},{"alias_kind":"pith_short_8","alias_value":"2TZFJEWB","created_at":"2026-07-05T06:09:03.412960+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.05272","citing_title":"Computing $H$-equations with 2-by-2 integral matrices","ref_index":32,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q","json":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q.json","graph_json":"https://pith.science/api/pith-number/2TZFJEWBW52D5GBJVCIFLTBA4Q/graph.json","events_json":"https://pith.science/api/pith-number/2TZFJEWBW52D5GBJVCIFLTBA4Q/events.json","paper":"https://pith.science/paper/2TZFJEWB"},"agent_actions":{"view_html":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q","download_json":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q.json","view_paper":"https://pith.science/paper/2TZFJEWB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2107.03154&json=true","fetch_graph":"https://pith.science/api/pith-number/2TZFJEWBW52D5GBJVCIFLTBA4Q/graph.json","fetch_events":"https://pith.science/api/pith-number/2TZFJEWBW52D5GBJVCIFLTBA4Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q/action/storage_attestation","attest_author":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q/action/author_attestation","sign_citation":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q/action/citation_signature","submit_replication":"https://pith.science/pith/2TZFJEWBW52D5GBJVCIFLTBA4Q/action/replication_record"}},"created_at":"2026-07-05T06:09:03.412960+00:00","updated_at":"2026-07-05T06:09:03.412960+00:00"}