{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:SR5ZJKCF6E2EYQSM673DY5KOG6","short_pith_number":"pith:SR5ZJKCF","schema_version":"1.0","canonical_sha256":"947b94a845f1344c424cf7f63c754e3787d920337c6f4612b1d82306b7ae524f","source":{"kind":"arxiv","id":"1604.02422","version":1},"attestation_state":"computed","paper":{"title":"A Jacobian module for disentanglements and applications to Mond's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"G. Pe\\~nafort-Sanchis, J. Fern\\'andez de Bobadilla, J. J. Nu\\~no-Ballesteros","submitted_at":"2016-04-08T18:36:28Z","abstract_excerpt":"Given a germ of holomorphic map $f$ from $\\mathbb C^n$ to $\\mathbb C^{n+1}$, we define a module $M(f)$ whose dimension over $\\mathbb C$ is an upper bound for the $\\mathscr A$-codimension of $f$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ of the module, for unfoldings $F$ of $f$. The main result is that if $(n,n+1)$ are nice dimensions, then the dimension of $M(f)$ over $\\mathbb C$ is an upper bound of the image Milnor number of $f$, with equality if and only if the relative module $M_y(F)$ is Cohen-Macaulay for some stable unfolding $F$. In particu"},"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":"1604.02422","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-04-08T18:36:28Z","cross_cats_sorted":[],"title_canon_sha256":"7bba2d236b3a3c17b861dd923fe1ea27a17e6da70bdc9bc7ffdd0eba2a078692","abstract_canon_sha256":"516e58b97b5e4415964e8eb4886296c7df9614162253c544ff20fe27d0b25208"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:26.595039Z","signature_b64":"mMrqNJtSCihu7e4kQ2RGD9GVUV2PiWCjfWjb/SGIcaoVGMf/thbSbGvge3qwsx+A6AeGtHzRqxk2KLlUsvvWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"947b94a845f1344c424cf7f63c754e3787d920337c6f4612b1d82306b7ae524f","last_reissued_at":"2026-05-18T01:17:26.594544Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:26.594544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Jacobian module for disentanglements and applications to Mond's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"G. Pe\\~nafort-Sanchis, J. Fern\\'andez de Bobadilla, J. J. Nu\\~no-Ballesteros","submitted_at":"2016-04-08T18:36:28Z","abstract_excerpt":"Given a germ of holomorphic map $f$ from $\\mathbb C^n$ to $\\mathbb C^{n+1}$, we define a module $M(f)$ whose dimension over $\\mathbb C$ is an upper bound for the $\\mathscr A$-codimension of $f$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ of the module, for unfoldings $F$ of $f$. The main result is that if $(n,n+1)$ are nice dimensions, then the dimension of $M(f)$ over $\\mathbb C$ is an upper bound of the image Milnor number of $f$, with equality if and only if the relative module $M_y(F)$ is Cohen-Macaulay for some stable unfolding $F$. In particu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02422","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":""},"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":"1604.02422","created_at":"2026-05-18T01:17:26.594638+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.02422v1","created_at":"2026-05-18T01:17:26.594638+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02422","created_at":"2026-05-18T01:17:26.594638+00:00"},{"alias_kind":"pith_short_12","alias_value":"SR5ZJKCF6E2E","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_16","alias_value":"SR5ZJKCF6E2EYQSM","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_8","alias_value":"SR5ZJKCF","created_at":"2026-05-18T12:30:44.179134+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/SR5ZJKCF6E2EYQSM673DY5KOG6","json":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6.json","graph_json":"https://pith.science/api/pith-number/SR5ZJKCF6E2EYQSM673DY5KOG6/graph.json","events_json":"https://pith.science/api/pith-number/SR5ZJKCF6E2EYQSM673DY5KOG6/events.json","paper":"https://pith.science/paper/SR5ZJKCF"},"agent_actions":{"view_html":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6","download_json":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6.json","view_paper":"https://pith.science/paper/SR5ZJKCF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.02422&json=true","fetch_graph":"https://pith.science/api/pith-number/SR5ZJKCF6E2EYQSM673DY5KOG6/graph.json","fetch_events":"https://pith.science/api/pith-number/SR5ZJKCF6E2EYQSM673DY5KOG6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6/action/storage_attestation","attest_author":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6/action/author_attestation","sign_citation":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6/action/citation_signature","submit_replication":"https://pith.science/pith/SR5ZJKCF6E2EYQSM673DY5KOG6/action/replication_record"}},"created_at":"2026-05-18T01:17:26.594638+00:00","updated_at":"2026-05-18T01:17:26.594638+00:00"}