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We show that $\\mathscr{A}_e$-$codim(X,f)=\\mu_I(f)$, where $\\mathscr{A}_e$-$codim(X,f)$ is the $\\mathscr{A}_e$-codimension, i.e., the minimum number of parameters in a versal deformation and $\\mu_I(f)$ is the image Milnor number, i.e., the number of vanishing cycles in the image of a stabilisation of $f$."},"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":"1709.09504","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-09-27T13:36:51Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"86c6f7b42bc79bd6abe42ac75e8c44d18c8a7af8a5eefed77ceb2864cad616d9","abstract_canon_sha256":"5ba3dd892635a8fde2db8fc2f25feccfabc3cfad708b64dd2fe29fd88b3fda74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:11.514985Z","signature_b64":"Pizf7Vzrk5NXZqsCP2Buy4aexozI+NNQP9lee3/1j8lSCHm0mE4tGGtdeT//yCGvrgFWZ7CRlD6z5172p62rDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16b5d98f48b5aee913cf98781a2b2f9d65370b8c38cf3f7c13fe34c94414e577","last_reissued_at":"2026-05-18T00:34:11.514229Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:11.514229Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Image Milnor number and $\\mathscr{A}_e$-codimension for maps between weighted homogeneous irreducible curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.AG","authors_text":"Daiane Alice Henrique Ament, Jo\\~ao Nivaldo Tomazella, Juan Jose Nu\\~no Ballesteros","submitted_at":"2017-09-27T13:36:51Z","abstract_excerpt":"Let $(X,0)\\subset (\\mathbb{C}^n,0)$ be an irreducible weighted homogeneous singularity curve and let $f:(X,0)\\to(\\mathbb{C}^2,0)$ be a map germ finite, one-to-one and weighted homogeneous with the same weights of $(X,0)$. We show that $\\mathscr{A}_e$-$codim(X,f)=\\mu_I(f)$, where $\\mathscr{A}_e$-$codim(X,f)$ is the $\\mathscr{A}_e$-codimension, i.e., the minimum number of parameters in a versal deformation and $\\mu_I(f)$ is the image Milnor number, i.e., the number of vanishing cycles in the image of a stabilisation of $f$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.09504","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":"1709.09504","created_at":"2026-05-18T00:34:11.514357+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.09504v1","created_at":"2026-05-18T00:34:11.514357+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.09504","created_at":"2026-05-18T00:34:11.514357+00:00"},{"alias_kind":"pith_short_12","alias_value":"C225TD2IWWXO","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"C225TD2IWWXOSE6P","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"C225TD2I","created_at":"2026-05-18T12:31:08.081275+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/C225TD2IWWXOSE6PTB4BUKZPTV","json":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV.json","graph_json":"https://pith.science/api/pith-number/C225TD2IWWXOSE6PTB4BUKZPTV/graph.json","events_json":"https://pith.science/api/pith-number/C225TD2IWWXOSE6PTB4BUKZPTV/events.json","paper":"https://pith.science/paper/C225TD2I"},"agent_actions":{"view_html":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV","download_json":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV.json","view_paper":"https://pith.science/paper/C225TD2I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.09504&json=true","fetch_graph":"https://pith.science/api/pith-number/C225TD2IWWXOSE6PTB4BUKZPTV/graph.json","fetch_events":"https://pith.science/api/pith-number/C225TD2IWWXOSE6PTB4BUKZPTV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV/action/storage_attestation","attest_author":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV/action/author_attestation","sign_citation":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV/action/citation_signature","submit_replication":"https://pith.science/pith/C225TD2IWWXOSE6PTB4BUKZPTV/action/replication_record"}},"created_at":"2026-05-18T00:34:11.514357+00:00","updated_at":"2026-05-18T00:34:11.514357+00:00"}