{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1997:MV7T6YVLANIR4MOVJ4BPDHPAZG","short_pith_number":"pith:MV7T6YVL","schema_version":"1.0","canonical_sha256":"657f3f62ab03511e31d54f02f19de0c9869fdb241479f5e126d3b1f32d5307fe","source":{"kind":"arxiv","id":"math/9702229","version":1},"attestation_state":"computed","paper":{"title":"Multiplicity of a zero of an analytic function on a trajectory of a vector field","license":"","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Andrei Gabrielov (Purdue University)","submitted_at":"1997-02-21T00:00:00Z","abstract_excerpt":"Let P(x) be a germ at the origin of an analytic function in C^n, where x = (x_1,..., x_n), and let\n  \\xi = \\xi_1(x) d/dx_1 + ... + \\xi_n(x) d/dx_n\n  be a germ at the origin of an analytic vector field. Suppose that \\xi(0) != 0, and let \\gamma be a trajectory of \\xi through the origin. Suppose that P|_\\gamma /\\equiv 0, and let \\mu(P|_\\gamma) be the multiplicity of a zero of P|_\\gamma at the origin. Let\n  \\xi P = \\xi_1 dP/dx_1 + ... + \\xi_n dP/dx_n\n  be derivative of P in the direction of \\xi, and let \\xi^kP be the kth iteration of this derivative.\n  We give a formula (Theorem 1) for \\mu(P|_\\gam"},"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":"math/9702229","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CV","submitted_at":"1997-02-21T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"152cb86c5a7093c3ccc05987c16032aae9ef1293541c7c6e9eee748d0eeb2b08","abstract_canon_sha256":"4edf707ba3ca2b17795151d0436263ee39e77424899f2f03b881a8f54908eba6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:37.105038Z","signature_b64":"AVXArSiCwpp+nPynmClBKxllS5mExgcfdizN4maw3Qa73y8FRFUfYCdqyB8hqaEPsMe7pmhe77iB3VCwj0qRCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"657f3f62ab03511e31d54f02f19de0c9869fdb241479f5e126d3b1f32d5307fe","last_reissued_at":"2026-05-18T01:05:37.104419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:37.104419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicity of a zero of an analytic function on a trajectory of a vector field","license":"","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Andrei Gabrielov (Purdue University)","submitted_at":"1997-02-21T00:00:00Z","abstract_excerpt":"Let P(x) be a germ at the origin of an analytic function in C^n, where x = (x_1,..., x_n), and let\n  \\xi = \\xi_1(x) d/dx_1 + ... + \\xi_n(x) d/dx_n\n  be a germ at the origin of an analytic vector field. Suppose that \\xi(0) != 0, and let \\gamma be a trajectory of \\xi through the origin. Suppose that P|_\\gamma /\\equiv 0, and let \\mu(P|_\\gamma) be the multiplicity of a zero of P|_\\gamma at the origin. Let\n  \\xi P = \\xi_1 dP/dx_1 + ... + \\xi_n dP/dx_n\n  be derivative of P in the direction of \\xi, and let \\xi^kP be the kth iteration of this derivative.\n  We give a formula (Theorem 1) for \\mu(P|_\\gam"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9702229","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":"math/9702229","created_at":"2026-05-18T01:05:37.104535+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9702229v1","created_at":"2026-05-18T01:05:37.104535+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9702229","created_at":"2026-05-18T01:05:37.104535+00:00"},{"alias_kind":"pith_short_12","alias_value":"MV7T6YVLANIR","created_at":"2026-05-18T12:25:48.327863+00:00"},{"alias_kind":"pith_short_16","alias_value":"MV7T6YVLANIR4MOV","created_at":"2026-05-18T12:25:48.327863+00:00"},{"alias_kind":"pith_short_8","alias_value":"MV7T6YVL","created_at":"2026-05-18T12:25:48.327863+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/MV7T6YVLANIR4MOVJ4BPDHPAZG","json":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG.json","graph_json":"https://pith.science/api/pith-number/MV7T6YVLANIR4MOVJ4BPDHPAZG/graph.json","events_json":"https://pith.science/api/pith-number/MV7T6YVLANIR4MOVJ4BPDHPAZG/events.json","paper":"https://pith.science/paper/MV7T6YVL"},"agent_actions":{"view_html":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG","download_json":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG.json","view_paper":"https://pith.science/paper/MV7T6YVL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9702229&json=true","fetch_graph":"https://pith.science/api/pith-number/MV7T6YVLANIR4MOVJ4BPDHPAZG/graph.json","fetch_events":"https://pith.science/api/pith-number/MV7T6YVLANIR4MOVJ4BPDHPAZG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG/action/storage_attestation","attest_author":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG/action/author_attestation","sign_citation":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG/action/citation_signature","submit_replication":"https://pith.science/pith/MV7T6YVLANIR4MOVJ4BPDHPAZG/action/replication_record"}},"created_at":"2026-05-18T01:05:37.104535+00:00","updated_at":"2026-05-18T01:05:37.104535+00:00"}