{"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"}