{"paper":{"title":"Fa\\`a di Bruno is Taylor Composition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Heinrich Hartmann","submitted_at":"2026-06-18T14:06:50Z","abstract_excerpt":"We prove that reduced Taylor polynomials compose: for $C^k$ maps $\\phi: E \\to F$ and $\\psi: F \\to G$ between Banach spaces, $T_{\\ast}^k(\\psi\\circ\\phi;\\, x) = \\pi_{\\leq k}\\bigl(T_{\\ast}^k(\\psi;\\, y) \\circ T_{\\ast}^k(\\phi;\\, x)\\bigr)$ where $y = \\phi(x)$.\n  The proof is a direct estimate of the Peano remainder and requires no combinatorics or partition arguments. From this we derive the multivariate Fa\\`a di Bruno formula in partition form (Levy 2006), by polarization, and in multi-index form (Constantine-Savits 1996) by coefficient extraction.\n  As an application we give a higher-order product "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.26133","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.26133/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"}