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When $\\Gamma_f$ contains the center of ${\\bf U}(n)$, we show that $f$ is spherically equivalent to a polynomial. When $f$ is minimal we show that there is a homomorphism $\\Phi:\\Gamma_f \\to T_f$ such that $f$ is equivariant with respect to $\\Phi$. To do so, we characterize minimality via the triviality of a third group $H_f$. We relate properties of ${\\rm Ker}("},"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":"1711.06539","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2017-11-17T14:02:27Z","cross_cats_sorted":[],"title_canon_sha256":"365029d040ce2a6abb479802deb1403840512477cebe5992e948c0d08e8c0c61","abstract_canon_sha256":"41c665ac24d3d218fc5f23bebf749476aae461cadfb2f264069f7a6c0a4009c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:20.772328Z","signature_b64":"xZkigF7z2/FhI24GCIoBEN6PTnL/rEWpuFDyo0Hz+ZozLIQzlR/zsKmc5uy889hoAIS97nydLnavutYo3wa1DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0555495a124141e1163710111307c72e5f80a999d62fe2880587cdd2360cc117","last_reissued_at":"2026-05-18T00:30:20.771617Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:20.771617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetries and regularity for holomorphic maps between balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"John P. 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