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We establish a Rolle type theorem for the generalized multiplicity (called {\\em cyclicity}) of zero of the family of univariate holomorphic functions ${\\sum_{j=1}^k f_{\\lambda;j}}_{\\lambda\\in V}$ at $0\\in\\Di$. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form $\\sum_{k=1}^m P_k(z)e^{Q_k(z)}$, $z\\in\\Co$, "},"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":"1308.2662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-08-12T19:53:13Z","cross_cats_sorted":[],"title_canon_sha256":"8b05410421ce6a3d18c209315d1528b397ce441d4714ba3691a5ea9a15e178d9","abstract_canon_sha256":"9276067a850069d85f19b6686e3eb8e91769abd1b58406a4b69d0cd525152d8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:06.904559Z","signature_b64":"QBbcGwIbja6VkpSAFvvNCEpn3geo2DneV9MbYJsdmZa7JyrEyF466QjDdqLVaMNP3FDL9AQFKA/yhsA/ZghwAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aeb98f47c2682da2771c3344e711e2322580c0dc4787876013d3622f8d8445ee","last_reissued_at":"2026-05-18T03:16:06.904113Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:06.904113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Rolle type theorem for cyclicity of zeros of families of analytic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Alexander Brudnyi","submitted_at":"2013-08-12T19:53:13Z","abstract_excerpt":"Let $\\{f_{\\lambda; j}\\}_{\\lambda\\in V; 1\\le j\\le k}$ be families of holomorphic functions in the open unit disk $\\Di\\subset\\Co$ depending holomorphically on a parameter $\\lambda\\in V\\subset \\Co^n$. We establish a Rolle type theorem for the generalized multiplicity (called {\\em cyclicity}) of zero of the family of univariate holomorphic functions ${\\sum_{j=1}^k f_{\\lambda;j}}_{\\lambda\\in V}$ at $0\\in\\Di$. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form $\\sum_{k=1}^m P_k(z)e^{Q_k(z)}$, $z\\in\\Co$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2662","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":"1308.2662","created_at":"2026-05-18T03:16:06.904177+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.2662v1","created_at":"2026-05-18T03:16:06.904177+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.2662","created_at":"2026-05-18T03:16:06.904177+00:00"},{"alias_kind":"pith_short_12","alias_value":"V24Y6R6CNAW2","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"V24Y6R6CNAW2E5Y4","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"V24Y6R6C","created_at":"2026-05-18T12:28:02.375192+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/V24Y6R6CNAW2E5Y4GNCOOEPCGI","json":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI.json","graph_json":"https://pith.science/api/pith-number/V24Y6R6CNAW2E5Y4GNCOOEPCGI/graph.json","events_json":"https://pith.science/api/pith-number/V24Y6R6CNAW2E5Y4GNCOOEPCGI/events.json","paper":"https://pith.science/paper/V24Y6R6C"},"agent_actions":{"view_html":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI","download_json":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI.json","view_paper":"https://pith.science/paper/V24Y6R6C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.2662&json=true","fetch_graph":"https://pith.science/api/pith-number/V24Y6R6CNAW2E5Y4GNCOOEPCGI/graph.json","fetch_events":"https://pith.science/api/pith-number/V24Y6R6CNAW2E5Y4GNCOOEPCGI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI/action/storage_attestation","attest_author":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI/action/author_attestation","sign_citation":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI/action/citation_signature","submit_replication":"https://pith.science/pith/V24Y6R6CNAW2E5Y4GNCOOEPCGI/action/replication_record"}},"created_at":"2026-05-18T03:16:06.904177+00:00","updated_at":"2026-05-18T03:16:06.904177+00:00"}