{"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"}