{"paper":{"title":"A refined realization theorem in the context of the Schur-Szeg\\H{o} composition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Vladimir Petrov Kostov","submitted_at":"2015-04-07T11:55:01Z","abstract_excerpt":"Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\\cdots +c_{n-1})$ is representable as Schur-Szeg\\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give necessary and sufficient conditions upon the possible values of the $8$-vector whose components are the number of positive, zero, negative and complex roots of a real polynomial $P$ and the number of positive, zero, negative and complex among the quantities $a_i$ corresponding to $P$. A similar result is proved about entire functions of the form $e^xR$, wher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01562","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"}