{"paper":{"title":"Convex entire noncommutative functions are polynomials of degree two or less","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"J. E. Pascoe, J. William Helton, Ryan Tully-Doyle, Victor Vinnikov","submitted_at":"2015-01-24T04:15:05Z","abstract_excerpt":"This paper concerns matrix \"convex\" functions of (free) noncommuting variables, $x = (x_1, \\ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result: that a function of $x$ that is matrix convex near $0$ and also that is \"analytic\" in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less.\n  More generally, we prove that a function $F$ in two classes of noncommuting variables, $a = (a_1, \\ldots, a_{\\tilde{g}})$ and $x = (x_1, \\ldots, x_g)$ that is \"an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06000","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"}