{"paper":{"title":"On the convexity of numerical range over certain fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"E. Ballico","submitted_at":"2016-11-18T14:03:27Z","abstract_excerpt":"Let $L$ be a degree $2$ Galois extension of the field $K$ and $M$ an $n\\times n$ matrix with coefficients in $L$. Let $\\langle \\ ,\\ \\rangle : L^n\\times L^n\\to L$ be the sesquilinear form associated to the involution $\\sigma: L\\to L$ fixing $K$. This sesquilinear form defines the numerical range $\\mathrm{Num}(M)$ of any $n\\times n$ matrix over $L$. In this paper we study the convexity of $\\mathrm{Num}(M)$ (under certain assumptions on $K$ and/or $M$). Many of the results are for ordered fields."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06082","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"}