{"paper":{"title":"Images of polynomials with involution on $2\\times 2$ matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Lucio Centrone, Thiago Castilho de Mello","submitted_at":"2026-05-22T17:24:55Z","abstract_excerpt":"Let $\\mathbb{F}$ be a field and let $M_2(\\mathbb{F})$ be the algebra of $2\\times 2$ matrices endowed with an involution of the first kind. We study the image of multilinear $*$-polynomials evaluated on $M_2(\\mathbb{F})$. For the transpose involution over $\\mathbb{R}$, we show that the image is either a proper vector subspace or contains a basis of $M_2(\\mathbb{R})$. For the symplectic involution over quadratically closed fields or over $\\mathbb{R}$, we prove that the image is always a vector space, namely one of $\\{0\\}$, $\\mathbb{F}$, $sl_2(\\mathbb{F})$ or $M_2(\\mathbb{F})$. As a byproduct, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23865","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.23865/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}