{"paper":{"title":"The images of multilinear polynomials evaluated on $3\\times 3$ matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alexei Kanel-Belov, Louis Rowen, Sergey Malev","submitted_at":"2013-06-18T23:25:23Z","abstract_excerpt":"Let $p$ be a multilinear polynomial in several noncommuting variables, with coefficients in a algebraically closed field $K$ of arbitrary characteristic. In this paper we classify the possible images of $p$ evaluated on $3\\times 3$ matrices. The image is one of the following: \\begin{itemize} \\item \\{0\\}, \\item the set of scalar matrices, \\item a (Zariski) dense subset of $\\sl_3(K)$, the matrices of trace 0,\n  \\item a dense subset of $M_3(K)$, \\item the set of $3-$scalar matrices (i.e., matrices having eigenvalues $(\\beta, \\beta \\varepsilon, \\beta \\varepsilon^2)$ where $\\varepsilon$ is a cube r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4389","kind":"arxiv","version":3},"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"}