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It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$. We prove the conjecture for $n=2$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1005.0191","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-05-03T06:16:25Z","cross_cats_sorted":[],"title_canon_sha256":"335495a098e272e7b363ea924a2df0a06a88093c284470b2425a5683be28b265","abstract_canon_sha256":"00ec4c1630eedf178033d18865f4b4692cbc424aa3759c2e6408780e885d105c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:06.177102Z","signature_b64":"aXvj49rD71+WZUX6tjwb6JFmU95YxbjmML0O2wCqtSaIfeq8bjozBqsf9fMceJEPhkBTDbOkETIa+EFlRFKbCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c59b69861a3cecbb9073bd7e7549bf06bcc3eac07002b9df7b13e1301098d98b","last_reissued_at":"2026-05-18T00:29:06.176573Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:06.176573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The images of non-commutative polynomials evaluated on $2\\times 2$ matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alexey Kanel-Belov, Louis Rowen, Sergey Malev","submitted_at":"2010-05-03T06:16:25Z","abstract_excerpt":"Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. 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