{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4UTAWDPHLPEIEUDW34EA6M2XNA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9ce93c45b2a0e3703517e35d0944e89267894ff6eac7267585f66ec82906cbd4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-31T15:55:45Z","title_canon_sha256":"72d2df3d0d6e9114d0f8cb04603f5c91aea100206a4c1f6d89320019e70fd9ab"},"schema_version":"1.0","source":{"id":"1310.8563","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.8563","created_at":"2026-05-18T03:04:36Z"},{"alias_kind":"arxiv_version","alias_value":"1310.8563v4","created_at":"2026-05-18T03:04:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.8563","created_at":"2026-05-18T03:04:36Z"},{"alias_kind":"pith_short_12","alias_value":"4UTAWDPHLPEI","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4UTAWDPHLPEIEUDW","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4UTAWDPH","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:767ff27c55b347a8165cd38743514e023cc3d3ad0a48e8ef4e3d40b8f1f9221b","target":"graph","created_at":"2026-05-18T03:04:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky 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 $M_n(K)$. This conjecture was proved for $n=2$ when $K$ is closed under quadratic extensions. In this paper the conjecture is verified for $K=\\mathbb{R}$ and $n=2$, also for semi-homogeneous polynomials $p$, with a partial solution for an arbitrary field $K$.","authors_text":"Sergey Malev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-31T15:55:45Z","title":"The images of non-commutative polynomials evaluated on $2\\times 2$ matrices over an arbitrary field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.8563","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b8a7ebc7e54595e0886b3764ab8fdc2ad6c192c30cc621ee248a8c7eac5facd4","target":"record","created_at":"2026-05-18T03:04:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9ce93c45b2a0e3703517e35d0944e89267894ff6eac7267585f66ec82906cbd4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-31T15:55:45Z","title_canon_sha256":"72d2df3d0d6e9114d0f8cb04603f5c91aea100206a4c1f6d89320019e70fd9ab"},"schema_version":"1.0","source":{"id":"1310.8563","kind":"arxiv","version":4}},"canonical_sha256":"e5260b0de75bc8825076df080f33576836907e5fdac532542301dae7d502689d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5260b0de75bc8825076df080f33576836907e5fdac532542301dae7d502689d","first_computed_at":"2026-05-18T03:04:36.767489Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:36.767489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g+A09K1+K3VEZuI51RMgtOx9AsoEVbT6Cgl/jhzaGREt6ApyJwIx7CBBLmUCqnh4Lf/LdqeFsHntFEbWGXilCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:36.768293Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.8563","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8a7ebc7e54595e0886b3764ab8fdc2ad6c192c30cc621ee248a8c7eac5facd4","sha256:767ff27c55b347a8165cd38743514e023cc3d3ad0a48e8ef4e3d40b8f1f9221b"],"state_sha256":"5ac9081d0aa3fc47ffe8e1362e8aae0717f1d00fd8571f6915972808743dec8d"}