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We show that in CMW's result it is possible to replace $\\mathbb{C}$ by any field of characteristic zero, and we conjecture the following 'two-dimensional Centralizer Conjecture over $D$': Suppose the Jacobian of $A$ and $B$ is invertible in $D[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \\in D[x,y]$, $D$ is an integral domain of charact"},"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":"1802.04685","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-12T18:58:44Z","cross_cats_sorted":[],"title_canon_sha256":"5b6c30ec2ee822bf69449a2909f24754b403a1469bed393cc8fe5f9148d614eb","abstract_canon_sha256":"dd214caafa174efc4d082921c7d38acc7067a64a41133230b064519d0914f928"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:54.438428Z","signature_b64":"HchDL9FJWDtRTck1SyAAczddGtahnmSe8zUbSWQ8qCR6rnzl/Z+wNaa9dGn7/UZSAY/c46BK8cveodGZ55tMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c53a9f61185f0af269e0c785cb5d6f4b8d87ed2a6961a28671cfa79902dd65a","last_reissued_at":"2026-05-18T00:22:54.437923Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:54.437923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The two-dimensional Centralizer Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Vered Moskowicz","submitted_at":"2018-02-12T18:58:44Z","abstract_excerpt":"A result by C. C.-A. Cheng, J. H. Mckay and S. S.-S. Wang says the following: Suppose the Jacobian of $A$ and $B$ is invertible in $\\mathbb{C}[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \\in \\mathbb{C}[x,y]$. Then $w \\in \\mathbb{C}[A]$. We show that in CMW's result it is possible to replace $\\mathbb{C}$ by any field of characteristic zero, and we conjecture the following 'two-dimensional Centralizer Conjecture over $D$': Suppose the Jacobian of $A$ and $B$ is invertible in $D[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \\in D[x,y]$, $D$ is an integral domain of charact"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.04685","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1802.04685","created_at":"2026-05-18T00:22:54.438013+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.04685v2","created_at":"2026-05-18T00:22:54.438013+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.04685","created_at":"2026-05-18T00:22:54.438013+00:00"},{"alias_kind":"pith_short_12","alias_value":"TRJ2T5QRQXYK","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"TRJ2T5QRQXYK6JU6","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"TRJ2T5QR","created_at":"2026-05-18T12:32:56.356000+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S","json":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S.json","graph_json":"https://pith.science/api/pith-number/TRJ2T5QRQXYK6JU6BR4FZNOW6S/graph.json","events_json":"https://pith.science/api/pith-number/TRJ2T5QRQXYK6JU6BR4FZNOW6S/events.json","paper":"https://pith.science/paper/TRJ2T5QR"},"agent_actions":{"view_html":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S","download_json":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S.json","view_paper":"https://pith.science/paper/TRJ2T5QR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.04685&json=true","fetch_graph":"https://pith.science/api/pith-number/TRJ2T5QRQXYK6JU6BR4FZNOW6S/graph.json","fetch_events":"https://pith.science/api/pith-number/TRJ2T5QRQXYK6JU6BR4FZNOW6S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S/action/storage_attestation","attest_author":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S/action/author_attestation","sign_citation":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S/action/citation_signature","submit_replication":"https://pith.science/pith/TRJ2T5QRQXYK6JU6BR4FZNOW6S/action/replication_record"}},"created_at":"2026-05-18T00:22:54.438013+00:00","updated_at":"2026-05-18T00:22:54.438013+00:00"}