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We prove that a function $D\\colon R\\to R$ is a derivation of order $n$ if and only if $D$ belongs to the closure of the set of differential operators of degree $n$ in the product topology of $R^R$, where the image space is endowed with the discrete topology. In other words, $f$ is a derivation of order $n$ if and only if, for every finite set $F\\subset R$, there is a differential operator $D$ of degree $n$ such that $f=D$ on $F$. We also prove that if $d_1, \\dots, d_n$ are nonzero derivations on $R$, then $d_1 \\circ \\ldots \\circ d_n$ is a d"},"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":"1803.01025","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-03-02T19:50:44Z","cross_cats_sorted":[],"title_canon_sha256":"66ea62b62bface719e775e3e4c97c6557318bd860f16283e7a686cfa69fd2cd3","abstract_canon_sha256":"42871184aabec8598910c47564b114bcf4e9557f98da339b75338aeb32fd848f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:08.262682Z","signature_b64":"72Xa+6WTQDOzBLvbCM0yNjl1VrfjDyeEaVSlaN3Dh9QUJIJeRKppZNWORNQAOYv0YBmZ7oGraNf4eS3qcOjfBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ef993df1d2493d79a6b51eadb1e10f73a38a17d79f26cc9f111c5114c5d4a93","last_reissued_at":"2026-05-18T00:19:08.261959Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:08.261959Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Derivations and differential operators on rings and fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Gergely Kiss, Mikl\\'os Laczkovich","submitted_at":"2018-03-02T19:50:44Z","abstract_excerpt":"Let $R$ be an integral domain of characteristic zero. We prove that a function $D\\colon R\\to R$ is a derivation of order $n$ if and only if $D$ belongs to the closure of the set of differential operators of degree $n$ in the product topology of $R^R$, where the image space is endowed with the discrete topology. In other words, $f$ is a derivation of order $n$ if and only if, for every finite set $F\\subset R$, there is a differential operator $D$ of degree $n$ such that $f=D$ on $F$. We also prove that if $d_1, \\dots, d_n$ are nonzero derivations on $R$, then $d_1 \\circ \\ldots \\circ d_n$ is a d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.01025","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":"1803.01025","created_at":"2026-05-18T00:19:08.262078+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.01025v2","created_at":"2026-05-18T00:19:08.262078+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.01025","created_at":"2026-05-18T00:19:08.262078+00:00"},{"alias_kind":"pith_short_12","alias_value":"P34ZHXY5ESJ5","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"P34ZHXY5ESJ5PGTL","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"P34ZHXY5","created_at":"2026-05-18T12:32:43.782077+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/P34ZHXY5ESJ5PGTLKHVNWHQQ64","json":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64.json","graph_json":"https://pith.science/api/pith-number/P34ZHXY5ESJ5PGTLKHVNWHQQ64/graph.json","events_json":"https://pith.science/api/pith-number/P34ZHXY5ESJ5PGTLKHVNWHQQ64/events.json","paper":"https://pith.science/paper/P34ZHXY5"},"agent_actions":{"view_html":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64","download_json":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64.json","view_paper":"https://pith.science/paper/P34ZHXY5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.01025&json=true","fetch_graph":"https://pith.science/api/pith-number/P34ZHXY5ESJ5PGTLKHVNWHQQ64/graph.json","fetch_events":"https://pith.science/api/pith-number/P34ZHXY5ESJ5PGTLKHVNWHQQ64/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64/action/storage_attestation","attest_author":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64/action/author_attestation","sign_citation":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64/action/citation_signature","submit_replication":"https://pith.science/pith/P34ZHXY5ESJ5PGTLKHVNWHQQ64/action/replication_record"}},"created_at":"2026-05-18T00:19:08.262078+00:00","updated_at":"2026-05-18T00:19:08.262078+00:00"}