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In this case we show that $K$ is polynomially convex if and only if for a fixed neighbourhood $U$, defined in terms of the defining functions of $M$, there exists a plurisubharmonic function $\\Psi$ on $\\mathbb{C}^n$ such that $K\\subset \\{\\Psi<0\\}\\subset U$."},"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":"1504.07049","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-04-27T12:21:50Z","cross_cats_sorted":[],"title_canon_sha256":"5764ce01d085cbf496982eda5cff1f16da83305b13241b55ea5f383f5385584f","abstract_canon_sha256":"900c5f93a7b99db42f25b40547cab91f37cff8831467633419a02f4711a7d9da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:43.815086Z","signature_b64":"3uEmbF75AzwB8y17BeafjjE/MUP96b081Pth0KAxA5oj+Sbey29+AmD4qgIC3Owww6ADlqsk2xbV5ZpZEjgeAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cffdcbe8dd75cc3275e5c06a681305615ddadd6d61a22805995b117a8210e42f","last_reissued_at":"2026-05-18T02:17:43.814683Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:43.814683Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On polynomial convexity of compact subsets of totally-real submanifold in $\\mathbb{C}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Sushil Gorai","submitted_at":"2015-04-27T12:21:50Z","abstract_excerpt":"Let $K$ be a compact subset of a totally-real manifold $M$, where $M$ is either a $\\mathcal{C}^2$-smooth graph in $\\mathbb{C}^{2n}$ over $\\mathbb{C}^n$, or $M=u^{-1}\\{0\\}$ for a $\\mathcal{C}^2$-smooth submersion $u$ from $\\mathbb{C}^n$ to $\\mathbb{R}^{2n-k}$, $k\\leq n$. In this case we show that $K$ is polynomially convex if and only if for a fixed neighbourhood $U$, defined in terms of the defining functions of $M$, there exists a plurisubharmonic function $\\Psi$ on $\\mathbb{C}^n$ such that $K\\subset \\{\\Psi<0\\}\\subset U$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07049","kind":"arxiv","version":1},"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":"1504.07049","created_at":"2026-05-18T02:17:43.814742+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.07049v1","created_at":"2026-05-18T02:17:43.814742+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07049","created_at":"2026-05-18T02:17:43.814742+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z764X2G5OXGD","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z764X2G5OXGDE5PF","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z764X2G5","created_at":"2026-05-18T12:29:52.810259+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/Z764X2G5OXGDE5PFYBVGQEYFMF","json":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF.json","graph_json":"https://pith.science/api/pith-number/Z764X2G5OXGDE5PFYBVGQEYFMF/graph.json","events_json":"https://pith.science/api/pith-number/Z764X2G5OXGDE5PFYBVGQEYFMF/events.json","paper":"https://pith.science/paper/Z764X2G5"},"agent_actions":{"view_html":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF","download_json":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF.json","view_paper":"https://pith.science/paper/Z764X2G5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.07049&json=true","fetch_graph":"https://pith.science/api/pith-number/Z764X2G5OXGDE5PFYBVGQEYFMF/graph.json","fetch_events":"https://pith.science/api/pith-number/Z764X2G5OXGDE5PFYBVGQEYFMF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF/action/storage_attestation","attest_author":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF/action/author_attestation","sign_citation":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF/action/citation_signature","submit_replication":"https://pith.science/pith/Z764X2G5OXGDE5PFYBVGQEYFMF/action/replication_record"}},"created_at":"2026-05-18T02:17:43.814742+00:00","updated_at":"2026-05-18T02:17:43.814742+00:00"}