{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:FNHFO7HXVZSAFHXEXQV6C4V7W3","short_pith_number":"pith:FNHFO7HX","schema_version":"1.0","canonical_sha256":"2b4e577cf7ae64029ee4bc2be172bfb6ecadc735c32d984ffd4e8c5f9c532284","source":{"kind":"arxiv","id":"1509.03211","version":2},"attestation_state":"computed","paper":{"title":"Structure of sets which are well approximated by zero sets of harmonic polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Matthew Badger, Max Engelstein, Tatiana Toro","submitted_at":"2015-09-10T16:12:44Z","abstract_excerpt":"The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how \"degree $k$ points\" sit inside zero sets of harmonic polynomials in $\\mathbb R^n$ of degree $d$ (for all $n\\geq 2$ and $1\\leq k\\leq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of s"},"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":"1509.03211","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-10T16:12:44Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"155bc3d449c07e2dcd6eca2c13b1a5a222140f213c4e23477a2c206ffff46868","abstract_canon_sha256":"f47f88442bc02948a50b5a74f191cc07193878119890cf74ec5671378be500d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:00.238610Z","signature_b64":"94FBEOnOjNqJkw1sToskFOjc7c5NsyyGVh8CAo7zGE4CGAIVzd3eHqqGUGFRaov0QNy7FQu0HzNXYoU3KoxeDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b4e577cf7ae64029ee4bc2be172bfb6ecadc735c32d984ffd4e8c5f9c532284","last_reissued_at":"2026-05-18T00:21:00.238098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:00.238098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structure of sets which are well approximated by zero sets of harmonic polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Matthew Badger, Max Engelstein, Tatiana Toro","submitted_at":"2015-09-10T16:12:44Z","abstract_excerpt":"The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how \"degree $k$ points\" sit inside zero sets of harmonic polynomials in $\\mathbb R^n$ of degree $d$ (for all $n\\geq 2$ and $1\\leq k\\leq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03211","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":"1509.03211","created_at":"2026-05-18T00:21:00.238180+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.03211v2","created_at":"2026-05-18T00:21:00.238180+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03211","created_at":"2026-05-18T00:21:00.238180+00:00"},{"alias_kind":"pith_short_12","alias_value":"FNHFO7HXVZSA","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"FNHFO7HXVZSAFHXE","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"FNHFO7HX","created_at":"2026-05-18T12:29:19.899920+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/FNHFO7HXVZSAFHXEXQV6C4V7W3","json":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3.json","graph_json":"https://pith.science/api/pith-number/FNHFO7HXVZSAFHXEXQV6C4V7W3/graph.json","events_json":"https://pith.science/api/pith-number/FNHFO7HXVZSAFHXEXQV6C4V7W3/events.json","paper":"https://pith.science/paper/FNHFO7HX"},"agent_actions":{"view_html":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3","download_json":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3.json","view_paper":"https://pith.science/paper/FNHFO7HX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.03211&json=true","fetch_graph":"https://pith.science/api/pith-number/FNHFO7HXVZSAFHXEXQV6C4V7W3/graph.json","fetch_events":"https://pith.science/api/pith-number/FNHFO7HXVZSAFHXEXQV6C4V7W3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/action/storage_attestation","attest_author":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/action/author_attestation","sign_citation":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/action/citation_signature","submit_replication":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/action/replication_record"}},"created_at":"2026-05-18T00:21:00.238180+00:00","updated_at":"2026-05-18T00:21:00.238180+00:00"}