{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:PV4RKIL3GAXGINKM7O5CR2ILB3","short_pith_number":"pith:PV4RKIL3","schema_version":"1.0","canonical_sha256":"7d7915217b302e64354cfbba28e90b0ec4a4cad5f6613c297bb620798ca7ac82","source":{"kind":"arxiv","id":"1502.07003","version":4},"attestation_state":"computed","paper":{"title":"Point-curve incidences in the complex plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Endre Szab\\'o, Joshua Zahl","submitted_at":"2015-02-24T23:18:50Z","abstract_excerpt":"We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\\big(m^{\\frac{k}{2k-1}}n^{\\frac{2k-2}{2k-1}}+m+n\\big)$. We establish the slightly weaker bound $O_\\varepsilon\\big(m^{\\frac{k}{2k-1}+\\varepsilon}n^{\\frac{2k-2}{2k-1}}+m+n\\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\\mathbb C}^2$ with $k$ degrees of freedom. We combine tools from algebraic geometry and differe"},"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":"1502.07003","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-24T23:18:50Z","cross_cats_sorted":[],"title_canon_sha256":"c7663658fa2da51a6ecba10fbc917f402ca18c26e83fb4cf0f3a836d63c56d51","abstract_canon_sha256":"640b82c64e1703d24b00e18e837d8a6c88c54a15f3a15ec7fb59a48b0526b3f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:41.813645Z","signature_b64":"yeB14n49Cp7sbhkCNzi6cMGdBZ2JFEN/tKGOVpOLH2819ah/NKopUFFHDk0CPDoBxHSbrWhQy5kDQf/zWToiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d7915217b302e64354cfbba28e90b0ec4a4cad5f6613c297bb620798ca7ac82","last_reissued_at":"2026-05-18T00:10:41.813054Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:41.813054Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Point-curve incidences in the complex plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Endre Szab\\'o, Joshua Zahl","submitted_at":"2015-02-24T23:18:50Z","abstract_excerpt":"We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\\big(m^{\\frac{k}{2k-1}}n^{\\frac{2k-2}{2k-1}}+m+n\\big)$. We establish the slightly weaker bound $O_\\varepsilon\\big(m^{\\frac{k}{2k-1}+\\varepsilon}n^{\\frac{2k-2}{2k-1}}+m+n\\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\\mathbb C}^2$ with $k$ degrees of freedom. We combine tools from algebraic geometry and differe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07003","kind":"arxiv","version":4},"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":"1502.07003","created_at":"2026-05-18T00:10:41.813142+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.07003v4","created_at":"2026-05-18T00:10:41.813142+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.07003","created_at":"2026-05-18T00:10:41.813142+00:00"},{"alias_kind":"pith_short_12","alias_value":"PV4RKIL3GAXG","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"PV4RKIL3GAXGINKM","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"PV4RKIL3","created_at":"2026-05-18T12:29:37.295048+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/PV4RKIL3GAXGINKM7O5CR2ILB3","json":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3.json","graph_json":"https://pith.science/api/pith-number/PV4RKIL3GAXGINKM7O5CR2ILB3/graph.json","events_json":"https://pith.science/api/pith-number/PV4RKIL3GAXGINKM7O5CR2ILB3/events.json","paper":"https://pith.science/paper/PV4RKIL3"},"agent_actions":{"view_html":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3","download_json":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3.json","view_paper":"https://pith.science/paper/PV4RKIL3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.07003&json=true","fetch_graph":"https://pith.science/api/pith-number/PV4RKIL3GAXGINKM7O5CR2ILB3/graph.json","fetch_events":"https://pith.science/api/pith-number/PV4RKIL3GAXGINKM7O5CR2ILB3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3/action/storage_attestation","attest_author":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3/action/author_attestation","sign_citation":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3/action/citation_signature","submit_replication":"https://pith.science/pith/PV4RKIL3GAXGINKM7O5CR2ILB3/action/replication_record"}},"created_at":"2026-05-18T00:10:41.813142+00:00","updated_at":"2026-05-18T00:10:41.813142+00:00"}