{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:YDHZV2F4FP3FY3DWBPCLA274ZC","short_pith_number":"pith:YDHZV2F4","schema_version":"1.0","canonical_sha256":"c0cf9ae8bc2bf65c6c760bc4b06bfcc8ad08e2f76a1feb9002bb2c87f9ec2efc","source":{"kind":"arxiv","id":"1606.04069","version":3},"attestation_state":"computed","paper":{"title":"Bounds on the individual Betti numbers of complex varieties, stability and algorithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.AT"],"primary_cat":"math.AG","authors_text":"Cordian Riener, Saugata Basu","submitted_at":"2016-06-13T19:06:45Z","abstract_excerpt":"We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\\mathrm{C}^k$, $\\mathbb{P}^k_{\\mathrm{C}}$, as well as products of projective spaces, defined by $r$ polynomials of degrees at most $d$ as a function of $p,d$ and $r$. Unlike previous bounds these bounds are independent of $k$, the dimension of the ambient space. We also prove as consequences of our technique certain homological and representational stability results for "},"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":"1606.04069","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-06-13T19:06:45Z","cross_cats_sorted":["cs.CC","math.AT"],"title_canon_sha256":"ff9612e1e9d1fbbab8e3047def3149f5cb205ab85098372a8c6ea362f21ace7c","abstract_canon_sha256":"5b6cd79dcf0676a29311466970aed05c0cfd34f7c8f404a98cd9b0b728fe3fb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:47.288402Z","signature_b64":"EYaqbHv64MAfu324CykKbfv7b42FjmbP/zHXXNrQOjLYhsC7a350PtLwnCOXCWm3RQtFHAh6PowhdMPVjv7zBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0cf9ae8bc2bf65c6c760bc4b06bfcc8ad08e2f76a1feb9002bb2c87f9ec2efc","last_reissued_at":"2026-05-18T01:10:47.287706Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:47.287706Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds on the individual Betti numbers of complex varieties, stability and algorithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.AT"],"primary_cat":"math.AG","authors_text":"Cordian Riener, Saugata Basu","submitted_at":"2016-06-13T19:06:45Z","abstract_excerpt":"We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\\mathrm{C}^k$, $\\mathbb{P}^k_{\\mathrm{C}}$, as well as products of projective spaces, defined by $r$ polynomials of degrees at most $d$ as a function of $p,d$ and $r$. Unlike previous bounds these bounds are independent of $k$, the dimension of the ambient space. We also prove as consequences of our technique certain homological and representational stability results for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.04069","kind":"arxiv","version":3},"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":"1606.04069","created_at":"2026-05-18T01:10:47.287806+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.04069v3","created_at":"2026-05-18T01:10:47.287806+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.04069","created_at":"2026-05-18T01:10:47.287806+00:00"},{"alias_kind":"pith_short_12","alias_value":"YDHZV2F4FP3F","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"YDHZV2F4FP3FY3DW","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"YDHZV2F4","created_at":"2026-05-18T12:30:53.716459+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/YDHZV2F4FP3FY3DWBPCLA274ZC","json":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC.json","graph_json":"https://pith.science/api/pith-number/YDHZV2F4FP3FY3DWBPCLA274ZC/graph.json","events_json":"https://pith.science/api/pith-number/YDHZV2F4FP3FY3DWBPCLA274ZC/events.json","paper":"https://pith.science/paper/YDHZV2F4"},"agent_actions":{"view_html":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC","download_json":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC.json","view_paper":"https://pith.science/paper/YDHZV2F4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.04069&json=true","fetch_graph":"https://pith.science/api/pith-number/YDHZV2F4FP3FY3DWBPCLA274ZC/graph.json","fetch_events":"https://pith.science/api/pith-number/YDHZV2F4FP3FY3DWBPCLA274ZC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC/action/storage_attestation","attest_author":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC/action/author_attestation","sign_citation":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC/action/citation_signature","submit_replication":"https://pith.science/pith/YDHZV2F4FP3FY3DWBPCLA274ZC/action/replication_record"}},"created_at":"2026-05-18T01:10:47.287806+00:00","updated_at":"2026-05-18T01:10:47.287806+00:00"}