{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:45QKDNVNZQHF7J2CWCVNY6HPLG","short_pith_number":"pith:45QKDNVN","schema_version":"1.0","canonical_sha256":"e760a1b6adcc0e5fa742b0aadc78ef59bd37dd57e01f983d282da8b292ca7016","source":{"kind":"arxiv","id":"math/0605517","version":2},"attestation_state":"computed","paper":{"title":"On the number of homotopy types of fibres of a definable map","license":"","headline":"","cross_cats":["math.LO"],"primary_cat":"math.AG","authors_text":"Nicolai Vorobjov, Saugata Basu","submitted_at":"2006-05-18T16:49:04Z","abstract_excerpt":"In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map, in terms of the format of its graph. In particular we show that if a semi-algebraic set $S \\subset {\\R}^{m+n}$, where $\\R$ is a real closed field, is defined by a Boolean formula with $s$ polynomials of degrees less than $d$, and $\\pi: {\\R}^{m+n} \\to {\\R}^n$ is the projection on a subspace, then the number of different homotopy types of fibres of $\\pi$ does not exceed $s^{2(m+1)n}(2^m nd)^{O(nm)}$. As applications of our main results we prove single exponential boun"},"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":"math/0605517","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2006-05-18T16:49:04Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"b3e15ff2d19aaa822898f6b1d0a0fd73a2d861fe76310f093d597154525e7802","abstract_canon_sha256":"89d28f9855dc78d5e0339290aa39a328387622fd336367ab20881c6a4aadbe2e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:30:12.835172Z","signature_b64":"rme8YA07vv0bMBglAUDgTXIaXSzFm6gUtbrPxFh6lR0lIUL4fXl9iOtjJEFH3woMx/rmI/X/T/s9vkobE/PPAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e760a1b6adcc0e5fa742b0aadc78ef59bd37dd57e01f983d282da8b292ca7016","last_reissued_at":"2026-05-18T04:30:12.834508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:30:12.834508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of homotopy types of fibres of a definable map","license":"","headline":"","cross_cats":["math.LO"],"primary_cat":"math.AG","authors_text":"Nicolai Vorobjov, Saugata Basu","submitted_at":"2006-05-18T16:49:04Z","abstract_excerpt":"In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map, in terms of the format of its graph. In particular we show that if a semi-algebraic set $S \\subset {\\R}^{m+n}$, where $\\R$ is a real closed field, is defined by a Boolean formula with $s$ polynomials of degrees less than $d$, and $\\pi: {\\R}^{m+n} \\to {\\R}^n$ is the projection on a subspace, then the number of different homotopy types of fibres of $\\pi$ does not exceed $s^{2(m+1)n}(2^m nd)^{O(nm)}$. As applications of our main results we prove single exponential boun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0605517","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":"math/0605517","created_at":"2026-05-18T04:30:12.834611+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0605517v2","created_at":"2026-05-18T04:30:12.834611+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0605517","created_at":"2026-05-18T04:30:12.834611+00:00"},{"alias_kind":"pith_short_12","alias_value":"45QKDNVNZQHF","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"45QKDNVNZQHF7J2C","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"45QKDNVN","created_at":"2026-05-18T12:25:53.939244+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/45QKDNVNZQHF7J2CWCVNY6HPLG","json":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG.json","graph_json":"https://pith.science/api/pith-number/45QKDNVNZQHF7J2CWCVNY6HPLG/graph.json","events_json":"https://pith.science/api/pith-number/45QKDNVNZQHF7J2CWCVNY6HPLG/events.json","paper":"https://pith.science/paper/45QKDNVN"},"agent_actions":{"view_html":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG","download_json":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG.json","view_paper":"https://pith.science/paper/45QKDNVN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0605517&json=true","fetch_graph":"https://pith.science/api/pith-number/45QKDNVNZQHF7J2CWCVNY6HPLG/graph.json","fetch_events":"https://pith.science/api/pith-number/45QKDNVNZQHF7J2CWCVNY6HPLG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG/action/storage_attestation","attest_author":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG/action/author_attestation","sign_citation":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG/action/citation_signature","submit_replication":"https://pith.science/pith/45QKDNVNZQHF7J2CWCVNY6HPLG/action/replication_record"}},"created_at":"2026-05-18T04:30:12.834611+00:00","updated_at":"2026-05-18T04:30:12.834611+00:00"}