{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:ZENA6H3DVL4OPH5CAA5OHLDTXK","short_pith_number":"pith:ZENA6H3D","schema_version":"1.0","canonical_sha256":"c91a0f1f63aaf8e79fa2003ae3ac73baa6477857623c9d9de31dc36e2b8e85e7","source":{"kind":"arxiv","id":"0704.0295","version":3},"attestation_state":"computed","paper":{"title":"On the number of topological types occurring in a parametrized family of arrangements","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Saugata Basu","submitted_at":"2007-04-03T05:51:58Z","abstract_excerpt":"Let ${\\mathcal S}(\\R)$ be an o-minimal structure over $\\R$, $T \\subset \\R^{k_1+k_2+\\ell}$ a closed definable set, and $$ \\displaylines{\\pi_1: \\R^{k_1+k_2+\\ell}\\to \\R^{k_1 + k_2}, \\pi_2: \\R^{k_1+k_2+\\ell}\\to \\R^{\\ell}, \\ \\pi_3: \\R^{k_1 + k_2} \\to \\R^{k_2}} $$ the projection maps.\n  For any collection ${\\mathcal A} = \\{A_1,...,A_n\\}$ of subsets of $\\R^{k_1+k_2}$, and $\\z \\in \\R^{k_2}$, let $\\A_\\z$ denote the collection of subsets of $\\R^{k_1}$, $\\{A_{1,\\z},..., A_{n,\\z}\\}$, where $A_{i,\\z} = A_i \\cap \\pi_3^{-1}(\\z), 1 \\leq i \\leq n$. We prove that there exists a constant $C = C(T) > 0,$ such tha"},"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":"0704.0295","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-04-03T05:51:58Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"7016a5fa196afd070048ffcf9a075a55a5d6e546d952eca99ec0cc64e8bdff8c","abstract_canon_sha256":"55fa45542cab1c6642d97ced3f1fc6d281c1850130434218cff2c922684affcc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:21.965625Z","signature_b64":"GwGy63xOgpeDO5eox/MKL9KvO9qAkAgI4seaGwUV9utiEZ1vyIelEBaeydWpBju/GqS+/OHFT7ctkqIn+rnFDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c91a0f1f63aaf8e79fa2003ae3ac73baa6477857623c9d9de31dc36e2b8e85e7","last_reissued_at":"2026-05-18T04:28:21.965083Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:21.965083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of topological types occurring in a parametrized family of arrangements","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Saugata Basu","submitted_at":"2007-04-03T05:51:58Z","abstract_excerpt":"Let ${\\mathcal S}(\\R)$ be an o-minimal structure over $\\R$, $T \\subset \\R^{k_1+k_2+\\ell}$ a closed definable set, and $$ \\displaylines{\\pi_1: \\R^{k_1+k_2+\\ell}\\to \\R^{k_1 + k_2}, \\pi_2: \\R^{k_1+k_2+\\ell}\\to \\R^{\\ell}, \\ \\pi_3: \\R^{k_1 + k_2} \\to \\R^{k_2}} $$ the projection maps.\n  For any collection ${\\mathcal A} = \\{A_1,...,A_n\\}$ of subsets of $\\R^{k_1+k_2}$, and $\\z \\in \\R^{k_2}$, let $\\A_\\z$ denote the collection of subsets of $\\R^{k_1}$, $\\{A_{1,\\z},..., A_{n,\\z}\\}$, where $A_{i,\\z} = A_i \\cap \\pi_3^{-1}(\\z), 1 \\leq i \\leq n$. We prove that there exists a constant $C = C(T) > 0,$ such tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.0295","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":"0704.0295","created_at":"2026-05-18T04:28:21.965260+00:00"},{"alias_kind":"arxiv_version","alias_value":"0704.0295v3","created_at":"2026-05-18T04:28:21.965260+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.0295","created_at":"2026-05-18T04:28:21.965260+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZENA6H3DVL4O","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZENA6H3DVL4OPH5C","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZENA6H3D","created_at":"2026-05-18T12:25:56.245647+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/ZENA6H3DVL4OPH5CAA5OHLDTXK","json":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK.json","graph_json":"https://pith.science/api/pith-number/ZENA6H3DVL4OPH5CAA5OHLDTXK/graph.json","events_json":"https://pith.science/api/pith-number/ZENA6H3DVL4OPH5CAA5OHLDTXK/events.json","paper":"https://pith.science/paper/ZENA6H3D"},"agent_actions":{"view_html":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK","download_json":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK.json","view_paper":"https://pith.science/paper/ZENA6H3D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0704.0295&json=true","fetch_graph":"https://pith.science/api/pith-number/ZENA6H3DVL4OPH5CAA5OHLDTXK/graph.json","fetch_events":"https://pith.science/api/pith-number/ZENA6H3DVL4OPH5CAA5OHLDTXK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK/action/storage_attestation","attest_author":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK/action/author_attestation","sign_citation":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK/action/citation_signature","submit_replication":"https://pith.science/pith/ZENA6H3DVL4OPH5CAA5OHLDTXK/action/replication_record"}},"created_at":"2026-05-18T04:28:21.965260+00:00","updated_at":"2026-05-18T04:28:21.965260+00:00"}