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Let $f:X\\to{\\mathbb R}$ be a subanalytic function on a subset $X\\in{\\mathfrak F}(M)$ such that the inverse image under $f$ of each interval of ${\\mathbb R}$ belongs to ${\\mathfrak F}(M)$. Let ${\\rm Max}(f)$ be the set of local maxima of $f$ and consider ${\\rm Max}_\\lambda(f):={\\rm Max}(f)\\cap\\{f=\\lambda\\}$ "},"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":"1803.06017","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-03-15T22:03:51Z","cross_cats_sorted":[],"title_canon_sha256":"d5bfe1288027aeaeec40f953880f1194c0238120274d00b624313d2f3673eed2","abstract_canon_sha256":"3b34fcdea9458f66924700881cdbcfd5d514544b92626b64d61b2f10d6073c92"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:51.025174Z","signature_b64":"Wz1bwTn3JPfH0MSa037hWvRrOaQbzjSryoRBzZMNANIDZ5hEWfL08M+Vm+oBPrQ3ApPJ9ECT5DWgtRSZ8NYrCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56617a9f6fe4f54da93b510eaa89ca67385054f887398b9d90d2d5332ea7ea82","last_reissued_at":"2026-05-18T00:20:51.024606Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:51.024606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the set of local extrema of a subanalytic function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jos\\'e F. 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Let ${\\rm Max}(f)$ be the set of local maxima of $f$ and consider ${\\rm Max}_\\lambda(f):={\\rm Max}(f)\\cap\\{f=\\lambda\\}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06017","kind":"arxiv","version":1},"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":"1803.06017","created_at":"2026-05-18T00:20:51.024673+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.06017v1","created_at":"2026-05-18T00:20:51.024673+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.06017","created_at":"2026-05-18T00:20:51.024673+00:00"},{"alias_kind":"pith_short_12","alias_value":"KZQXVH3P4T2U","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_16","alias_value":"KZQXVH3P4T2U3KJ3","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_8","alias_value":"KZQXVH3P","created_at":"2026-05-18T12:32:33.847187+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/KZQXVH3P4T2U3KJ3KEHKVCOKM4","json":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4.json","graph_json":"https://pith.science/api/pith-number/KZQXVH3P4T2U3KJ3KEHKVCOKM4/graph.json","events_json":"https://pith.science/api/pith-number/KZQXVH3P4T2U3KJ3KEHKVCOKM4/events.json","paper":"https://pith.science/paper/KZQXVH3P"},"agent_actions":{"view_html":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4","download_json":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4.json","view_paper":"https://pith.science/paper/KZQXVH3P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.06017&json=true","fetch_graph":"https://pith.science/api/pith-number/KZQXVH3P4T2U3KJ3KEHKVCOKM4/graph.json","fetch_events":"https://pith.science/api/pith-number/KZQXVH3P4T2U3KJ3KEHKVCOKM4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4/action/storage_attestation","attest_author":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4/action/author_attestation","sign_citation":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4/action/citation_signature","submit_replication":"https://pith.science/pith/KZQXVH3P4T2U3KJ3KEHKVCOKM4/action/replication_record"}},"created_at":"2026-05-18T00:20:51.024673+00:00","updated_at":"2026-05-18T00:20:51.024673+00:00"}