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We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height."},"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":"1309.5682","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-23T02:10:42Z","cross_cats_sorted":[],"title_canon_sha256":"7050f9f6f932a6a560e829f7c5de7ad7d08c1268ca071450e3b032940063ad0d","abstract_canon_sha256":"829356804eecc7d6339c42bb779e507bf02012a72374b96a7cfe7455e7bff8fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:34.508235Z","signature_b64":"n2knWk7LITNBj0609SPWdTioMYJZSYHknm7yrS70q1/FGkC1c9v708zvyWXV0hNEeqwtGM3quUhz5qqIms/JCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c83d277f55566b31f34f294d1b1755dd66357c46155cdd449b57a35899a8ae15","last_reissued_at":"2026-05-18T03:12:34.507432Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:34.507432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Variation of the canonical height in a family of rational maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dragos Ghioca, Niki Myrto Mavraki","submitted_at":"2013-09-23T02:10:42Z","abstract_excerpt":"Let $d\\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the rational map $f_t$. We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5682","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":"1309.5682","created_at":"2026-05-18T03:12:34.507557+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.5682v1","created_at":"2026-05-18T03:12:34.507557+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5682","created_at":"2026-05-18T03:12:34.507557+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZA6SO72VKZVT","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZA6SO72VKZVTD42P","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZA6SO72V","created_at":"2026-05-18T12:28:09.283467+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/ZA6SO72VKZVTD42PFFGRWF2V3V","json":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V.json","graph_json":"https://pith.science/api/pith-number/ZA6SO72VKZVTD42PFFGRWF2V3V/graph.json","events_json":"https://pith.science/api/pith-number/ZA6SO72VKZVTD42PFFGRWF2V3V/events.json","paper":"https://pith.science/paper/ZA6SO72V"},"agent_actions":{"view_html":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V","download_json":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V.json","view_paper":"https://pith.science/paper/ZA6SO72V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.5682&json=true","fetch_graph":"https://pith.science/api/pith-number/ZA6SO72VKZVTD42PFFGRWF2V3V/graph.json","fetch_events":"https://pith.science/api/pith-number/ZA6SO72VKZVTD42PFFGRWF2V3V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V/action/storage_attestation","attest_author":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V/action/author_attestation","sign_citation":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V/action/citation_signature","submit_replication":"https://pith.science/pith/ZA6SO72VKZVTD42PFFGRWF2V3V/action/replication_record"}},"created_at":"2026-05-18T03:12:34.507557+00:00","updated_at":"2026-05-18T03:12:34.507557+00:00"}