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Poonen and Faber classified the portraits that occur for infinitely many $c$'s.\n  Given a portrait $P$, we prove an asymptotic formula for counting the number of $c \\in F$'s by height, such that $Preper(f_c, F) \\cong P$. We also prove an asymptotic formula for the analogous counting problem, where $Preper"},"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":"2409.18074","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-09-26T17:21:22Z","cross_cats_sorted":["math.AG","math.DS"],"title_canon_sha256":"0a3bc8073bb8b985ff4c038dfc5ed05107bcdf849b05e3d30655c950df616200","abstract_canon_sha256":"ec0a807eaecc80f0b5a5f3b9dfd5c556d95483de7221278aad551e7b8a048c5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:16:21.895974Z","signature_b64":"kCtpJOu1NiPcIO5XiLg9LB7xEk9cjCR+N2fdinjJ8efeAPM4FCkUkXIl7oZhbgpqFSFysAenG4mpYEZn/mKRDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b6a3af5e1892e05c9a2ead01128b3a5656aab748371c859dfafd05168db6fd2","last_reissued_at":"2026-07-05T09:16:21.895486Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:16:21.895486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of quadratic polynomials with a given portrait","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.DS"],"primary_cat":"math.NT","authors_text":"Ho Chung Siu","submitted_at":"2024-09-26T17:21:22Z","abstract_excerpt":"Let $F$ be a number field. Given a quadratic polynomial $f_c(z) = z^2 + c \\in F[z]$, we can construct a directed graph $Preper(f_c, F)$ (also called a portrait), whose vertices are $F$-rational preperiodic points for $f_c$, with an edge $\\alpha \\to \\beta$ if and only if $f_c(\\alpha) = \\beta$. Poonen and Faber classified the portraits that occur for infinitely many $c$'s.\n  Given a portrait $P$, we prove an asymptotic formula for counting the number of $c \\in F$'s by height, such that $Preper(f_c, F) \\cong P$. We also prove an asymptotic formula for the analogous counting problem, where $Preper"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.18074","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.18074/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2409.18074","created_at":"2026-07-05T09:16:21.895543+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.18074v2","created_at":"2026-07-05T09:16:21.895543+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.18074","created_at":"2026-07-05T09:16:21.895543+00:00"},{"alias_kind":"pith_short_12","alias_value":"BNVDV5PBREXA","created_at":"2026-07-05T09:16:21.895543+00:00"},{"alias_kind":"pith_short_16","alias_value":"BNVDV5PBREXALSNC","created_at":"2026-07-05T09:16:21.895543+00:00"},{"alias_kind":"pith_short_8","alias_value":"BNVDV5PB","created_at":"2026-07-05T09:16:21.895543+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2411.13522","citing_title":"Heights and morphisms in number fields","ref_index":35,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV","json":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV.json","graph_json":"https://pith.science/api/pith-number/BNVDV5PBREXALSNC5LIBCKFTUV/graph.json","events_json":"https://pith.science/api/pith-number/BNVDV5PBREXALSNC5LIBCKFTUV/events.json","paper":"https://pith.science/paper/BNVDV5PB"},"agent_actions":{"view_html":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV","download_json":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV.json","view_paper":"https://pith.science/paper/BNVDV5PB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.18074&json=true","fetch_graph":"https://pith.science/api/pith-number/BNVDV5PBREXALSNC5LIBCKFTUV/graph.json","fetch_events":"https://pith.science/api/pith-number/BNVDV5PBREXALSNC5LIBCKFTUV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV/action/storage_attestation","attest_author":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV/action/author_attestation","sign_citation":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV/action/citation_signature","submit_replication":"https://pith.science/pith/BNVDV5PBREXALSNC5LIBCKFTUV/action/replication_record"}},"created_at":"2026-07-05T09:16:21.895543+00:00","updated_at":"2026-07-05T09:16:21.895543+00:00"}