{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:HFG7E5MAAUESEWFPFLQYJ66JAN","short_pith_number":"pith:HFG7E5MA","schema_version":"1.0","canonical_sha256":"394df2758005092258af2ae184fbc90352d059ca60d16a70aa60faa6bfed10cb","source":{"kind":"arxiv","id":"1107.2816","version":1},"attestation_state":"computed","paper":{"title":"Periods of rational maps modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Benjamin Hutz, Dragos Ghioca, P\\\"ar Kurlberg, Robert L. Benedetto, Thomas J. Tucker, Thomas Scanlon","submitted_at":"2011-07-14T13:35:46Z","abstract_excerpt":"Let $K$ be a number field, let $\\phi \\in K(t)$ be a rational map of degree at least 2, and let $\\alpha, \\beta \\in K$. We show that if $\\alpha$ is not in the forward orbit of $\\beta$, then there is a positive proportion of primes ${\\mathfrak p}$ of $K$ such that $\\alpha \\mod {\\mathfrak p}$ is not in the forward orbit of $\\beta \\mod {\\mathfrak p}$.\n  Moreover, we show that a similar result holds for several maps and several points.\n  We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $\\alpha$ by a hypersurface, su"},"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":"1107.2816","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-07-14T13:35:46Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"21755c62c049a89fe63879cc9de25b055d283770c5a5522be47d3e28a98f0171","abstract_canon_sha256":"02fc006e7f2b56bc638a05c7f91833b8d52022ad20f8669afe8477144563348a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:15.644762Z","signature_b64":"T5+sfC06lczqH1U0qVAs+VouDWoLckXEz+HtHTmOOpOcHz+qqNEiqpPHmEL0LbV/ktd+6DLIKplRqoSYm3ytCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"394df2758005092258af2ae184fbc90352d059ca60d16a70aa60faa6bfed10cb","last_reissued_at":"2026-05-18T04:18:15.644274Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:15.644274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Periods of rational maps modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Benjamin Hutz, Dragos Ghioca, P\\\"ar Kurlberg, Robert L. Benedetto, Thomas J. Tucker, Thomas Scanlon","submitted_at":"2011-07-14T13:35:46Z","abstract_excerpt":"Let $K$ be a number field, let $\\phi \\in K(t)$ be a rational map of degree at least 2, and let $\\alpha, \\beta \\in K$. We show that if $\\alpha$ is not in the forward orbit of $\\beta$, then there is a positive proportion of primes ${\\mathfrak p}$ of $K$ such that $\\alpha \\mod {\\mathfrak p}$ is not in the forward orbit of $\\beta \\mod {\\mathfrak p}$.\n  Moreover, we show that a similar result holds for several maps and several points.\n  We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $\\alpha$ by a hypersurface, su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2816","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":"1107.2816","created_at":"2026-05-18T04:18:15.644356+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.2816v1","created_at":"2026-05-18T04:18:15.644356+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2816","created_at":"2026-05-18T04:18:15.644356+00:00"},{"alias_kind":"pith_short_12","alias_value":"HFG7E5MAAUES","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"HFG7E5MAAUESEWFP","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"HFG7E5MA","created_at":"2026-05-18T12:26:30.835961+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/HFG7E5MAAUESEWFPFLQYJ66JAN","json":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN.json","graph_json":"https://pith.science/api/pith-number/HFG7E5MAAUESEWFPFLQYJ66JAN/graph.json","events_json":"https://pith.science/api/pith-number/HFG7E5MAAUESEWFPFLQYJ66JAN/events.json","paper":"https://pith.science/paper/HFG7E5MA"},"agent_actions":{"view_html":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN","download_json":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN.json","view_paper":"https://pith.science/paper/HFG7E5MA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.2816&json=true","fetch_graph":"https://pith.science/api/pith-number/HFG7E5MAAUESEWFPFLQYJ66JAN/graph.json","fetch_events":"https://pith.science/api/pith-number/HFG7E5MAAUESEWFPFLQYJ66JAN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN/action/storage_attestation","attest_author":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN/action/author_attestation","sign_citation":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN/action/citation_signature","submit_replication":"https://pith.science/pith/HFG7E5MAAUESEWFPFLQYJ66JAN/action/replication_record"}},"created_at":"2026-05-18T04:18:15.644356+00:00","updated_at":"2026-05-18T04:18:15.644356+00:00"}