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In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a simila"},"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":"1210.7715","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-29T16:28:14Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"9cde00abefd83b259c22249535503c425d2bcf7a3fea8e7487283b614aa2821e","abstract_canon_sha256":"a63c6b5b168595e17fe0eb15a5bc4991f9e644a2127e7970f36e0669a2f92d2c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:53:27.440433Z","signature_b64":"T8yVm38xPBoTmAhho1rm9go0Hwp9HHGMC/JjHTSnwPX2sN67qGGRtxdO8XrOphPPf7r0EHyPVOEv0Pg7bna5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"312366bb5ed4001a1f6c336328f811c74704b49cca03457742e7e60ec75c67a9","last_reissued_at":"2026-05-18T01:53:27.439851Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:53:27.439851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Preperiodic points for families of rational map","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Dragos Ghioca, Liang-Chung Hsia, Thomas J. Tucker","submitted_at":"2012-10-29T16:28:14Z","abstract_excerpt":"Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. 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