{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:TAK32QLXENS34WCP3J35MGBP4L","short_pith_number":"pith:TAK32QLX","schema_version":"1.0","canonical_sha256":"9815bd41772365be584fda77d6182fe2ed8d6172588b539a6146a6e85e774598","source":{"kind":"arxiv","id":"1808.01421","version":1},"attestation_state":"computed","paper":{"title":"Rational Solutions of the Painlev\\'e-III Equation: Large Parameter Asymptotics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP","nlin.SI"],"primary_cat":"math.CA","authors_text":"Peter D. Miller, Thomas Bothner","submitted_at":"2018-08-04T04:15:30Z","abstract_excerpt":"The Painlev\\'e-III equation with parameters $\\Theta_0=n+m$ and $\\Theta_\\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\\infty;m)=1$ whenever $n\\in\\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \\cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\\to+\\infty$ with $m\\in\\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is "},"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":"1808.01421","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-08-04T04:15:30Z","cross_cats_sorted":["math-ph","math.CV","math.MP","nlin.SI"],"title_canon_sha256":"e35284150e3bdfe127e974508d1e4f1610ecd39b5c58e4198d9bdec84fa8b1c5","abstract_canon_sha256":"bf3915576b005e8ac72a35ea0b1b8b1fa631e7a3905f4b9676f4c94a07242b29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:53.043956Z","signature_b64":"Cd5zgjNY0Fe8QhKByBl5poG4rgE7LzvRp1l4TIS3Tr2cwbhDK/vW/OQXHf52wymGaK0cc+a+bLzgbJ0H779eBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9815bd41772365be584fda77d6182fe2ed8d6172588b539a6146a6e85e774598","last_reissued_at":"2026-05-18T00:08:53.043264Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:53.043264Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational Solutions of the Painlev\\'e-III Equation: Large Parameter Asymptotics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP","nlin.SI"],"primary_cat":"math.CA","authors_text":"Peter D. Miller, Thomas Bothner","submitted_at":"2018-08-04T04:15:30Z","abstract_excerpt":"The Painlev\\'e-III equation with parameters $\\Theta_0=n+m$ and $\\Theta_\\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\\infty;m)=1$ whenever $n\\in\\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \\cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\\to+\\infty$ with $m\\in\\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01421","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":"1808.01421","created_at":"2026-05-18T00:08:53.043383+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.01421v1","created_at":"2026-05-18T00:08:53.043383+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.01421","created_at":"2026-05-18T00:08:53.043383+00:00"},{"alias_kind":"pith_short_12","alias_value":"TAK32QLXENS3","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"TAK32QLXENS34WCP","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"TAK32QLX","created_at":"2026-05-18T12:32:53.628368+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/TAK32QLXENS34WCP3J35MGBP4L","json":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L.json","graph_json":"https://pith.science/api/pith-number/TAK32QLXENS34WCP3J35MGBP4L/graph.json","events_json":"https://pith.science/api/pith-number/TAK32QLXENS34WCP3J35MGBP4L/events.json","paper":"https://pith.science/paper/TAK32QLX"},"agent_actions":{"view_html":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L","download_json":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L.json","view_paper":"https://pith.science/paper/TAK32QLX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.01421&json=true","fetch_graph":"https://pith.science/api/pith-number/TAK32QLXENS34WCP3J35MGBP4L/graph.json","fetch_events":"https://pith.science/api/pith-number/TAK32QLXENS34WCP3J35MGBP4L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L/action/storage_attestation","attest_author":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L/action/author_attestation","sign_citation":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L/action/citation_signature","submit_replication":"https://pith.science/pith/TAK32QLXENS34WCP3J35MGBP4L/action/replication_record"}},"created_at":"2026-05-18T00:08:53.043383+00:00","updated_at":"2026-05-18T00:08:53.043383+00:00"}