{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ORLGGZPEJILH4BZ4APGPYHDXPL","short_pith_number":"pith:ORLGGZPE","canonical_record":{"source":{"id":"1410.0134","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-10-01T07:52:07Z","cross_cats_sorted":[],"title_canon_sha256":"cdcad128c3c9e360d82b61dacd6a59e6a80293d9c089b252290ed98b5b10e42c","abstract_canon_sha256":"cbc1034fc2562e806a486035fc8fd8cc0a17a064e1cfa759670a36b63cf7d5af"},"schema_version":"1.0"},"canonical_sha256":"74566365e44a167e073c03ccfc1c777af6ca0e0199ee858e5e50a45c363fb786","source":{"kind":"arxiv","id":"1410.0134","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.0134","created_at":"2026-05-18T01:33:49Z"},{"alias_kind":"arxiv_version","alias_value":"1410.0134v2","created_at":"2026-05-18T01:33:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.0134","created_at":"2026-05-18T01:33:49Z"},{"alias_kind":"pith_short_12","alias_value":"ORLGGZPEJILH","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"ORLGGZPEJILH4BZ4","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"ORLGGZPE","created_at":"2026-05-18T12:28:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ORLGGZPEJILH4BZ4APGPYHDXPL","target":"record","payload":{"canonical_record":{"source":{"id":"1410.0134","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-10-01T07:52:07Z","cross_cats_sorted":[],"title_canon_sha256":"cdcad128c3c9e360d82b61dacd6a59e6a80293d9c089b252290ed98b5b10e42c","abstract_canon_sha256":"cbc1034fc2562e806a486035fc8fd8cc0a17a064e1cfa759670a36b63cf7d5af"},"schema_version":"1.0"},"canonical_sha256":"74566365e44a167e073c03ccfc1c777af6ca0e0199ee858e5e50a45c363fb786","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:49.110188Z","signature_b64":"m3YF7TmDDA5Wz2JbGlulNrmGTt7EoM0bgoLOBRUenAVCTs5NRJvBqLndiDBgqJcrD9gE2dFm5jWD3JMvVt2EAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74566365e44a167e073c03ccfc1c777af6ca0e0199ee858e5e50a45c363fb786","last_reissued_at":"2026-05-18T01:33:49.109616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:49.109616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.0134","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:33:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FIhdKErvb4mWZF/OQ4pj0uv66+EK5yYcmUk1IyIt2gqvXxmVuaRyMsZt/DYHoaWKbA0zF13acklyO+DwLVT3CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:05:09.897949Z"},"content_sha256":"28caefe4d290e4722a4d1f38cd3f51a3445f2932b3a4024926a1e1c2d7ce47eb","schema_version":"1.0","event_id":"sha256:28caefe4d290e4722a4d1f38cd3f51a3445f2932b3a4024926a1e1c2d7ce47eb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ORLGGZPEJILH4BZ4APGPYHDXPL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Note on the Maximum Number of Zeros of $r(z) - \\bar{z}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"J\\\"org Liesen, Olivier S\\`ete, Robert Luce","submitted_at":"2014-10-01T07:52:07Z","abstract_excerpt":"An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4), 2006) states that the complex harmonic function $r(z) - \\bar{z}$, where $r$ is a rational function of degree $n \\geq 2$, has at most $5 (n - 1)$ zeros. In this note we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form $r(z) - \\bar{z}$ no more than $5 (n - 1) - 1$ zeros can occur. Moreover, we show that $r(z) - \\bar{z}$ is regular, if it has the maximal number of zeros."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0134","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:33:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"11km6PVQIgsg+fSROB7A6Ru+mdsNg505Mg1LyzDsjwnH+pTyeW1PdQoEwyYgrJniRQRKCz3jDE+o5R+vV8SECg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:05:09.898287Z"},"content_sha256":"0594de4f018ea02eee9c20fee236862b755045fdacbb33a96be9ccaa25de462a","schema_version":"1.0","event_id":"sha256:0594de4f018ea02eee9c20fee236862b755045fdacbb33a96be9ccaa25de462a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ORLGGZPEJILH4BZ4APGPYHDXPL/bundle.json","state_url":"https://pith.science/pith/ORLGGZPEJILH4BZ4APGPYHDXPL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ORLGGZPEJILH4BZ4APGPYHDXPL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T04:05:09Z","links":{"resolver":"https://pith.science/pith/ORLGGZPEJILH4BZ4APGPYHDXPL","bundle":"https://pith.science/pith/ORLGGZPEJILH4BZ4APGPYHDXPL/bundle.json","state":"https://pith.science/pith/ORLGGZPEJILH4BZ4APGPYHDXPL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ORLGGZPEJILH4BZ4APGPYHDXPL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ORLGGZPEJILH4BZ4APGPYHDXPL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"cbc1034fc2562e806a486035fc8fd8cc0a17a064e1cfa759670a36b63cf7d5af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-10-01T07:52:07Z","title_canon_sha256":"cdcad128c3c9e360d82b61dacd6a59e6a80293d9c089b252290ed98b5b10e42c"},"schema_version":"1.0","source":{"id":"1410.0134","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.0134","created_at":"2026-05-18T01:33:49Z"},{"alias_kind":"arxiv_version","alias_value":"1410.0134v2","created_at":"2026-05-18T01:33:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.0134","created_at":"2026-05-18T01:33:49Z"},{"alias_kind":"pith_short_12","alias_value":"ORLGGZPEJILH","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"ORLGGZPEJILH4BZ4","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"ORLGGZPE","created_at":"2026-05-18T12:28:43Z"}],"graph_snapshots":[{"event_id":"sha256:0594de4f018ea02eee9c20fee236862b755045fdacbb33a96be9ccaa25de462a","target":"graph","created_at":"2026-05-18T01:33:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4), 2006) states that the complex harmonic function $r(z) - \\bar{z}$, where $r$ is a rational function of degree $n \\geq 2$, has at most $5 (n - 1)$ zeros. In this note we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form $r(z) - \\bar{z}$ no more than $5 (n - 1) - 1$ zeros can occur. Moreover, we show that $r(z) - \\bar{z}$ is regular, if it has the maximal number of zeros.","authors_text":"J\\\"org Liesen, Olivier S\\`ete, Robert Luce","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-10-01T07:52:07Z","title":"A Note on the Maximum Number of Zeros of $r(z) - \\bar{z}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0134","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:28caefe4d290e4722a4d1f38cd3f51a3445f2932b3a4024926a1e1c2d7ce47eb","target":"record","created_at":"2026-05-18T01:33:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"cbc1034fc2562e806a486035fc8fd8cc0a17a064e1cfa759670a36b63cf7d5af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-10-01T07:52:07Z","title_canon_sha256":"cdcad128c3c9e360d82b61dacd6a59e6a80293d9c089b252290ed98b5b10e42c"},"schema_version":"1.0","source":{"id":"1410.0134","kind":"arxiv","version":2}},"canonical_sha256":"74566365e44a167e073c03ccfc1c777af6ca0e0199ee858e5e50a45c363fb786","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74566365e44a167e073c03ccfc1c777af6ca0e0199ee858e5e50a45c363fb786","first_computed_at":"2026-05-18T01:33:49.109616Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:49.109616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m3YF7TmDDA5Wz2JbGlulNrmGTt7EoM0bgoLOBRUenAVCTs5NRJvBqLndiDBgqJcrD9gE2dFm5jWD3JMvVt2EAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:49.110188Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.0134","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:28caefe4d290e4722a4d1f38cd3f51a3445f2932b3a4024926a1e1c2d7ce47eb","sha256:0594de4f018ea02eee9c20fee236862b755045fdacbb33a96be9ccaa25de462a"],"state_sha256":"08375d2e8d2103ff1ec74f74e775d7b0e329fafd70c8743d345a98c6ae1d5d0a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AqnIhhMp74ErLi15iWrk8yGvShEWsneLpoQ7XJYNa7sXqIY3sP2wbbK42eOVKN71xAkRf9OKv0iAuGjsFKuPCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T04:05:09.900208Z","bundle_sha256":"e035d3388eb771299e06c6c1967e4731e562baea338b398ded902f2e8777c54e"}}