{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:LMCITYTWI6DQ7SZNARU2LXIWK6","short_pith_number":"pith:LMCITYTW","canonical_record":{"source":{"id":"1706.02840","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-09T05:59:26Z","cross_cats_sorted":[],"title_canon_sha256":"fc8dba9c003e2e3f5fc1758ccdf1810e6c0be52b60bd58aec67c3ab866628945","abstract_canon_sha256":"c0b6fcf81b9cf876714143edee51e6c11145147a72532660d9325a6a06c3d24d"},"schema_version":"1.0"},"canonical_sha256":"5b0489e27647870fcb2d0469a5dd1657886578b7b627f35f524c5ac51ec3137d","source":{"kind":"arxiv","id":"1706.02840","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.02840","created_at":"2026-05-18T00:42:40Z"},{"alias_kind":"arxiv_version","alias_value":"1706.02840v1","created_at":"2026-05-18T00:42:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.02840","created_at":"2026-05-18T00:42:40Z"},{"alias_kind":"pith_short_12","alias_value":"LMCITYTWI6DQ","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"LMCITYTWI6DQ7SZN","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"LMCITYTW","created_at":"2026-05-18T12:31:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:LMCITYTWI6DQ7SZNARU2LXIWK6","target":"record","payload":{"canonical_record":{"source":{"id":"1706.02840","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-09T05:59:26Z","cross_cats_sorted":[],"title_canon_sha256":"fc8dba9c003e2e3f5fc1758ccdf1810e6c0be52b60bd58aec67c3ab866628945","abstract_canon_sha256":"c0b6fcf81b9cf876714143edee51e6c11145147a72532660d9325a6a06c3d24d"},"schema_version":"1.0"},"canonical_sha256":"5b0489e27647870fcb2d0469a5dd1657886578b7b627f35f524c5ac51ec3137d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:40.787912Z","signature_b64":"sccAt9GCGHBs/qIXr6P9zMemfMn8I0AySnq9QTDElIrect42uXRBOcse5LD+ZRBLmWcNxoB8TTibltDZsRT/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b0489e27647870fcb2d0469a5dd1657886578b7b627f35f524c5ac51ec3137d","last_reissued_at":"2026-05-18T00:42:40.787268Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:40.787268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.02840","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:42:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XlryyeRMkwzWyltsoi2K66mCE2oMF9qdwDSj7tEEl3gqhZAI+VyZtUPsR2N7BF2CkxM6jhdmGLn8H+88LeXOCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T17:22:53.505698Z"},"content_sha256":"2bfdde4ee70e27aed894b9cc6f9c4c3de1b376c47f3a53b9b3b1383953065995","schema_version":"1.0","event_id":"sha256:2bfdde4ee70e27aed894b9cc6f9c4c3de1b376c47f3a53b9b3b1383953065995"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:LMCITYTWI6DQ7SZNARU2LXIWK6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Remark on the roots of generalized Lens equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Mutsuo Oka","submitted_at":"2017-06-09T05:59:26Z","abstract_excerpt":"We consider roots of a generalized Lens polynomial $L(z,\\bar z)={\\bar z}^m q(z)-p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\\bar z)=r(\\bar z)q(z)-p(z)$ and with ${\\rm deg}\\,q(z)=n$, ${\\rm deg}\\,r(\\bar z)=m$ and ${\\rm deg}\\,p(z)\\le n$. We have shown that there exists a harmonically splitting polynomial $r(\\bar z)q(z)-p(z)$ which takes $5n+m-6$ roots, using a bifurcation family of polynomials. In this note, we show that this number can be taken by a generalized Lens polynomial ${\\bar z}^mq(z)-p(z)$ after a slight modification of the bifurcation family of a Rhie polynomia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02840","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:42:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZQF3l0Ri7LFVMoA0f2mT4hnruc+KIwrTNER7R23hNh4UCgeWoW5+gXrVKPqT3N6z0xcCZ+QYjRKG1YNxZKzdBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T17:22:53.506030Z"},"content_sha256":"092e23958086337281b0674b8a1cae09c69a5c4c3e1581c35d7aa4f6a839d5ad","schema_version":"1.0","event_id":"sha256:092e23958086337281b0674b8a1cae09c69a5c4c3e1581c35d7aa4f6a839d5ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/bundle.json","state_url":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/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":"2026-06-20T17:22:53Z","links":{"resolver":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6","bundle":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/bundle.json","state":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:LMCITYTWI6DQ7SZNARU2LXIWK6","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":"c0b6fcf81b9cf876714143edee51e6c11145147a72532660d9325a6a06c3d24d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-09T05:59:26Z","title_canon_sha256":"fc8dba9c003e2e3f5fc1758ccdf1810e6c0be52b60bd58aec67c3ab866628945"},"schema_version":"1.0","source":{"id":"1706.02840","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.02840","created_at":"2026-05-18T00:42:40Z"},{"alias_kind":"arxiv_version","alias_value":"1706.02840v1","created_at":"2026-05-18T00:42:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.02840","created_at":"2026-05-18T00:42:40Z"},{"alias_kind":"pith_short_12","alias_value":"LMCITYTWI6DQ","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"LMCITYTWI6DQ7SZN","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"LMCITYTW","created_at":"2026-05-18T12:31:28Z"}],"graph_snapshots":[{"event_id":"sha256:092e23958086337281b0674b8a1cae09c69a5c4c3e1581c35d7aa4f6a839d5ad","target":"graph","created_at":"2026-05-18T00:42:40Z","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":"We consider roots of a generalized Lens polynomial $L(z,\\bar z)={\\bar z}^m q(z)-p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\\bar z)=r(\\bar z)q(z)-p(z)$ and with ${\\rm deg}\\,q(z)=n$, ${\\rm deg}\\,r(\\bar z)=m$ and ${\\rm deg}\\,p(z)\\le n$. We have shown that there exists a harmonically splitting polynomial $r(\\bar z)q(z)-p(z)$ which takes $5n+m-6$ roots, using a bifurcation family of polynomials. In this note, we show that this number can be taken by a generalized Lens polynomial ${\\bar z}^mq(z)-p(z)$ after a slight modification of the bifurcation family of a Rhie polynomia","authors_text":"Mutsuo Oka","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-09T05:59:26Z","title":"Remark on the roots of generalized Lens equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02840","kind":"arxiv","version":1},"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":"sha256:2bfdde4ee70e27aed894b9cc6f9c4c3de1b376c47f3a53b9b3b1383953065995","target":"record","created_at":"2026-05-18T00:42:40Z","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":"c0b6fcf81b9cf876714143edee51e6c11145147a72532660d9325a6a06c3d24d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-09T05:59:26Z","title_canon_sha256":"fc8dba9c003e2e3f5fc1758ccdf1810e6c0be52b60bd58aec67c3ab866628945"},"schema_version":"1.0","source":{"id":"1706.02840","kind":"arxiv","version":1}},"canonical_sha256":"5b0489e27647870fcb2d0469a5dd1657886578b7b627f35f524c5ac51ec3137d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b0489e27647870fcb2d0469a5dd1657886578b7b627f35f524c5ac51ec3137d","first_computed_at":"2026-05-18T00:42:40.787268Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:40.787268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sccAt9GCGHBs/qIXr6P9zMemfMn8I0AySnq9QTDElIrect42uXRBOcse5LD+ZRBLmWcNxoB8TTibltDZsRT/Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:40.787912Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.02840","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2bfdde4ee70e27aed894b9cc6f9c4c3de1b376c47f3a53b9b3b1383953065995","sha256:092e23958086337281b0674b8a1cae09c69a5c4c3e1581c35d7aa4f6a839d5ad"],"state_sha256":"e911631c781d2b7595424f4be74cfefe7131952fdd2008f5257a984d9b8ff75f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nSGpxGLog0I2QsLvDya5Hfwuvmclc3sToujygatHV2wTm/M0nlgZ/mMDDI9oBv2qLAmCdGGg4YIiI32TEeGzCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T17:22:53.507863Z","bundle_sha256":"08de7e96e051f8b93c79126ee1883ffcc78ae4613a96330d9e4a9335a01af5f5"}}