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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"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1706.02840","created_at":"2026-05-18T00:42:40.787370+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.02840v1","created_at":"2026-05-18T00:42:40.787370+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.02840","created_at":"2026-05-18T00:42:40.787370+00:00"},{"alias_kind":"pith_short_12","alias_value":"LMCITYTWI6DQ","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_16","alias_value":"LMCITYTWI6DQ7SZN","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_8","alias_value":"LMCITYTW","created_at":"2026-05-18T12:31:28.150371+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/LMCITYTWI6DQ7SZNARU2LXIWK6","json":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6.json","graph_json":"https://pith.science/api/pith-number/LMCITYTWI6DQ7SZNARU2LXIWK6/graph.json","events_json":"https://pith.science/api/pith-number/LMCITYTWI6DQ7SZNARU2LXIWK6/events.json","paper":"https://pith.science/paper/LMCITYTW"},"agent_actions":{"view_html":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6","download_json":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6.json","view_paper":"https://pith.science/paper/LMCITYTW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.02840&json=true","fetch_graph":"https://pith.science/api/pith-number/LMCITYTWI6DQ7SZNARU2LXIWK6/graph.json","fetch_events":"https://pith.science/api/pith-number/LMCITYTWI6DQ7SZNARU2LXIWK6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/action/storage_attestation","attest_author":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/action/author_attestation","sign_citation":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/action/citation_signature","submit_replication":"https://pith.science/pith/LMCITYTWI6DQ7SZNARU2LXIWK6/action/replication_record"}},"created_at":"2026-05-18T00:42:40.787370+00:00","updated_at":"2026-05-18T00:42:40.787370+00:00"}