{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:RILBUD7ZQFEA3NT4FNEYIGRIGY","short_pith_number":"pith:RILBUD7Z","schema_version":"1.0","canonical_sha256":"8a161a0ff981480db67c2b49841a28363bd930266ef37452b6d47933286549ab","source":{"kind":"arxiv","id":"math/0702005","version":1},"attestation_state":"computed","paper":{"title":"Semidefinite Representation of the $k$-Ellipse","license":"","headline":"","cross_cats":["math.OC"],"primary_cat":"math.AG","authors_text":"Bernd Sturmfels, Jiawang Nie, Pablo A. Parrilo","submitted_at":"2007-01-31T23:57:31Z","abstract_excerpt":"The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming."},"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":"math/0702005","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2007-01-31T23:57:31Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"c6b39872ffe09261c995de0b4a8c232dca43ba31f29f799a706b8923e8fb598a","abstract_canon_sha256":"6b70ee0e9bc2b33c41ce378add28a1fc98fef330963ee48438f5b7fedd191c13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:19.365464Z","signature_b64":"3T/FGLO9Hwi+mAhnHnOHjou1YIDRzsndQh1kWjHP68sqiPIYNZrjuL/lmWSOyjvlMi2m8p+vzzwHxVdJuje3CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a161a0ff981480db67c2b49841a28363bd930266ef37452b6d47933286549ab","last_reissued_at":"2026-05-18T04:12:19.364832Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:19.364832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semidefinite Representation of the $k$-Ellipse","license":"","headline":"","cross_cats":["math.OC"],"primary_cat":"math.AG","authors_text":"Bernd Sturmfels, Jiawang Nie, Pablo A. Parrilo","submitted_at":"2007-01-31T23:57:31Z","abstract_excerpt":"The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702005","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":"math/0702005","created_at":"2026-05-18T04:12:19.364911+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0702005v1","created_at":"2026-05-18T04:12:19.364911+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702005","created_at":"2026-05-18T04:12:19.364911+00:00"},{"alias_kind":"pith_short_12","alias_value":"RILBUD7ZQFEA","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"RILBUD7ZQFEA3NT4","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"RILBUD7Z","created_at":"2026-05-18T12:25:56.245647+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/RILBUD7ZQFEA3NT4FNEYIGRIGY","json":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY.json","graph_json":"https://pith.science/api/pith-number/RILBUD7ZQFEA3NT4FNEYIGRIGY/graph.json","events_json":"https://pith.science/api/pith-number/RILBUD7ZQFEA3NT4FNEYIGRIGY/events.json","paper":"https://pith.science/paper/RILBUD7Z"},"agent_actions":{"view_html":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY","download_json":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY.json","view_paper":"https://pith.science/paper/RILBUD7Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0702005&json=true","fetch_graph":"https://pith.science/api/pith-number/RILBUD7ZQFEA3NT4FNEYIGRIGY/graph.json","fetch_events":"https://pith.science/api/pith-number/RILBUD7ZQFEA3NT4FNEYIGRIGY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY/action/storage_attestation","attest_author":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY/action/author_attestation","sign_citation":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY/action/citation_signature","submit_replication":"https://pith.science/pith/RILBUD7ZQFEA3NT4FNEYIGRIGY/action/replication_record"}},"created_at":"2026-05-18T04:12:19.364911+00:00","updated_at":"2026-05-18T04:12:19.364911+00:00"}