{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:RILBUD7ZQFEA3NT4FNEYIGRIGY","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":"6b70ee0e9bc2b33c41ce378add28a1fc98fef330963ee48438f5b7fedd191c13","cross_cats_sorted":["math.OC"],"license":"","primary_cat":"math.AG","submitted_at":"2007-01-31T23:57:31Z","title_canon_sha256":"c6b39872ffe09261c995de0b4a8c232dca43ba31f29f799a706b8923e8fb598a"},"schema_version":"1.0","source":{"id":"math/0702005","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0702005","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"arxiv_version","alias_value":"math/0702005v1","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702005","created_at":"2026-05-18T04:12:19Z"},{"alias_kind":"pith_short_12","alias_value":"RILBUD7ZQFEA","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"RILBUD7ZQFEA3NT4","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"RILBUD7Z","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:4c7e17c43fc1766fed5731a5df5a8357cb42dd0879256f75ba26b06b4507d757","target":"graph","created_at":"2026-05-18T04:12:19Z","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":"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.","authors_text":"Bernd Sturmfels, Jiawang Nie, Pablo A. Parrilo","cross_cats":["math.OC"],"headline":"","license":"","primary_cat":"math.AG","submitted_at":"2007-01-31T23:57:31Z","title":"Semidefinite Representation of the $k$-Ellipse"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702005","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":"sha256:9380d3a6c0a2e5e869b8ec443e69ec78acd0dca66edd4f187f0e138ceb68641e","target":"record","created_at":"2026-05-18T04:12:19Z","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":"6b70ee0e9bc2b33c41ce378add28a1fc98fef330963ee48438f5b7fedd191c13","cross_cats_sorted":["math.OC"],"license":"","primary_cat":"math.AG","submitted_at":"2007-01-31T23:57:31Z","title_canon_sha256":"c6b39872ffe09261c995de0b4a8c232dca43ba31f29f799a706b8923e8fb598a"},"schema_version":"1.0","source":{"id":"math/0702005","kind":"arxiv","version":1}},"canonical_sha256":"8a161a0ff981480db67c2b49841a28363bd930266ef37452b6d47933286549ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8a161a0ff981480db67c2b49841a28363bd930266ef37452b6d47933286549ab","first_computed_at":"2026-05-18T04:12:19.364832Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:19.364832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3T/FGLO9Hwi+mAhnHnOHjou1YIDRzsndQh1kWjHP68sqiPIYNZrjuL/lmWSOyjvlMi2m8p+vzzwHxVdJuje3CA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:19.365464Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0702005","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9380d3a6c0a2e5e869b8ec443e69ec78acd0dca66edd4f187f0e138ceb68641e","sha256:4c7e17c43fc1766fed5731a5df5a8357cb42dd0879256f75ba26b06b4507d757"],"state_sha256":"0f6d913f40d8c4fd82c1ca608ea4f31a0aea31ff82bd1ddb22a8f55b1e73db3f"}