{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:MKXTTUHFFTIBPHLBLIXM2F7NMQ","short_pith_number":"pith:MKXTTUHF","schema_version":"1.0","canonical_sha256":"62af39d0e52cd0179d615a2ecd17ed64284621fb2834e26be327793f9c7bbdae","source":{"kind":"arxiv","id":"1806.08656","version":3},"attestation_state":"computed","paper":{"title":"Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO","math.MG"],"primary_cat":"math.OC","authors_text":"Gennadiy Averkov","submitted_at":"2018-06-22T13:36:55Z","abstract_excerpt":"The abbreviations LMI and SOS stand for `linear matrix inequality' and `sum of squares', respectively. The cone $\\Sigma_{n,2d}$ of SOS polynomials in $n$ variables of degree at most $2d$ is known to have a semidefinite extended formulation with one LMI of size $\\binom{n+d}{n}$. In other words, $\\Sigma_{n,2d}$ is a linear image of a set described by one LMI of size $\\binom{n+d}{n}$. We show that $\\Sigma_{n,2d}$ has no semidefinite extended formulation with finitely many LMIs of size less than $\\binom{n+d}{n}$. Thus, the standard extended formulation of $\\Sigma_{n,2d}$ is optimal in terms of the"},"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":"1806.08656","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-06-22T13:36:55Z","cross_cats_sorted":["math.AG","math.CO","math.MG"],"title_canon_sha256":"0949887df72762c2f1c335bb7d1d73c4fc644ac1fbc43ae469e2bfe85de736e3","abstract_canon_sha256":"ab79cb651b45bd3db6761e3885d6c4dac12ee1c4280153d81c7a7595c073eb15"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:29.718459Z","signature_b64":"hlh6Q/Rtckw/yQx0kphueZPcs/l3tk0k8RkKDqCbTQsNnE7gjL1f09VQWam0JUb/SWGoEjJNJcXzVJJaUqStBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62af39d0e52cd0179d615a2ecd17ed64284621fb2834e26be327793f9c7bbdae","last_reissued_at":"2026-05-17T23:56:29.717963Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:29.717963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO","math.MG"],"primary_cat":"math.OC","authors_text":"Gennadiy Averkov","submitted_at":"2018-06-22T13:36:55Z","abstract_excerpt":"The abbreviations LMI and SOS stand for `linear matrix inequality' and `sum of squares', respectively. The cone $\\Sigma_{n,2d}$ of SOS polynomials in $n$ variables of degree at most $2d$ is known to have a semidefinite extended formulation with one LMI of size $\\binom{n+d}{n}$. In other words, $\\Sigma_{n,2d}$ is a linear image of a set described by one LMI of size $\\binom{n+d}{n}$. We show that $\\Sigma_{n,2d}$ has no semidefinite extended formulation with finitely many LMIs of size less than $\\binom{n+d}{n}$. 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