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We establish that an even symmetric $n$-ary $2d$-ic psd form is sos if and only if $n=2$ or $d=1$ or $(n,2d)=(n,4)_{n \\geq 3}$ or $(n,2d)= (3,8)$."},"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":"1509.07482","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-09-24T19:17:24Z","cross_cats_sorted":[],"title_canon_sha256":"43a08b88f7501c19e1c61926a54590914aa0bf012549ae7cbb95e759ec00b5fc","abstract_canon_sha256":"7f211d86790142e96183ad9326fe3500c1f11d0666f9f7533eac81d01751bf9c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:40.367874Z","signature_b64":"NGfRncYpb9kJB1R1XK0+BHf/YmeDhURanCL7ZrD+UYIbT+TR7hVqUe+UxErNoYZC3D9qIvuwryBgx2QP3mrOAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18ee7f4162dfa49f2a658d97263a9c011e3bc77da98cc82661a0ebc5730c1ed1","last_reissued_at":"2026-05-18T01:00:40.367428Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:40.367428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The analogue of Hilbert's 1888 theorem for Even Symmetric Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Bruce Reznick, Charu Goel, Salma Kuhlmann","submitted_at":"2015-09-24T19:17:24Z","abstract_excerpt":"Hilbert proved in 1888 that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or $d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the degree of the form. We study the analogue for even symmetric forms. We establish that an even symmetric $n$-ary $2d$-ic psd form is sos if and only if $n=2$ or $d=1$ or $(n,2d)=(n,4)_{n \\geq 3}$ or $(n,2d)= (3,8)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07482","kind":"arxiv","version":2},"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":"1509.07482","created_at":"2026-05-18T01:00:40.367492+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.07482v2","created_at":"2026-05-18T01:00:40.367492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07482","created_at":"2026-05-18T01:00:40.367492+00:00"},{"alias_kind":"pith_short_12","alias_value":"DDXH6QLC36SJ","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"DDXH6QLC36SJ6KTF","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"DDXH6QLC","created_at":"2026-05-18T12:29:17.054201+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/DDXH6QLC36SJ6KTFRWLSMOU4AE","json":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE.json","graph_json":"https://pith.science/api/pith-number/DDXH6QLC36SJ6KTFRWLSMOU4AE/graph.json","events_json":"https://pith.science/api/pith-number/DDXH6QLC36SJ6KTFRWLSMOU4AE/events.json","paper":"https://pith.science/paper/DDXH6QLC"},"agent_actions":{"view_html":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE","download_json":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE.json","view_paper":"https://pith.science/paper/DDXH6QLC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.07482&json=true","fetch_graph":"https://pith.science/api/pith-number/DDXH6QLC36SJ6KTFRWLSMOU4AE/graph.json","fetch_events":"https://pith.science/api/pith-number/DDXH6QLC36SJ6KTFRWLSMOU4AE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE/action/storage_attestation","attest_author":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE/action/author_attestation","sign_citation":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE/action/citation_signature","submit_replication":"https://pith.science/pith/DDXH6QLC36SJ6KTFRWLSMOU4AE/action/replication_record"}},"created_at":"2026-05-18T01:00:40.367492+00:00","updated_at":"2026-05-18T01:00:40.367492+00:00"}