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We show that $py(3,2d) = d+1$ for $2d = 8,10,12$. The main technical tool is Diesel's characterization of height 3 Gorenstein algebras."},"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":"2410.17123","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-10-22T15:54:58Z","cross_cats_sorted":[],"title_canon_sha256":"359dc0e3e28d65e93c535cc19df3f8e80c174e57f10e82c1ff50fddd1a843723","abstract_canon_sha256":"955304d0d4a84f3366e324d609affd2617b294112f6de49399e60e03387058e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:29:50.027136Z","signature_b64":"4uhb75nYmAoLSvqcEYnJba5pTUCp6t9ullJ0EPVswGuMNCixgqLfYgqHWuBlRS1u+OgrA9ToiLHT5HL9wa5yDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a886f70f309117b3aaea893cfc7c1db0b5eff1c5be5a6e8134ef67b88b24d42","last_reissued_at":"2026-07-05T09:29:50.026696Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:29:50.026696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pythagoras Numbers for Ternary Forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alex Dunbar, Grigoriy Blekherman, Rainer Sinn","submitted_at":"2024-10-22T15:54:58Z","abstract_excerpt":"We study the Pythagoras numbers $py(3,2d)$ of real ternary forms, defined for each degree $2d$ as the minimal number $r$ such that every degree $2d$ ternary form which is a sum of squares can be written as the sum of at most $r$ squares of degree $d$ forms. Scheiderer showed that $d+1\\leq py(3,2d)\\leq d+2$. We show that $py(3,2d) = d+1$ for $2d = 8,10,12$. 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