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If $X=(X_1,...,X_g)$ is a $g$ tuple of symmetric $n\\times n$ matrices, then the evaluation $p(X)$ is naturally defined and further $p^*(X)=p(X)^*$. In particular, if $p$ is symmetric, then $p(X)^*=p(X)$. The main result of this article says if $p$ is symmetric, $p(0)=0$ and for each $n$ and each symmetric positive definite $n"},"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":"1208.3582","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-08-17T12:11:30Z","cross_cats_sorted":[],"title_canon_sha256":"6fe34274f0435deeb1636119f1551d9d9b201fd2c20d8601dae482a2c273de74","abstract_canon_sha256":"ac795dd632d64e9dcd8ef86e435687f734af69a176705db92b1932ab777474f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:34.818000Z","signature_b64":"FN3czhXlvNw0MBBGCXveTsO0IjVjaFiIcNiu7g5lu2NXWctSakUqq+iE2a33dOAKrHdtzfu3ZqTJfXf1u/HJBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"002bf8d17ee7091c6dd4be277c0d98a5a7357f653faa6baa85b1e23f03a183c6","last_reissued_at":"2026-05-18T03:48:34.817474Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:34.817474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quasi-Convex Free Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Scott McCullough, Sriram Balasubramanian","submitted_at":"2012-08-17T12:11:30Z","abstract_excerpt":"Let $\\Rx$ denote the ring of polynomials in $g$ freely non-commuting variables $x=(x_1,...,x_g)$. 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