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The main tools for the proof are the Li-Sze characterization of higher-rank numerical ranges, Weyl's perturbation theorem for eigenvalues of Hermitian matrices and Bezout's theorem for the number of common zeros for two homogeneous polynomials."},"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":"1208.1333","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-08-07T04:19:02Z","cross_cats_sorted":[],"title_canon_sha256":"8498e4d1c8254736dae53a972aeba753d3bc9086929a0fcfe0a8bfebdad2b365","abstract_canon_sha256":"54887064a83d71ff6a40dd49cd9f20026575c70668f58cb07826390715911772"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:59.908227Z","signature_b64":"P4nSg2lDoGqt6kX1HDVbOFeHXtds6tiv72PfNysCTKroY4Jk4Qw4MP0JVKJXRHFyzl5qJWuNXXs7FNVaPGLvDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2583a97c7a9a3d2260440a7d1253edc3b16fe211076ec59a11010d304ec4a5d0","last_reissued_at":"2026-05-18T03:09:59.907675Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:59.907675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher-rank Numerical Ranges and Kippenhahn Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hwa-Long Gau, Pei Yuan Wu","submitted_at":"2012-08-07T04:19:02Z","abstract_excerpt":"We prove that two n-by-n matrices A and B have their rank-k numerical ranges $\\Lambda_k(A)$ and $\\Lambda_k(B)$ equal to each other for all k, $1\\le k\\le \\lfloor n/2\\rfloor+1$, if and only if their Kippenhahn polynomials $p_A(x,y,z)\\equiv\\det(x Re A+y Im A+zI_n)$ and $p_B(x,y,z)\\equiv\\det(x Re B+y Im B+zI_n)$ coincide. 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