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We prove that the polynomial $f(z)$ does not vanish in the sector $$ \\left\\{z\\in\\mathbb{C}: |\\arg (z)| < \\frac{\\pi}{M}\\right\\} $$ whenever the matrix $H_M$ is totally nonnegative. This result generalizes the classical Hurwitz' Theorem on stable polynomials ($M=2$), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials with negative real roots ($M=1$), and the Cowling-Thron theorem ($M="},"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":"1506.07379","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-06-24T14:24:14Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"ba8fd747c49b7fa41300f3da71377d06e33e68c56a77fcdc74a0806d053fe2d2","abstract_canon_sha256":"162efbcca7c4029ad66339e607544d4e5f3dd8bbede32ee9ed8dc9ce501e0554"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:55.282334Z","signature_b64":"W0+bfKeHJ/49+uMfPhfNTVFKtf2diSrsLV7mXi1eTq3/CvPO0zTKbA3C1gAISu6wMiorH542o+Z8LSndjsddBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6bd48c4aaff35baf8a5d57af91be6782feb0b3ecee8ce644ed4c99793a54822","last_reissued_at":"2026-05-18T01:09:55.281747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:55.281747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Hurwitz matrices, generalized Euclidean algorithm, and forbidden sectors of the complex plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CA","authors_text":"Olga Holtz, Olga Kushel, Sergey Khrushchev","submitted_at":"2015-06-24T14:24:14Z","abstract_excerpt":"Given a polynomial \\[ f(x)=a_0x^n+a_1x^{n-1}+\\cdots +a_n \\] with positive coefficients $a_k$, and a positive integer $M\\leq n$, we define a(n infinite) generalized Hurwitz matrix $H_M(f):=(a_{Mj-i})_{i,j}$. We prove that the polynomial $f(z)$ does not vanish in the sector $$ \\left\\{z\\in\\mathbb{C}: |\\arg (z)| < \\frac{\\pi}{M}\\right\\} $$ whenever the matrix $H_M$ is totally nonnegative. 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