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In this paper, we prove that if $\\mathcal{A}$ is a nonnegative weakly irreducible tensor with spectral radius $\\rho$, then $\\mathrm{am}(\\lambda) \\ge |\\"},"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":"2410.20830","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-10-28T08:28:50Z","cross_cats_sorted":[],"title_canon_sha256":"868e717df11e66c565f4c2e085f5d8eafbb9300891861ffe41e1d10c0a22deb8","abstract_canon_sha256":"7c830071c4a0035b471af3b53b3582cb857570aa19d7e124be2380865c714c0e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:43:35.783149Z","signature_b64":"AbZK8mafmxzrnzwwV4zTSj3i6Ev3kxauvIq4DrLTf27+Do4IEV902U9dHmy2l10i5BTiXjXXmZethKVcE2yMCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c317b3937d36338e4f3f3236b6e8fe37a4b03b344860afdc4c1744b59d06aa58","last_reissued_at":"2026-07-05T09:43:35.782634Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:43:35.782634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The multiplicity of eigenvalues of nonnegative weakly irreducible tensors and uniform hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yi-Zheng Fan","submitted_at":"2024-10-28T08:28:50Z","abstract_excerpt":"Hu and Ye conjectured that for a $k$-th order and $n$-dimensional tensor $\\mathcal{A}$ with an eigenvalue $\\lambda$ and the corresponding eigenvariety $\\mathcal{V}_\\lambda(\\mathcal{A})$, $$\\mathrm{am}(\\lambda) \\ge \\sum_{i=1}^\\kappa \\mathrm{dim}(V_i)(k-1)^{\\mathrm{dim}(V_i)-1},$$ where $\\mathrm{am}(\\lambda)$ is the algebraic multiplicity of $\\lambda$, and $V_1,\\ldots,V_\\kappa$ are all irreducible components of $\\mathcal{V}_\\lambda(\\mathcal{A})$. 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