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In this paper, we consider the generalized Liouville system: \\label{abeq1} \\Delta_g u_i+\\sum_{j=1}^n a_{ij}\\rho_j(\\frac{h_j e^{u_j}}{\\int h_j e^{u_j}}-1)=0\\quad\\text{in \\,}M,  where $0< h_j\\in C^1(M)$ and $\\rho_j\\in \\mathbb R^+$, and prove that, under the assumptions of $(H_1)$ and $(H_2)$\\,(see Introduction), the Leray-Schauder degree of \\eqref{abeq1} is equal to \\frac{(-\\chi(M)+1)... (-\\chi(M)+N)}{N!} if $\\rho=(\\rho_1,..., \\rho_n)$ satisfies 8\\pi N\\sum_{i=1}^n\\rho_i<\\sum_{1\\leq i,j\\l"},"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":"1009.0259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-01T19:50:01Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"96248fab8c8b460fe4f1778a42189f00cf91271374358d203e41512015ce81a0","abstract_canon_sha256":"d9d63e862dd6b3f8f0f0b4194d460a601702730cb804a61a52732dfea71ec856"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:35.282567Z","signature_b64":"Mem2HQ/VE22JLO9YXKcnW/1TB5EwVp1nASRVy8OVWWMJyUDq+ZfMEit++TfakbWyj6nS0KE/8fGs58BxQxsjDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d92ffcb226a4d8929639cd5cec130cd61ea9b8ebd40514c1c65b77b9cfdea75","last_reissued_at":"2026-05-18T04:41:35.282137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:35.282137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Topological Degree Counting for some Liouville Systems of Mean Field Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Lei Zhang","submitted_at":"2010-09-01T19:50:01Z","abstract_excerpt":"Let $A=(a_{ij})_{n\\times n}$ be an invertible matrix and $A^{-1}=(a^{ij})_{n\\times n}$ be the inverse of $A$. In this paper, we consider the generalized Liouville system: \\label{abeq1} \\Delta_g u_i+\\sum_{j=1}^n a_{ij}\\rho_j(\\frac{h_j e^{u_j}}{\\int h_j e^{u_j}}-1)=0\\quad\\text{in \\,}M,  where $0< h_j\\in C^1(M)$ and $\\rho_j\\in \\mathbb R^+$, and prove that, under the assumptions of $(H_1)$ and $(H_2)$\\,(see Introduction), the Leray-Schauder degree of \\eqref{abeq1} is equal to \\frac{(-\\chi(M)+1)... 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