{"paper":{"title":"Degree counting theorems for singular Liouville systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lei Zhang, Yi Gu","submitted_at":"2018-11-01T02:37:03Z","abstract_excerpt":"Let $(M,g)$ be a compact Riemann surface with no boundary and $u=(u_1,...,u_n)$ be a solution of the following singular Liouville system: \\begin{equation*} \\Delta_g u_i+\\sum_{j=1}^na_{ij}\\rho_j(\\frac{h_je^{u_j}}{\\int_M h_j e^{u_j}dV_g}-\\frac{1}{vol_g(M)})=\\sum_{t=1}^N4\\pi \\gamma_t( \\delta_{p_t}-\\frac{1}{vol_g(M)}), \\end{equation*} where $i=1,...,n$,\n  $h_1,...,h_n$ are positive smooth functions, $p_1,...,p_N$ are distinct points on $M$, $\\delta_{p_t}$ are Dirac masses, $\\rho=(\\rho_1,...,\\rho_n)$ ($\\rho_i\\ge 0)$ and $(\\gamma_{1},...,\\gamma_{N})$ ($\\gamma_{t}>-1$ ) are constant vectors. If the c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00190","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}