Abstract
AbstractA Riemann-Roch theorem on an edge-weighted infinite graph with local finiteness was established by the present authors in [1], where the spectral gap of Laplacian associated determined by the edge-weight was investigated as the corner stone of the proof. On the other hand, as for non-archimedean metric space, the Laplacians in the construction of Hunt processes such as in [3, 5] based on the Dirichlet space theory can be highlighted. However, in those studies, a positive edge-weight was given substantially between each pair of balls with an identical diameter with respect to the ultrametric and the spectral gap is infeasible. In the present article, we rethink the notion of chip-firing and show an upper bound of function given by accumulation of chip-firing to materialize a counterpart of the dimension of linear system in ultrametric space. In the final section of this article, we establish a Riemann-Roch theorem on an ultrametric space.KeywordsDirichlet spaceLaplacianRiemann-Roch theoremUltrametric spaceWeighted graph
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