Linear systems on edge-weighted graphs

Abstract

Let $R$ be any subring of the reals. We present a generalization of linear systems on graphs where divisors are $R$-valued functions on the set of vertices and graph edges are permitted to have nonnegative weights in $R$. Using this generalization, we provide an independent proof of a Riemann-Roch formula, which implies the Riemann-Roch formula of Baker and Norine.