Linear Systems on Edge-Weighted Graphs

TL;DR

This paper generalizes linear systems on graphs by allowing real-valued divisors and edge weights, providing a new proof of the Riemann-Roch formula that supports prior results by Baker and Norine.

Contribution

It introduces a generalized framework for linear systems on edge-weighted graphs with real-valued divisors, offering an independent proof of the Riemann-Roch formula.

Findings

Generalization of linear systems on graphs with R-valued divisors and nonnegative edge weights

Independent proof of the Riemann-Roch formula for graphs

Supports and extends the Riemann-Roch theorem of Baker and Norine

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 nonegative 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.