Riemann-Roch theory on finite sets

TL;DR

This paper extends Riemann-Roch theory to a broader combinatorial setting beyond graphs, generalizing previous results on divisors and linear systems to real-valued functions.

Contribution

It introduces a Riemann-Roch theorem applicable to general finite sets, not limited to graph structures, broadening the scope of the theory.

Findings

Established a Riemann-Roch theorem for functions on finite sets.

Unified previous graph-based and real-valued function theories.

Demonstrated the theorem's implications in a more general combinatorial context.

Abstract

In [1] M. Baker and S. Norine developed a theory of divisors and linear systems on graphs, and proved a Riemann-Roch Theorem for these objects (conceived as integer-valued functions on the vertices). In [2] and [3] the authors generalized these concepts to real-valued functions, and proved a corresponding Riemann-Roch Theorem in that setting, showing that it implied the Baker-Norine result. In this article we prove a Riemann-Roch Theorem in a more general combinatorial setting that is not necessarily driven by the existence of a graph.