Riemann-Roch for Toric Rank Functions

In this thesis we study toric rank functions for chip firing games and prove special cases of a conjectural Riemann-Roch. The original motivation for an investigation into this area of study came for the adaptation (due to Matt Baker) of Riemann-Roch into a graph theoretic analogue through the use of chip-firing games. Here, we collect known results and present new observations that indicate Riemann–Roch holds for trees and polygons. We also prove an asymptotic case of Riemann–Roch (i.e.~Riemann–Roch for divisors of large degree). Finally, we also provide magma code and computational evidence that Riemann–Roch holds for the toric rank function.