Simplicial Dollar Game

TL;DR

This paper extends the dollar game and Riemann-Roch theorem from graphs to higher-dimensional simplicial complexes, introducing new notions of degree and proving generalized results about divisors and chip-firing.

Contribution

It generalizes the dollar game and Riemann-Roch theorem to simplicial complexes, defining a chain degree and proving key properties in higher dimensions.

Findings

Generalization of the dollar game to simplicial complexes

Extension of the Riemann-Roch theorem to higher dimensions

Proven that large enough degree chains are winnable in the generalized game

Abstract

The dollar game is a chip-firing game introduced by Baker and Norine (2007) as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these is Theorem 18, generalizing the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second is Corollary 34, generalizing the fact that trees (graphs of genus 0) are exactly the graphs on which every divisor of degree 0, interpreted as an instance of the dollar game, is winnable.