Simplicial Dollar Game

Abstract

The dollar game is a chip-firing game introduced by Baker 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 generalizes 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 generalizes 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.