Simplicial Dollar Game
Jesse Kim, David Perkinson

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;…
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