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.
@article{10_37236_9821,
author = {Jesse Kim and David Perkinson},
title = {Simplicial dollar game},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {2},
doi = {10.37236/9821},
zbl = {1491.05133},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9821/}
}
TY - JOUR
AU - Jesse Kim
AU - David Perkinson
TI - Simplicial dollar game
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/9821/
DO - 10.37236/9821
ID - 10_37236_9821
ER -
%0 Journal Article
%A Jesse Kim
%A David Perkinson
%T Simplicial dollar game
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/9821/
%R 10.37236/9821
%F 10_37236_9821
Jesse Kim; David Perkinson. Simplicial dollar game. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/9821
