Let G be a finite, connected, weighted, directed graph with a
vertex v that is accessible, i.e., each vertex besides
v has a directed path to v. The homogeneous toppling
ideal for G is the lattice ideal associated with the Laplacian
matrix for G. The first paper on the subject is
Polynomial Ideals for Sandpiles and their Grobner Bases,
INRIA (2000) by Cori, Rossin, and Salvy.
I have been working on this subject with several Reed College
undergraduates. Our result so far:
An eprint of our results appears in the form of a
primer. The online
documentation for Sage Sandpiles contains a description of many of these
results with examples.
- The set of zeros of the ideal is isomorphic to the sandpile group
for G with sink v.
- Every lattice ideal defining a finite set of points in projective
space arises as the toppling ideal of some graph.
- Description of a Gröbner basis for the ideal (generalizing the work of
Cori, et al. to directed graphs)
- A description of the minimal free resolution of the ideal in terms
of simplicial complexes associated to linear systems on the graph, in
general, and in terms of certain connectivity properties in the the case
of an undirected graph.
- A characterization of complete intersection toppling ideals.
- A method of constructing arithmetically Gorenstein toppling ideals.