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Algebraic GeometryLet 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:
