The full toppling Gröbner fan in the tree case

Definitions and mathematical statement

Let $G$ be a finite connected loopless undirected graph with vertex set $V=\{1,\ldots,n\}$, allowing parallel edges. Let $\Delta$ be its integer Laplacian, with diagonal degrees and off-diagonal entries minus the numbers of joining edges. The Laplacian lattice is $L_G=\Delta\mathbb Z^n\subseteq\mathbb Z^n$. For $u\in L_G$, write $u=u^+-u^-$ with $u_i^+=\max(u_i,0)$ and $u_i^-=\max(-u_i,0)$. The homogeneous toppling ideal over $\mathbb Q$ is \[ I_G=\langle x^{u^+}-x^{u^-}:u\in L_G\rangle \subseteq\mathbb Q[x_1,\ldots,x_n], \qquad x^a=\prod_i x_i^{a_i}. \] It encodes equality of divisor classes under integer chip firing.

For a weight vector $w\in\mathbb R^n$, the initial form of a polynomial is the sum of its terms having maximal scalar product $w\cdot a$. Let $\operatorname{in}_w(I_G)$ be the ideal generated by these initial forms over all elements of $I_G$. The Gröbner fan partitions weight space into relatively open cones on which this initial ideal is constant, together with their closed cones and faces. Describe this entire fan combinatorially in terms of $G$: all cones, their face relations, and their associated initial ideals. The target includes nongeneric weights and nonmonomial initial ideals, not only the familiar sink-dependent monomial ideals. Since every lattice vector has coordinate sum zero, adding a multiple of $(1,\ldots,1)$ to $w$ leaves the initial ideal unchanged; the description may therefore use the quotient by this lineality space.

Short English statement

Describe every initial ideal and cone of the full Gröbner fan of a graph’s homogeneous toppling ideal.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000988 (AIM-COMBINATORICS-0113), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Generalizations of chip-firing and the critical group, item 23. Original workshop question underlying AIM-COMBINATORICS-0113. https://aimath.org/pastworkshops/chipfiringproblems.pdf.