文摘
For a finite index sublattice L of the root lattice of type A, we construct a deterministic algorithm to deform the lattice ideal \(I_\mathrm{L}\) to a nearby generic lattice ideal, answering a question posed by Miller and Sturmfels. Our algorithm is based on recent results of Perkinson, Perlman, and Wilmes concerning commutative algebraic aspects of chip-firing on directed graphs. As an application of our deformation algorithm, we construct a cellular resolution of the lattice ideal \(I_\mathrm{L}\) by degenerating the Scarf complex of its deformation. Keywords Lattice ideals Binomial ideals Chip-firing on graphs Directed graphs Laplacian matrix Scarf complex Free resolutions of ideals Explicit deformation