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Explicit deformation of lattice ideals via chip-firing games on directed graphs. (English) Zbl 1328.05119
Summary: 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 E. Miller and B. Sturmfels [Combinatorial commutative algebra. New York, NY: Springer (2005; Zbl 1066.13001)]. Our algorithm is based on recent results of D. Perkinson et al. [Contemp. Math. 605, 211–256 (2013; Zbl 1320.05060)] 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.

##### MSC:
 05C57 Games on graphs (graph-theoretic aspects) 05C20 Directed graphs (digraphs), tournaments 06B99 Lattices
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##### References:
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