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Polynomial ideals for sandpiles and their Gröbner bases. (English) Zbl 1002.68105
Summary: A polynomial ideal encoding topplings in the abelian sandpile model on a graph is introduced. A Gröbner basis of this ideal is interpreted combinatorially in terms of well-connected subgraphs. This gives rise to algorithms to determine the identity and the operation in the group of recurrent configurations.

MSC:
68R10 Graph theory (including graph drawing) in computer science
13B10 Morphisms of commutative rings
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[1] Bak, P.; Tang, Chao; Wiesenfeld, K., Self-organized criticality, Phys. rev. A third ser., 38, 1, 364-374, (1988) · Zbl 1230.37103
[2] Biggs, N.L., Chip-firing and the critical group of a graph, J. algebraic combin., 9, 1, 25-45, (1999) · Zbl 0919.05027
[3] N. Biggs, Chip firing on distance-regular graphs, Tech. Report LSE-CDAM-96-11, CDAM Research Report Series, June 1996.
[4] Björner, A.; Lovász, L.; Shor, P.W., Chip-firing games on graphs, European J. combin., 12, 4, 283-291, (1991) · Zbl 0729.05048
[5] Cori, R.; Rossin, D., On the sandpile group of dual graphs, European J. combin., 21, 447-459, (2000) · Zbl 0969.05034
[6] Cox, D.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms, (1997), Springer New York
[7] Creutz, M., Abelian sandpiles, Comput. phys., 5, 2, 198-203, (1991)
[8] Dhar, D.; Ruelle, P.; Sen, S.; Verma, D.-N., Algebraic aspects of abelian sandpile models, J. phys. A, 28, 4, 805-831, (1995) · Zbl 0848.68062
[9] Dhar, D., Self-organized critical state of sandpile automaton models, Phys. rev. lett., 64, 14, 1613-1616, (1990) · Zbl 0943.82553
[10] Eisenbud, D.; Sturmfels, B., Binomial ideals, Duke math. J., 84, 1, 1-45, (1996) · Zbl 0873.13021
[11] Mayr, E.W.; Meyer, A.R., The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in math., 46, 3, 305-329, (1982) · Zbl 0506.03007
[12] Moore, C.; Nilsson, M., The computational complexity of sandpiles, J. statist. phys., 96, 1-2, 205-224, (1999) · Zbl 0964.82037
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