Primer for the algebraic geometry of sandpiles

Perkinson, David; Perlman, Jacob; Wilmes, John

doi:10.48550/arxiv.1112.6163

Cited by 13 publications

(37 citation statements)

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“…Let G be a connected simple graph. Two prime examples of graded lattice ideals of dimension one are the vanishing ideal I(X) of a set X parameterized by the edges of G [32] and the toppling ideal I G of G [31].…”

Section: Problems and Related Resultsmentioning

confidence: 99%

Computing the degree of a lattice ideal of dimension one

López

Villarreal

2014

Journal of Symbolic Computation

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Section: Problems and Related Resultsmentioning

confidence: 99%

Computing the degree of a lattice ideal of dimension one

López

Villarreal

2014

Journal of Symbolic Computation

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“…Laplacian matrices of complete graphs are PCB matrices; this type of matrix occurs in [23]. The matrix ideal [23,32]. If G is connected, the toppling ideal has dimension 1.…”

Section: Introductionmentioning

confidence: 99%

Degree and Algebraic Properties of Lattice and Matrix Ideals

O’Carroll¹,

Planas-Vilanova²,

Villarreal³

2014

SIAM J. Discrete Math.

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Abstract. We study the degree of non-homogeneous lattice ideals over arbitrary fields, and give formulae to compute the degree in terms of the torsion of certain factor groups of Z s and in terms of relative volumes of lattice polytopes. We also study primary decompositions of lattice ideals over an arbitrary field using the Eisenbud-Sturmfels theory of binomial ideals over algebraically closed fields. We then use these results to study certain families of integer matrices (PCB, GPCB, CB, GCB matrices) and the algebra of their corresponding matrix ideals. In particular, the family of generalized positive critical binomial matrices (GPCB matrices) is shown to be closed under transposition, and previous results for PCB ideals are extended to GPCB ideals. Then, more particularly, we give some applications to the theory of 1-dimensional binomial ideals. If G is a connected graph, we show as a further application that the order of its sandpile group is the degree of the Laplacian ideal and the degree of the toppling ideal. We also use our earlier results to give a structure theorem for graded lattice ideals of dimension 1 in 3 variables and for homogeneous lattices in Z 3 in terms of critical binomial ideals (CB ideals) and critical binomial matrices, respectively, thus complementing a well-known theorem of Herzog on the toric ideal of a monomial space curve.

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“…Wilmes' conjecture concerns the Betti numbers of a certain polynomial ideal closely related to the chip-firing game on a graph. Chip-firing on a graph has been studied in various contexts and under various names, including graphical parking functions [15], the Abelian sandpile model [14], and discrete Riemann-Roch theory [1]. For a comprehensive introduction to the sandpile theory behind the problem, see [14], especially §7.…”

Section: Introduction: the G-parking Function Idealmentioning

confidence: 99%

“…Chip-firing on a graph has been studied in various contexts and under various names, including graphical parking functions [15], the Abelian sandpile model [14], and discrete Riemann-Roch theory [1]. For a comprehensive introduction to the sandpile theory behind the problem, see [14], especially §7. We state the conjecture here concisely and without broader context; however, a few definitions are required first:…”

Section: Introduction: the G-parking Function Idealmentioning

confidence: 99%

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Another proof of Wilmes’ conjecture

Hopkins¹

2014

Discrete Mathematics

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We present a new proof of the monomial case of Wilmes' conjecture, which gives a formula for the coarsely-graded Betti numbers of the G-parking function ideal in terms of maximal parking functions of contractions of G. Our proof is via poset topology and relies on a theorem of Gasharov, Peeva, and Welker [6] that connects the Betti numbers of a monomial ideal to the topology of its lcm-lattice. , 2010 Mathematics Subject Classification. 05E40, 05C25, 13D02, 06B30.

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Primer for the algebraic geometry of sandpiles

Cited by 13 publications

References 0 publications

Computing the degree of a lattice ideal of dimension one

Computing the degree of a lattice ideal of dimension one

Degree and Algebraic Properties of Lattice and Matrix Ideals

Another proof of Wilmes’ conjecture

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