Relative volumes and minors in monomial subrings

Escobar, Cesar A.; Martinez-Bernal, Jose; Villarreal, Rafael H.

doi:10.1016/s0024-3795(03)00612-8

Cited by 18 publications

(12 citation statements)

References 13 publications

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“…For any toric or lattice ideal, we 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. A general reference for connections between monomial subrings, Ehrhart rings, polyhedra and volume is [5, Chapters 5 and 6] (see also [12,34,37]).…”

Section: The Degree Of Lattice and Toric Idealsmentioning

confidence: 99%

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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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Section: The Degree Of Lattice and Toric Idealsmentioning

confidence: 99%

“…First, we exhibit a formula for the degree that holds for any toric ideal (Theorem 4.5), the graded case was shown in [34,Theorem 4.16,p. 36] and [12]. If S has the standard grading and I(L) is a graded lattice ideal of dimension 1, the degree of S/I(L) is the order of T (Z s /L) [21].…”

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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“…(a) The ideal I is normal. Then by [8,Theorem 3.15] the set B = {(v i , 1)} s i=1 is a Hilbert basis. Therefore, using [19,Theorem 3.5], we obtain that (q − 1) n−1 divides |X|.…”

Section: Upper Bounds For the Minimum Distancementioning

confidence: 99%

On the Vanishing Ideal of an Algebraic Toric Set and Its Parametrized Linear Codes

2012

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Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).

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“…where "vol" is the relative volume in the sense of [15,36] (see Corollary 3.18). If an integral polytope in R s has dimension s, then its relative volume agrees with its usual volume [36, p. 239].…”

Section: Introductionmentioning

confidence: 99%