Binomial arithmetical rank of lattice ideals

Eto, Kazufumi

doi:10.1007/s00229-002-0317-5

Cited by 6 publications

(14 citation statements)

References 12 publications

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“…, q} such that (u i ) + ∈ Z E and (u i ) − / ∈ Z E or (u i ) − ∈ Z E and (u i ) + / ∈ Z E . The next theorem improves and generalizes results of S. Eliahou-R. Villarreal [9] and K. Eto [10,11]. It shows how the binomial generation of the radical of a lattice ideal is related with the geometry of the cone σ L and the algebra of the lattice L. Theorem 3.5.…”

Section: Introductionsupporting

confidence: 67%

Binomial generation of the radical of a lattice ideal

2010

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Section: Introductionsupporting

confidence: 67%

Binomial generation of the radical of a lattice ideal

2010

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“…Also, both of them can serve as a measure of the "size" of a binomial ideal, see [4], and the complexity of the problem of computing ara(I L ). In the cases preceding this work, the numbers ara(I L ), bar(I L ) and ara A (I L ), were either identical or very close to each other, see for example [2,3,8,14,18]. Recently Barile and Lyubeznik [1] used techniques from [3] and étale cohomology to give a class of lattice ideals such that htI L = ara(I L ) = bar(I L ) only over fields of one positive characteristic p.…”

Section: Introductionmentioning

confidence: 93%

Stanley–Reisner rings and the radicals of lattice ideals

Katsabekis

Morales

Thoma

2006

Journal of Pure and Applied Algebra

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In this article we associate to every lattice ideal I L, ⊂ K[x 1 , . . . , x m ] a cone and a simplicial complex with vertices the minimal generators of the Stanley-Reisner ideal of . We assign a simplicial subcomplex (F ) of to every polynomial F. If F 1 , . . . , F s generate I L, or they generate rad(I L, ) up to radical, then s i=1 (F i ) is a spanning subcomplex of . This result provides a lower bound for the minimal number of generators of I L, which improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally, we show by a family of examples that the given bounds are sharp. (A. Katsabekis), Marcel.Morales@ujf-grenoble.fr (M. Morales), athoma@cc.uoi.gr (A. Thoma).

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“…Another related problem that drew the attention of a number of authors over the last years was the generation of a lattice ideal by binomials up to radical [1][2][3][6][7][8][9]. In 2002 K. Eto [8] characterized complete intersection finitely generated, abelian semigroups with no invertible elements or equivalently complete intersection lattice ideals as those that are set-theoretic complete intersection on binomials in characteristic zero.…”

Section: Introductionmentioning

confidence: 99%