An Algorithm for Computing the Multigraded Hilbert Depth of a Module

Ichim, Bogdan; Zarojanu, Andrei

doi:10.1080/10586458.2014.908753

Cited by 10 publications

(15 citation statements)

References 22 publications

(46 reference statements)

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“…. , x n ] for some degrees [4,12], results on the Stanley depth of complete intersection monomial ideals [13,15,22], and CoCoA implementations of the algorithm [10,21].…”

Section: Introductionmentioning

confidence: 99%

Depth and Stanley depth of the edge ideals of square paths and square cycles

Iqbal

Ishaq

Aamir

2017

Communications in Algebra

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ABSTRACT. We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal I and the Stanley depth of its compliment, S/I. Using these results we are able to prove that if S is a polynomial ring with at most 5 indeterminates and I is a square-free monomial ideal, then the Stanley depth of S/I is strictly larger than the Stanley depth of I. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.

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“…. , x n ] for some degrees [4,12], results on the Stanley depth of complete intersection monomial ideals [13,15,22], and CoCoA implementations of the algorithm [10,21].…”

Section: Introductionmentioning

confidence: 99%

Depth and Stanley depth of the edge ideals of square paths and square cycles

Iqbal

Ishaq

Aamir

2017

Communications in Algebra

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“…Every edge of this complex is contained in two triangles and is thus not meet-irreducible, and every element of rank 3 is in the Scarf complex. So we cannot apply Lemma 6.4 for spdim Q L. We also tried to compute a Stanley decomposition using an implementation of the algorithm given in [IZ14], but unfortunately this seems infeasible.…”

Section: Name Facetsmentioning

confidence: 99%

Stanley depth and simplicial spanning trees

Katthän

2015

J Algebr Comb

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Abstract. We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex.We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators we obtain a partial result. IntroductionLet K be a field, S an N n -graded K-algebra and M a finitely generated Z n -graded S-module. The Stanley depth of M, denoted sdepth M, is a combinatorial invariant of M related to a conjecture of Stanley from 1982 [Sta82, Conjecture 5.1] (nowadays called the Stanley conjecture), which states that depth M ≤ sdepth M. We refer the reader to [Pou+09] for an introduction to the subject and to the survey by Herzog [Her13] for a comprehensive account of the known results. Most of the research concentrates on the particular case where S is a polynomial ring and M is either a monomial ideal I ⊂ S or a quotient S/I. In the present paper we will also work in this setting.The main result of the present paper is that for proving the Stanley conjecture for M = S/I or M = I, it is sufficient to consider certain very special ideals. These ideals (more precisely their lcm lattices) are in bijection with certain simplicial complexes which we call stoss complexes (short for Spanning Tree Of a Skeleton of a Simplex ):These complexes can be seen as high dimensional analogues of trees. They have already been studied by Kalai [Kal83] in 1983, and more recently by Duval, Klivans and Martin [DKM09] and others. For k ∈ N we denote by B(k) the boolean algebra on k generators, i.e. the set of all subsets of

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“…there are 8 cases for k = 5, which should be compared with the total number |L k | = 7443. Then, for each of these "extremal" lattices, we verify the Stanley conjecture using a fast C++ implementation of the algorithms described in [IMF14] and [IZ14]. By Corollary 4.1, that is enough to prove the Stanley conjecture for all elements of L k ; this is the content of the already mentioned Theorem 1.2:…”

Section: Computational Experiments -Lcm-lattices With Few Generatorsmentioning

confidence: 99%

LCM Lattices and Stanley Depth: A First Computational Approach

Ichim

Katthän

Moyano‐Fernández

2015

Experimental Mathematics

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Let K be a field, and let S = K[X 1 , . . . , X n ] be the polynomial ring. Let I be a monomial ideal of S with up to 5 generators. In this paper, we present a computational experiment which allows us to prove that depth S S/I = sdepth S S/I < sdepth S I.This shows that the Stanley conjecture is true for S/I and I, if I can be generated by at most 5 monomials. The result also brings additional computational evidence for a conjecture made by Herzog.

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An Algorithm for Computing the Multigraded Hilbert Depth of a Module

Cited by 10 publications

References 22 publications

Depth and Stanley depth of the edge ideals of square paths and square cycles

Depth and Stanley depth of the edge ideals of square paths and square cycles

Stanley depth and simplicial spanning trees

LCM Lattices and Stanley Depth: A First Computational Approach

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