Stanley depth of monomial ideals with small number of generators

Cimpoeaş, Mircea

doi:10.2478/s11533-009-0037-0

Cited by 18 publications

(19 citation statements)

References 11 publications

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“…On the one hand, we have the depth, which is an algebraic invariant (homological in nature), and on the other hand we have the Stanley depth, which is a purely combinatorial invariant. Nevertheless, several parallel results for the depth and the Stanley depth have been found, see for example [Rau07,Cim09,HJZ10,Ish12,SF17]. A counterexample to the original Stanley conjecture was recently given by Duval, Goeckner, Klivans and Martin in [DGKM16].…”

Section: Introductionmentioning

confidence: 95%

Stanley depth and the lcm-lattice

Ichim

Katthän

Moyano‐Fernández

2017

Journal of Combinatorial Theory, Series A

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ABSTRACT. In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients I/J of monomial ideals J ⊂ I, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in [IKMF15]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.

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Section: Introductionmentioning

confidence: 95%

Stanley depth and the lcm-lattice

Ichim

Katthän

Moyano‐Fernández

2017

Journal of Combinatorial Theory, Series A

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“…Thus

depth (I) = depth ((I : x_{k}))

, which implies that

depth (S / I) = depth (S / (I : x_{k}))

. On the other hand, it follows from [, Theorem 1.1] that

sdepth (I) = sdepth ((I : x_{k}))

and

sdepth (S / I) = sdepth (S / (I : x_{k}))

. Hence, the induction hypothesis implies

\begin{matrix} true \\ right sdepth (I) = sdepth ((I : x_{k})) \geq depth ((I : x_{k})) = depth (I) . \end{matrix}

Similarly,

sdepth (S / I) \geq depth (S / I)

.…”

Section: Monomial Ideals With Small Lcm Number and Order Dimensionmentioning

confidence: 93%

“…Let

μ (I) : = | G (I) |

. Cimpoeaş proved that

sdepth (S / I) \geq n - μ (I)

(see [, Proposition 1.2]). It is completely clear that the bound given in Corollary for the Stanley depth of

S / I

is better than the bound given by Cimpoeaş.…”

Section: Stanley Depth and The Lcm Numbermentioning

confidence: 99%

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Two lower bounds for the Stanley depth of monomial ideals

Katthän

Fakhari

2015

Math. Nachr.

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Abstract. Let J I be two monomial ideals of the polynomial ring S = K[x 1 , . . . , x n ]. In this paper, we provide two lower bounds for the Stanley depth of I/J. On the one hand, we introduce the notion of lcm number of I/J, denoted by l(I/J), and prove that the inequality sdepth(I/J) ≥ n − l(I/J) + 1 hold. On the other hand, we show that sdepth(I/J) ≥ n−dim L I/J , where dim L I/J denotes the order dimension of the lcm lattice of I/J. We show that I and S/I satisfy Stanley's conjecture, if either the lcm number of I or the order dimension of the lcm lattice of I is small enough. Among other results, we also prove that the Stanley-Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.

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“…If Stanley conjecture holds for S/I then I is called a Stanley ideal. The conjecture is still open but true in some special cases [1], [2], [4], [5], [6], [7], [9], [11], [12], [13], [14].…”

Section: Introductionmentioning

confidence: 99%