Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

Ahn, Jea Man; Cho, Young Hyun; Park, Jung Pil

doi:10.1016/j.jalgebra.2007.09.016

Cited by 16 publications

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References 10 publications

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“…where the first and second inequalities follow from Lemma 1.5 (2) and (3), respectively, and the third one follows from the condition (1) …”

Section: Resultsmentioning

confidence: 99%

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Conditions for generic initial ideals to be almost reverse lexicographic

Cho

Park

2008

Journal of Algebra

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Let I be a homogeneous Artinian ideal in a polynomial ring R = k[x 1 , . . . , x n ] over a field k of characteristic 0. We study an equivalent condition for the generic initial ideal gin(I ) with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that the Moreno-Socias conjecture implies the Fröberg conjecture and the Pardue conjecture. And for the case codim I 3, we show that R/I has the strong Lefschetz property if and only if gin(I ) is almost reverse lexicographic. Finally we give a positive partial answer to the Moreno-Socias conjecture, and hence to the Pardue conjecture.

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“…where the first and second inequalities follow from Lemma 1.5 (2) and (3), respectively, and the third one follows from the condition (1) …”

Section: Resultsmentioning

confidence: 99%

“…In the paper[2], Ahn et al gave the following tool detecting whether I has the strong Lefschetz (or Stanley) property, from the view point of the minimal system of generators of gin(I ). This tool gives us the chance to prove the main theorems easily.…”

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confidence: 99%

Conditions for generic initial ideals to be almost reverse lexicographic

Cho

Park

2008

Journal of Algebra

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“…Proof. If one of a, b, c and d is equal to one, the theorem is reduced to the case of three variables, and follows from [6], [1], or [2]. We consider the case where a, b, c, d ≥ 2.…”

Section: (4)mentioning

confidence: 99%

“…In the polynomial rings with one or two variables, generic initial ideals are trivially determined, since Borel-fixed ideals are unique for Hilbert functions. In the case of three variables, due to the result of Harima and Wachi [6], Ahn, Cho, and Park [1] or Cimpoeaş [2], the generic initial ideals of Artinian monomial complete intersections are determined. In this note we focus on the case of four variables.…”

Section: Introductionmentioning

confidence: 99%

Generic initial ideals of some monomial complete intersections in four variables

2010

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Let R = K[x1, x2, x3, x4] be the polynomial ring over a field of characteristic zero. For the ideal (where at least one of a, b, c and d is equal to two, we prove that its generic initial ideal with respect to the reverse lexicographic order is the almost revlex ideal corresponding to the same Hilbert function. Mathematics Subject Classification (2000). Primary 13A02; Secondary 13C40, 13F20, 13D40.

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“…Computing generic initial ideals is generally challenging because they are defined by an existence theorem rather than an explicit construction (see Galligo's Theorem, Theorem 2.1). As a result, there are few classes of ideals for which generic initial ideals have been explicitly computed (see [Gre98] for a survey, or [Cim06], [ACP07], [CP08], and [CR10] for more recent results).…”

Section: Introductionmentioning

confidence: 99%

The generic initial ideals of powers of a 2-complete intersection

Mayes¹

2015

J. Commut. Algebra

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We compute the reverse lexicographic generic initial ideals of the powers of a 2-complete intersection ideal I. In particular, we give six algorithms to compute these generic initial ideals, the choice of which depends on the power and on the relative degrees of the minimal generators of I. 1 arXiv:1206.5750v2 [math.AC] 30 Sep 2012 λ nα−1 = β − α + 1.Note that we can write λ 0 and λ k−1 in terms of l := β − α and α as follows:3.2. The Hilbert function of gin(I n ). The following result tells us that the invariants of gin(I n ) are completely determined by H gin(I n ) (t); this observation will be the key to computing these invariants.Lemma 3.6. Suppose that we have an ideal J of the formwhere the µ i s are strictly decreasing. If H J (t) = H I n (t) for a type (α, β) complete intersection ideal I then gin(I n ) = J.This lemma is an immediate consequence of the following well-known result. Although it is used in the literature (for example, it has the same content as Lemma 4.2 of [Gre98]), we record a complete proof here. 8 SARAH MAYES Lemma 3.7. An ideal of the form J = (x k , x k−1 y λ k−1 , . . . , xy λ 1 , y λ 0 )where λ 0 > λ 1 > · · · > λ k−1 is uniquely determined by its Hilbert function.Proof. The key observation here is thatis the Hilbert function of an ideal J as in the statement of the lemma.First note that x k is the smallest degree element of J so that k = min{t|H J (t) = 0}.Consider the ideal L k = (x k ) ⊂ J and its Hilbert function H L k (t). Set S k = min{t|H J (t) = H L k (t)} so that the smallest degree monomial that is in J but not in L k is of degree

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Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

Cited by 16 publications

References 10 publications

Conditions for generic initial ideals to be almost reverse lexicographic

Conditions for generic initial ideals to be almost reverse lexicographic

Generic initial ideals of some monomial complete intersections in four variables

The generic initial ideals of powers of a 2-complete intersection

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