A study of the lex plus powers conjecture

Richert, Benjamin

doi:10.1016/s0022-4049(03)00130-0

Cited by 25 publications

(20 citation statements)

References 6 publications

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“…This conjecture is equivalent (see [8]) to the statement that lex-pluspowers ideals determine the collection of Hilbert functions possible for ideals which minimally contain an regular sequence. In particular, an equivalent version of EGH is the following.…”

Section: Conjecture 2 (Egh) Suppose That I ⊂ R Contains Anmentioning

confidence: 99%

A Proof of Evans’ Convexity Conjecture

Richert

2015

Communications in Algebra

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Section: Conjecture 2 (Egh) Suppose That I ⊂ R Contains Anmentioning

confidence: 99%

A Proof of Evans’ Convexity Conjecture

Richert

2015

Communications in Algebra

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“…Charalambous and Evans have shown that there may be incomparable minimal elements in the partial order for a particular Hilbert function [2]. The structure of the partially ordered set is an active area of research; see, for example, the work of Richert [23].…”

Section: Resolutions For a Given Hilbert Functionmentioning

confidence: 99%

New approaches to bounding the multiplicity of an ideal

Francisco¹

2006

Journal of Algebra

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We use the theory of resolutions for a given Hilbert function to investigate the multiplicity conjectures of Huneke and Srinivasan, and Herzog and Srinivasan. To prove the conjectures for all modules with a particular Hilbert function, we show that it is enough to prove the statements only for elements at the bottom of the partially ordered set of resolutions with that Hilbert function. This enables us to test the conjectured upper bound for the multiplicity efficiently with the computer algebra system Macaulay 2, and we verify the upper bound for many Artinian modules in three variables with small socle degree. Moreover, with this approach, we show that though numerical techniques have been sufficient in several of the known special cases, they are insufficient to prove the conjectures in general. Finally, we apply a result of Herzog and Srinivasan on ideals with a quasipure resolution to prove the upper bound for Cohen-Macaulay quotients by ideals with generators in high degrees relative to the regularity.

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“…Gasharov, Hibi, and Peeva have introduced the idea of a-stable ideals, and they study their homological properties in [9]. Richert [18] has proven the LPP Conjecture (and Conjecture 1.2) in the case n = 2 and for monomial ideals in three variables. He also has some reductions that we hope will make it possible to understand connections between the Eisenbud-Green-Harris and LPP Conjectures more clearly in the future.…”

Section: Proofmentioning

confidence: 99%

Almost complete intersections and the Lex-Plus-Powers Conjecture

Francisco

2004

Journal of Algebra

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We prove the almost complete intersection case of the Lex-Plus-Powers Conjecture on graded Betti numbers. We show that the resolution of a lex-plus-powers almost complete intersection provides an upper bound for the graded Betti numbers of any other ideal with regular sequence in the same degrees and the same Hilbert function. A key ingredient is finding an explicit comparison map between two Koszul complexes. Finally, we obtain bounds on the Hilbert function of an almost complete intersection, including a special case of a conjecture of Eisenbud-Green-Harris.

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A study of the lex plus powers conjecture

Cited by 25 publications

References 6 publications

A Proof of Evans’ Convexity Conjecture

A Proof of Evans’ Convexity Conjecture

New approaches to bounding the multiplicity of an ideal

Almost complete intersections and the Lex-Plus-Powers Conjecture

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