The Lex-Plus-Powers Conjecture holds for pure powers

Mermin, Jeffrey; Murai, Shinji

doi:10.1016/j.aim.2010.08.022

Cited by 25 publications

(18 citation statements)

References 25 publications

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“…, x a n n ). In this special case, Conjecture 1.2 (2) was proved in a series of papers [16,19,20], and Conjecture 1.2(1) was proved in [22] when the characteristic of K is 0. On the other hand, little is known for other Macaulay-Lex rings.…”

Section: Conjecture 12 (Mermin-peeva) Suppose That R = A/m Is Macaulmentioning

confidence: 93%

Betti numbers of lex ideals over some Macaulay-Lex rings

Mermin

Murai

2009

J Algebr Comb

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. . . , x n ] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R = A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Füredi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.

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Section: Conjecture 12 (Mermin-peeva) Suppose That R = A/m Is Macaulmentioning

confidence: 93%

Betti numbers of lex ideals over some Macaulay-Lex rings

Mermin

Murai

2009

J Algebr Comb

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“…Mermin and Murai, for instance, have shown that LPP2 holds in the case that I contains an regular sequence of monomials [7]. A survey on both EGH and LPP can be found in [6].…”

Section: Conjecture 5 (Lpp) Suppose That I ⊂ R Contains Anmentioning

confidence: 97%

A Proof of Evans’ Convexity Conjecture

Richert

2015

Communications in Algebra

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“…The conjecture has been proven in the cases where L is an almost complete intersection [10], and when n D 2 [22]. Mermin-Murai [20] and Mermin-Peeva-Stillman [21] have interesting results related to the conjecture for ideals containing a monomial regular sequence. Recently, Caviglia-Maclagan [3] have proven that the EGH Conjecture is true if d j > P j 1 iD1 .d i 1/ for j D 1; : : : ; n. Thus, when n D 3 the outstanding cases for the EGH Conjecture are the "tight degrees" D D ¹d 1 ; d 2 ; d 3 º where…”

Section: Conjecture 25 (Eisenbud-green-harris (Egh) Conjecturementioning

confidence: 99%

Subsets of Complete Intersections and the EGH Conjecture

Cooper¹

2012

Progress in Commutative Algebra 1

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We give a "Macaulay-type" characterization of the Hilbert functions of finite sets of distinct points which are subsets of complete intersections in projective space P 2 and a family of complete intersections in P 3. Doing so, we prove that the Hilbert functions of subsets of complete intersections of type ¹d 1 ; d 2 º; ¹2; d; d º and ¹3; d; d º are the same as those for rectangular complete intersections of the same type. This verifies special cases of the Eisenbud-Green-Harris Conjecture, which is a statement that lex-plus-powers ideals exhibit extremal conditions among all homogeneous ideals containing a homogeneous regular sequence in fixed degrees. The characterization is then applied to give a family of points which has the Cayley-Bacharach Property.

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The Lex-Plus-Powers Conjecture holds for pure powers

Abstract: We prove Evans' Lex-Plus-Powers Conjecture for ideals containing a monomial regular sequence.

Cited by 25 publications

References 25 publications

Betti numbers of lex ideals over some Macaulay-Lex rings

Betti numbers of lex ideals over some Macaulay-Lex rings

A Proof of Evans’ Convexity Conjecture

Subsets of Complete Intersections and the EGH Conjecture

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