A study of the lex plus powers conjecture.
type: Item
instance of: scholarly article
title: A study of the lex plus powers conjecture.
zbMATH Open document ID: 1052.13008
DOI: 10.1016/S0022-4049(03)00130-0
author: Benjamin P. Richert
published in: Journal of Pure and Applied Algebra
publication date: 2004-01-26T00:00:00Z
review text: Let \(R:=k[x_1, \dots, x_n]\), fix \(n\) integers \(a_1, \dots, a_n\) such that \(1 \leq a_1 \leq \cdots \leq a_n\). A monomial ideal \(I \subseteq R\) is called an \(\{a_1, \dots, a_n\}\)-\textit{lex plus powers ideal} if there exist monomials \(m_1, \dots, m_l \in I\) such that: \(x_1^{a_1}, \dots, x_n^{a_n}, m_1, \dots, m_l\) is a minimal set of generators of \(I\); if \(m\) is a monomial such that \(\deg(m) = \deg(m_j)\) for some \(j \in \{ 1, \dots, l\}\) and \(m > m_j\), then \(m \in I\). Conjectures by Eisenbud and others and by Charalambous and Evans claim that \(\{a_1, \dots, a_n\}\)-lex plus powers ideals have extremal properties among ideals containing a regular sequence of degree \(\{a_1, \dots, a_n\}\). These extremal properties are similar to those of lex-ideals. In more details, the first conjecture (LPPH) claims that, given a degree \(d\) and an ideal \(I\) containing an \(\{a_1, \dots, a_n\}\)-regular sequence, then if there exists an \(\{a_1, \dots, a_n\}\)-lex plus powers ideal \(L\) such that the Hilbert functions of \(L\) and \(I\) agree in degree \(d\), then there exists a \(\{a_1, \dots, a_n\}\)-lex plus powers ideal \(L'\) such that the Hilbert functions of \(L'\) and \(I\) agree in degree \(d\) and the Hilbert function of \(L'\) bounds that of \(I\) in degree \(d+1\). The second conjecture (LPP) claims that, after fixing a Hilbert function \({\mathcal H}\) and a list of degrees \(\{a_1, \dots, a_n\}\), then the graded Betti numbers of the \(\{a_1, \dots, a_n\}\)-lex plus powers ideal attaining \({\mathcal H}\) should be everywhere larger then the graded Betti numbers of any \(I \subseteq R\) such that \(I\) contains an \(\{a_1, \dots, a_n\}\)-regular sequence, and \(H(R/I)= {\mathcal H}\). The first part of the paper contains several equivalent reformulations of the conjectures. For instance it is proved that LPPH is equivalent to the statement that for a fixed Hilbert function, lex plus powers ideals have the most minimal generators in each…
reviewed by: Alessandro Logar
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13E15
zbMATH DE Number: 2032649
zbMATH Keywords: Hilbert function
zbMATH Keywords: Betti numbers
zbMATH Keywords: lex plus powers ideal
zbMATH Keywords: monomial ideal
zbMATH Keywords: LPPH
zbMATH Keywords: LPP
Wikidata QID: Q122958894
describes a project that uses: Macaulay2
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2032649