Algebraic interpretation of a theorem of Clements and Lindström
type: Item
instance of: scholarly article
title: Algebraic interpretation of a theorem of Clements and Lindström
zbMATH Open document ID: 1209.13018
DOI: 10.1216/JCA-2009-1-3-361
published in: Journal of Commutative Algebra
publication date: 2010-02-19T00:00:00Z
review text: Let \(R=k[x_1,\ldots,x_n]\) be a polynomial ring over a field \(k\), \(I\subset R\) a homogeneous ideal and \(S=R/I\). The Hilbert function of \(S\) is defined by \(H_S(i)=\dim_k S_i\) for every \(i\geq 0\). Macaulay characterized all the sequences of integers admissible as Hilbert functions, the so-called \(O\)-sequences. Clements and Lindström generalized Macaulay's characterization to quotients of truncated polynomial rings of the form \(R_{\underline{e}}= k[x_1,\ldots,x_n]/(x_1^{e_2+1}, \dots, x_n^{e_n+1})\) where \(\underline{e}=\{ e_1,e_2,\dots, e_n\}\) is a set of positive integers. The purpose of the paper is to give an algebraic interpretation of the work of \textit{G. F. Clemens} and \textit{B. Lindström} [J. Comb. Theory 7, 230--238 (1969; Zbl 0186.01704)], which is written in a combinatorial language. One of the main tool is the concept of order ideal, in particular the study of the rev-lex segment order ideals in \(R_{\underline{e}}\). The authors count the numbers of elements in rev-lex segments, obtaining binomial expansions similar to the ones of Macaulay's characterization. Moreover they give an algorithm using these binomial expansions to work with the analogue of \(O\)-sequence in the truncated case.
reviewed by: Emanuela De Negri
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 5673537
zbMATH Keywords: Hilbert function
zbMATH Keywords: O-sequences: truncated polynomial ring
zbMATH Keywords: rev-lex segments
author: Susan Marie Cooper
author: Leslie G. Roberts
MaRDI profile type: Publication
cites work: A generalization of a combinatorial theorem of macaulay
cites work: Q4196436
cites work: Compressed ideals
cites work: Lexlike sequences
cites work: The residuals of lex plus powers ideals and the Eisenbud-Green-Harris conjecture
cites work: Hilbert functions of graded algebras
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description: scientific article; zbMATH DE number 5673537