Algebraic interpretation of a theorem of Clements and Lindström
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.
scientific article; zbMATH DE number 5673537
scientific article