Lexifying ideals
The authors investigate a generalization of Macaulay's theorem about the characterization of Hilbert functions of graded ideals: For each such ideal there is a lexicographic ideal with the same Hilbert function. Let \(B=k[x_1,\dots,x_n]\) be a polynomial ring over the field \(k\) with standard grading \(\deg(x_i)=1\) and \(M\subset B\) a monomial ideal. A graded ideal in \(B/M\) is called \textit{lexifiable} if there exists a lexicographic ideal in \(B/M\) with the same Hilbert function. \(M\) is called \textit{Macaulay-Lex} if every graded ideal in \(B/M\) is lexifiable. \textit{F. S. Macaulay}'s theorem [Proceedings L. M. S. (2) 26, 531--555 (1927; JFM 53.0104.01)] says that \(0\) is a Macaulay-Lex ideal, \textit{J. B. Kruskal} - \textit{G. Katona}'s theorem [Math. Optimization Tech. 251--278 (1963; Zbl 0116.35102), Theory of Graphs, Proc. Colloquium Tihany, Hungary 1966, 187--207 (1968; Zbl 0313.05003)] says that \(P=(x_1^2,\dots,x_n^2)\) is a Macaulay-Lex ideal, and Clements-Lindström's theorem [\textit{G. F. Clements} and \textit{B. Lindström}, J. Comb. Theory 7, 230--238 (1969; Zbl 0186.01704)] says that \((x_1^{e_1},\dots,x_n^{e_n})\) with \(e_1\leq\dots\leq e_n\leq\infty\) is a Macaulay-Lex ideal. The authors prove several results and give counterexamples about Macaulay-Lex ideals and a slightly weaker condition (pro-lex ideals), in particular for \(B/(M+P)\) where \(M\) is a compressed monomial ideal and \(P\) as above. The proofs use the compression method that can be traced back up to Macaulay's original paper.
scientific article; zbMATH DE number 5127176