A characterization of stable and Borel ideals
The authors study special classes of monomial ideals in the polynomial ring over a field of characteristic zero. For every monomial ideal \(a\) they consider the order ideal \({\mathcal N}(a)\) associated to \(a\), that is, the set of the terms outside \(a\). The main result of the paper is a characterization of stable and Borel-fixed ideals in terms of the corresponding order ideals. Moreover the authors associate to every monomial ideal \(a\) the so-called \(\varepsilon\)-vectors, denoted by \(\varepsilon(a)\), consisting of a sequence of symbols \(\infty\), positive integers and finite sets of /'s. Such a vector provides strong and compact information on the ``geometric configuration'' of the order ideal \({\mathcal N}(a)\) and on the generators of \(a\). In particular, if \(a\) is stable then the knowledge of \(\varepsilon(a)\) allows to write the minimal system of generators of \(a\). In the special case of lex-segment ideals a full characterization of the possible \(\varepsilon\)-vectors is given.
scientific article; zbMATH DE number 5004483
scientific article