Monomial ideals and $n$-lists

Abstract

This paper generalizes a construction of Geramita, Harima, and Shin (Illinois J. Math. \textbf{45} (2001), 1--23). They give an inductive description of a certain set of elements called $n$-type vectors, and use these objects to prove various results about Hilbert functions of sets of points. We extend their notation by inductively describing the monomial ideals in $R$ and identifying certain interesting subsets. We demonstrate that this new notation is useful by using it to calculate multiplicity and the degree of the Hilbert polynomial for quotients of Borel fixed ideals, and by giving another proof of the result of Geramita, Harima, and Shin: The set of $n$-type vectors is in bijective correspondence with all Hilbert functions of finite length cyclic $R$-modules over the polynomial ring $R=\poly{n}$, where $k$ is a field.