Powers of monomial ideals and the Ratliff-Rush operation

Abstract

Powers of (monomial) ideals is a subject that still calls attraction in various ways. In this paper we present a nice presentation of high powers of ideals in a certain class in $\mathbb K[x_1, \ldots, x_n]$ and $\mathbb K[[x_1, \ldots, x_n]]$. As an interesting application it leads to an algorithm for computation of the Ratliff--Rush operation on ideals in that class. The Ratliff--Rush operation itself has several applications, for instance, if $I$ is a regular $\mathfrak m$-primary ideal in a local ring $(R,m)$, then the Ratliff--Rush associated ideal $\tilde I$ is the unique largest ideal containing $I$ and having the same Hilbert polynomial as $I$.