Powers of monomial ideals and the Ratliff-Rush operation

TL;DR

This paper provides a new presentation of high powers of monomial ideals in polynomial and power series rings, and introduces an algorithm for computing the Ratliff-Rush operation with applications in algebraic geometry.

Contribution

It offers a novel presentation of high powers of monomial ideals and develops an algorithm for the Ratliff-Rush operation in that class.

Findings

New presentation of high powers of monomial ideals

Algorithm for computing Ratliff-Rush operation

Applications to Hilbert polynomial analysis

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 $I$.