On the number of generators of powers of an ideal

TL;DR

This paper investigates the growth in the number of generators of powers of ideals in regular rings, providing bounds and properties related to their minimal generating sets, especially for monomial ideals.

Contribution

It offers new lower bounds for the number of generators of ideal powers and explores the Cohen-Macaulay type of squared ideals in polynomial rings.

Findings

Lower bounds for generators of ideal powers

CM-type of $I^2$ is at least 3 for certain monomial ideals

Non-principal ideals have increasing minimal generators in their powers

Abstract

We study the number of generators of ideals in regular rings and ask the question whether $\mu(I)<\mu(I^2)$ if $I$ is not a principal ideal, where $\mu(J)$ denotes the number of generators of an ideal $J$. We provide lower bounds for the number of generators for the powers of an ideal and also show that the CM-type of $I^2$ is $\geq 3$ if $I$ is a monomial ideal of height $n$ in $K[x_1,\ldots,x_n]$ and $n\geq 3$.