# Depth functions of powers of homogeneous ideals

## TL;DR

This paper proves that the depth function of powers of homogeneous ideals can realize any convergent numerical sequence and resolves a longstanding question about the associated primes of such powers.

## Contribution

It confirms Herzog and Hibi's conjecture and answers Ratliff's open question, advancing understanding of the algebraic properties of ideal powers.

## Key findings

- Depth functions can be any convergent numerical sequence.
- Associated primes of powers of ideals have been characterized.
- Resolved two longstanding open problems in commutative algebra.

## Abstract

We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing open question of Ratliff on the associated primes of powers of ideals.