Predicted decay ideal

TL;DR

This paper constructs a family of monomial ideals with a prescribed number of associated primes for each power, demonstrating a method to control the prime structure of ideal powers.

Contribution

It introduces a novel construction of monomial ideals where the number of associated primes of each power matches a given sequence of positive integers.

Findings

Explicit construction of monomial ideals with prescribed associated primes.

Demonstration of control over the associated prime structure of ideal powers.

Explicit formulas for associated primes of the constructed ideals.

Abstract

For arbitrary positive integers $q_1 \ge q_2 \ge q_3 \ge \cdots$ we construct a family of monomial ideals such that for each positive integer $e$ and for each ideal $I$ in the family, the number of associated primes of $I^e$ is the $q_e$. We present the associated primes explicitly.