Many associated primes of powers of primes

TL;DR

This paper constructs specific prime ideals in polynomial rings where the number of associated primes of their powers grows exponentially, providing new bounds on associated primes in algebraic geometry.

Contribution

It introduces families of prime ideals with exponentially many associated primes in their powers and establishes a lower bound on the Ananyan-Hochster constant.

Findings

Number of associated primes can grow exponentially in polynomial rings.

Provides a lower bound on the Ananyan-Hochster constant.

Constructs explicit examples of prime ideals with complex associated prime structures.

Abstract

We construct families of prime ideals in polynomial rings for which the number of associated primes of the second power (or higher powers) is exponential in the number of variables in the ring. We give a lower bound on the Ananyan-Hochster constant for the number of associated primes.