Associated prime ideals of powers of ideals

Let R be a multivariate polynomial ring over a field and I a (monomial) ideal of R. A prime ideal P of R is called associated to I is a prime ideal of the form P = Ann R (x) for a non-zero element x ∈ R/I. It is known that the number of associated primes of an ideal I of R is finite. It is, however, hard to compute this number for a general ideal, even in the case of monomial ideals. For a given ideal I, we aim to understand the sequence (a n ) n∈N of numbers of associated prime ideals of the powers I n of the ideal I. Such sequences are called realizable by the ideal I. It is an open question which sequences of positive integers are realizable by (monomial) ideals. This talk serves as an introduction to the topic.