Monomial Ideals in Few Variables: Generators, Betti Numbers, and Regularity

Abstract

We study monomial ideals in polynomial rings with few variables, focusing on two-variable rings k[x, y] where the staircase geometry yields strong structural results. We prove that for two-generator ideals I = (x^a, y^b), the number of minimal generators μ(I^s) stabilizes at a + b - 1 for s ≥ a + b - 1. For powers I^s with s ≤ 3 we give an explicit algorithm to compute all graded Betti numbers in O(d²μ(I)²) time, where d bounds generator degrees. We derive closed formulas for Betti numbers of powers of two-generator ideals. We show that every monomial ideal in k[x,y] has the persistence property for associated primes, and that persistence of associated primes holds in two variables but fails in three. Our main new result is an explicit counterexample showing persistence fails in three variables: for I = (x², y², z², xy, yz) ⊂ k[x,y,z], we have (x,y,z) ∈ Ass(R/I²) but (x,y,z) ∉ Ass(R/I³). Computational verification via Macaulay2 is provided.