Minimal generating sets of large powers of bivariate monomial ideals

It is known that for a monomial ideal I , the number of minimal generators, μ ( I n ) , eventually follows a polynomial pattern for increasing n . In general, little is known about the power at which this pattern emerges. Even less is known about the exact form of the minimal generators after this power. Let s ≥ μ ( I ) ( d 2 − 1 ) + 1 , where d is a constant bounded above by the maximal x - or y -degree appearing in the set G ( I ) of minimal generators of I . We show that every higher power I s + ℓ for any ℓ ≥ 0 can be constructed from certain subideals of I s . This provides an explicit description of G ( I s + ℓ ) in terms of G ( I s ) . Given G ( I s ) , this construction significantly reduces computational complexity in determining larger powers of I . This further enables us to explicitly compute μ ( I n ) for all n ≥ s in terms of a linear polynomial in n . We include runtime measurements for the attached implementation in SageMath.